Fractional Schroedinger equation
International Nuclear Information System (INIS)
Laskin, Nick
2002-01-01
Some properties of the fractional Schroedinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schroedinger equation we find the energy spectra of a hydrogenlike atom (fractional 'Bohr atom') and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schroedinger equations
The forced nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Kaup, D.J.; Hansen, P.J.
1985-01-01
The nonlinear Schroedinger equation describes the behaviour of a radio frequency wave in the ionosphere near the reflexion point where nonlinear processes are important. A simple model of this phenomenon leads to the forced nonlinear Schroedinger equation in terms of a nonlinear boundary value problem. A WKB analysis of the time evolution equations for the nonlinear Schroedinger equation in the inverse scattering transform formalism gives a crude order of magnitude estimation of the qualitative behaviour of the solutions. This estimation is compared with the numerical solutions. (D.Gy.)
International Nuclear Information System (INIS)
Gligoric, G; Hadzievski, Lj; Maluckov, A; Malomed, B A
2009-01-01
A model of the Bose-Einstein condensate (BEC) of dipolar atoms, confined in a combination of a cigar-shaped trap and optical lattice acting in the axial direction, is studied in the framework of the one-dimensional (1D) nonpolynomial Schroedinger equation (NPSE) with additional terms describing long-range dipole-dipole (DD) interactions. The NPSE makes it possible to describe the collapse of localized modes, which was experimentally observed in the self-attractive BEC confined in tight traps, in the framework of the 1D description. We study the influence of the DD interactions on the dynamics of bright solitons, especially concerning their collapse-induced instability. Both attractive and repulsive contact and DD interactions are considered. The results are summarized in the form of stability/collapse diagrams in a respective parametric space. In particular, it is shown that the attractive DD interactions may prevent the collapse instability in the condensate with attractive contact interactions.
Effective Schroedinger equations on submanifolds
Energy Technology Data Exchange (ETDEWEB)
Wachsmuth, Jakob
2010-02-11
In this thesis the time dependent Schroedinger equation is considered on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When the potential is scaled in the directions normal to C by a small parameter epsilon, the solutions concentrate in an epsilon-neighborhood of the submanifold. An effective Schroedinger equation on the submanifold C is derived and it is shown that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order {epsilon}{sup 3} vertical stroke t vertical stroke at time t. Furthermore, it is proved that, under reasonable conditions, the eigenvalues of the corresponding Hamiltonians below a certain energy coincide upto errors of order {epsilon}{sup 3}. These results holds in the situation where tangential and normal energies are of the same order, and where exchange between normal and tangential energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were needed, to ensure that the separation of energies is maintained during the time evolution. The most important consequence of this thesis is that now constraining potentials that change their shape along the submanifold can be treated, which is the typical situation in applications like molecular dynamics and quantum waveguides.
P-adic Schroedinger type equation
International Nuclear Information System (INIS)
Vladimirov, V.S.; Volovich, I.V.
1988-12-01
In p-adic quantum mechanics a Schroedinger type equation is considered. We discuss the appropriate notion of differential operators. A solution of the Schroedinger type equation is given. A new set of vacuum states for the p-adic quantum harmonic oscillator is presented. The correspondence principle with the standard quantum mechanics is discussed. (orig.)
Two routes to the one-dimensional discrete nonpolynomial Schroedinger equation
International Nuclear Information System (INIS)
Gligoric, G.; Hadzievski, Lj.; Maluckov, A.; Salasnich, L.; Malomed, B. A.
2009-01-01
The Bose-Einstein condensate (BEC), confined in a combination of the cigar-shaped trap and axial optical lattice, is studied in the framework of two models described by two versions of the one-dimensional (1D) discrete nonpolynomial Schroedinger equation (NPSE). Both models are derived from the three-dimensional Gross-Pitaevskii equation (3D GPE). To produce 'model 1' (which was derived in recent works), the 3D GPE is first reduced to the 1D continual NPSE, which is subsequently discretized. 'Model 2,' which was not considered before, is derived by first discretizing the 3D GPE, which is followed by the reduction in the dimension. The two models seem very different; in particular, model 1 is represented by a single discrete equation for the 1D wave function, while model 2 includes an additional equation for the transverse width. Nevertheless, numerical analyses show similar behaviors of fundamental unstaggered solitons in both systems, as concerns their existence region and stability limits. Both models admit the collapse of the localized modes, reproducing the fundamental property of the self-attractive BEC confined in tight traps. Thus, we conclude that the fundamental properties of discrete solitons predicted for the strongly trapped self-attracting BEC are reliable, as the two distinct models produce them in a nearly identical form. However, a difference between the models is found too, as strongly pinned (very narrow) discrete solitons, which were previously found in model 1, are not generated by model 2--in fact, in agreement with the continual 1D NPSE, which does not have such solutions either. In that respect, the newly derived model provides for a more accurate approximation for the trapped BEC.
Reparametrization invariance and the Schroedinger equation
International Nuclear Information System (INIS)
Tkach, V.I.; Pashnev, A.I.; Rosales, J.J.
1999-01-01
A time-dependent Schroedinger equation for systems invariant under the reparametrization of time is considered. We develop the two-stage procedure of construction such systems from a given initial ones, which are not invariant under the time reparametrization. One of the first-class constraints of the systems in such description becomes the time-dependent Schroedinger equation. The procedure is applicable in the supersymmetric theories as well. The n = 2 supersymmetric quantum mechanics is coupled to world-line supergravity, and the local supersymmetric action is constructed leading to the square root representation of the time-dependent Schroedinger equation
Stochastic effects on the nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Flessas, G P; Leach, P G L; Yannacopoulos, A N
2004-01-01
The aim of this article is to provide a brief review of recent advances in the field of stochastic effects on the nonlinear Schroedinger equation. The article reviews rigorous and perturbative results. (review article)
Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation
International Nuclear Information System (INIS)
Ren Ji; Ruan Hangyu
2008-01-01
We study solutions of the nonlinear Schroedinger equation (NLSE) and higher-order nonlinear Schroedinger equation (HONLSE) with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method (GLGRM), the abundant solutions of NLSE and HONLSE are obtained
Simulation of the Schroedinger equation on SHAC
International Nuclear Information System (INIS)
Stewart, A.
1976-01-01
A simulation of the Schroedinger wave equation for the hydrogen atom, on SHAC, a simple homogeneous analogue computer primarily intended for use in schools, is described. Due to the incorporation of FET switches very high speed switching from initial conditions to compute modes is possible. The techniques employed in the multiplier and divider are discussed and the flow diagram for the Schroedinger program shown. Results and photographs are discussed. (U.K.)
A reliable treatment for nonlinear Schroedinger equations
International Nuclear Information System (INIS)
Khani, F.; Hamedi-Nezhad, S.; Molabahrami, A.
2007-01-01
Exp-function method is used to find a unified solution of nonlinear wave equation. Nonlinear Schroedinger equations with cubic and power law nonlinearity are selected to illustrate the effectiveness and simplicity of the method. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equation
Some physical applications of fractional Schroedinger equation
International Nuclear Information System (INIS)
Guo Xiaoyi; Xu Mingyu
2006-01-01
The fractional Schroedinger equation is solved for a free particle and for an infinite square potential well. The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained. In the barrier penetration problem, the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined. In the quantum scattering problem, according to the fractional Schroedinger equation, the Green's function of the Lippmann-Schwinger integral equation is given
Quantum derivatives and the Schroedinger equation
International Nuclear Information System (INIS)
Ben Adda, Faycal; Cresson, Jacky
2004-01-01
We define a scale derivative for non-differentiable functions. It is constructed via quantum derivatives which take into account non-differentiability and the existence of a minimal resolution for mean representation. This justify heuristic computations made by Nottale in scale-relativity. In particular, the Schroedinger equation is derived via the scale-relativity principle and Newton's fundamental equation of dynamics
Spectral problem for the radial Schroedinger equation
International Nuclear Information System (INIS)
Vshivtsev, A.S.; Tatarintsev, A.V.; Prokopov, A.V.; Sorokin, V. N.
1998-01-01
For the first time, a procedure for determining spectra on the basis of generalized integral transformations is implemented for a wide class of radial Schroedinger equations. It is shown that this procedure works well for known types of potentials. Concurrently, this method makes it possible to obtain new analytic results for the Cornell potential. This may prove important for hadron physics
Measurement theory and the Schroedinger equation
International Nuclear Information System (INIS)
Schwarz, A.S.; Tyupkin, Yu.S.
1987-01-01
The paper is an analysis of the measuring process in quantum mechanics based on the Schroedinger equation. The arguments employed use an assumption reflecting, to some extent, the statistical properties of the vacuum. A description is given of the cases in which different incoherent superpositions of pure states in quantum mechanics are physically equivalent. The fundamental difference between quantum and classical mechanics as explained by the existence of unobservable variables is discussed. (U.K.)
Hidden Statistics of Schroedinger Equation
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
Inversion transformation in the Schroedinger equation
International Nuclear Information System (INIS)
Demkov, Yu.N.; Semenova, N.V.
1984-01-01
Using the inversion with respect to a sphere in the coordinate space, the equivalence between the Schroedinger equations with different potentials is established. It is shown that the zero-energy equation for a spherically symmetric potential is equivalent to the equation with an axially symmetric potential of a special form. The particular exact solutions of the zero-energy problem for the motion of a particle in the field of two Maxwell ''fish-eye'' potentials and potentials with the two Coulomb singularities are found
Dual Schroedinger Equation as Global Optimization Algorithm
International Nuclear Information System (INIS)
Huang Xiaofei; eGain Communications, Mountain View, CA 94043
2011-01-01
The dual Schroedinger equation is defined as replacing the imaginary number i by -1 in the original one. This paper shows that the dual equation shares the same stationary states as the original one. Different from the original one, it explicitly defines a dynamic process for a system to evolve from any state to lower energy states and eventually to the lowest one. Its power as a global optimization algorithm might be used by nature for constructing atoms and molecules. It shall be interesting to verify its existence in nature.
The Schroedinger equation and canonical perturbation theory
International Nuclear Information System (INIS)
Graffi, S.; Paul, T.
1987-01-01
Let T 0 (ℎ,ω)+εV be the Schroedinger operator corresponding to the classical Hamiltonian H 0 (ω)+εV, where H 0 (ω) is the d-dimensional harmonic oscillator with non-resonant frequencies ω=(ω 1 ..., ω d ) and the potential V(q 1 , ..., q d ) is an entire function of order (d+l) -1 . We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schroedinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schroedinger series near any eigenvalue of T 0 (ℎ,ω) admits a convergent expansion in powers of ℎ of initial point the corresponding term of the classical Birkhoff expansion. Moreover if V is an even polynomial, the above result and the KAM theorem show that all eigenvalues λ n (ℎ,ε) of T 0 +εV such that nℎ coincides with a KAM torus are given, up to order ε ∞ , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in ℎ. (orig.)
An implicit spectral formula for generalized linear Schroedinger equations
International Nuclear Information System (INIS)
Schulze-Halberg, A.; Garcia-Ravelo, J.; Pena Gil, Jose Juan
2009-01-01
We generalize the semiclassical Bohr–Sommerfeld quantization rule to an exact, implicit spectral formula for linear, generalized Schroedinger equations admitting a discrete spectrum. Special cases include the position-dependent mass Schroedinger equation or the Schroedinger equation for weighted energy. Requiring knowledge of the potential and the solution associated with the lowest spectral value, our formula predicts the complete spectrum in its exact form. (author)
Solving the Schroedinger equation using Smolyak interpolants
International Nuclear Information System (INIS)
Avila, Gustavo; Carrington, Tucker Jr.
2013-01-01
In this paper, we present a new collocation method for solving the Schroedinger equation. Collocation has the advantage that it obviates integrals. All previous collocation methods have, however, the crucial disadvantage that they require solving a generalized eigenvalue problem. By combining Lagrange-like functions with a Smolyak interpolant, we device a collocation method that does not require solving a generalized eigenvalue problem. We exploit the structure of the grid to develop an efficient algorithm for evaluating the matrix-vector products required to compute energy levels and wavefunctions. Energies systematically converge as the number of points and basis functions are increased
Scale calculus and the Schroedinger equation
International Nuclear Information System (INIS)
Cresson, Jacky
2003-01-01
This paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f(t,ε) such that f(t,ε)→f(t) when ε goes to zero. This led to several new notions as representations: fractal functions and ε-differentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativity theory developed by Nottale. We obtain a nonlinear Schroedinger equation via the classical Newton's equation of dynamics using the scale operator. Under special assumptions we recover the classical Schroedinger equation and we discuss the relevance of these assumptions
International Nuclear Information System (INIS)
Kotel'nikov, G.A.
1994-01-01
An algorithm id proposed for research the symmetries of mathematical physics equation. The application of this algorithm to the Schroedinger equation permitted to establish, that in addition to the known symmetry the Schroedinger equation possesses also the relativistic symmetry
Feynman path integral related to stochastic schroedinger equation
International Nuclear Information System (INIS)
Belavkin, V.P.; Smolyanov, O.G.
1998-01-01
The derivation of the Schroedinger equation describing the continuous measurement process is presented. The representation of the solution of the stochastic Schroedinger equation for continuous measurements is obtained by means of the Feynman path integral. The connection with the heuristic approach to the description of continuous measurements is considered. The connection with the Senon paradox is established [ru
Spectrum of the ballooning Schroedinger equation
International Nuclear Information System (INIS)
Dewar, R.L.
1997-01-01
The ballooning Schroedinger equation (BSE) is a model equation for investigating global modes that can, when approximated by a Wentzel-Kramers-Brillouin (WKB) ansatz, be described by a ballooning formalism locally to a field line. This second order differential equation with coefficients periodic in the independent variable θ k is assumed to apply even in cases where simple WKB quantization conditions break down, thus providing an alternative to semiclassical quantization. Also, it provides a test bed for developing more advanced WKB methods: e.g. the apparent discontinuity between quantization formulae for open-quotes trappedclose quotes and open-quotes passingclose quotes modes, whose ray paths have different topologies, is removed by extending the WKB method to include the phenomena of tunnelling and reflection. The BSE is applied to instabilities with shear in the real part of the local frequency, so that the dispersion relation is inherently complex. As the frequency shear is increased, it is found that trapped modes go over to passing modes, reducing the maximum growth rate by averaging over θ k
On the Schroedinger equation for the minisuperspace models
International Nuclear Information System (INIS)
Tkach, V.I.; Pashnev, A.I.; Rosales, J.J.
2000-01-01
We obtain a time-dependent Schroedinger equation for the Friedmann-Robertson-Walker (FRW) model interacting with a homogeneous scalar matter field. We show that for this purpose it is necessary to include an additional action invariant under the reparametrization of time. The last one does not change the equations of motion of the system, but changes only the constraint which at the quantum level becomes time-dependent Schroedinger equation. The same procedure is applied to the supersymmetric case and the supersymmetric quantum constraints are obtained, one of them is a square root of the Schroedinger operator
Structural invariance of the Schroedinger equation and chronoprojective geometry
International Nuclear Information System (INIS)
Burdet, G.; Perrin, M.
1983-07-01
We describe an extension of the chronoprojective geometry and show how its automorphisms are related to the invariance properties of the Schroedinger equation describing a quantum test particle in any Newton-Cartan structure
Wigner function and Schroedinger equation in phase-space representation
International Nuclear Information System (INIS)
Chruscinski, Dariusz; Mlodawski, Krzysztof
2005-01-01
We discuss a family of quasidistributions (s-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schroedinger equation. It turns out that although Wigner functions satisfy the Schroedinger equation in phase space, they have a completely different interpretation
Discrete transparent boundary conditions for Schroedinger-type equations
International Nuclear Information System (INIS)
Schmidt, F.; Yevick, D.
1997-01-01
We present a general technique for constructing nonlocal transparent boundary conditions for one-dimensional Schroedinger-type equations. Our method supplies boundary conditions for the θ-family of implicit one-step discretizations of Schroedinger's equation in time. The use of Mikusinski's operator approach in time avoids direct and inverse transforms between time and frequency domains and thus implements the boundary conditions in a direct manner. 14 refs., 9 figs
Linearised collective Schroedinger equation for nuclear quadrupole surface vibrations
International Nuclear Information System (INIS)
Greiner, M.; Heumann, D.; Scheid, W.
1990-11-01
The linearisation of the Schroedinger equation for nuclear quadrupole surface vibrations yields a new spin degree of freedom, which is called collective spin and has a value of 3/2. With the introduction of collective spin dependent potentials, this linearised Schroedinger equation is then used for the description of low energy spectra and electromagnetic transition probabilities of some even-odd Xe, Ir and Au nuclei which have a spin 3/2 in their groundstate. (orig.)
Exact solutions to two higher order nonlinear Schroedinger equations
International Nuclear Information System (INIS)
Xu Liping; Zhang Jinliang
2007-01-01
Using the homogeneous balance principle and F-expansion method, the exact solutions to two higher order nonlinear Schroedinger equations which describe the propagation of femtosecond pulses in nonlinear fibres are obtained with the aid of a set of subsidiary higher order ordinary differential equations (sub-equations for short)
Dispersive estimates for the Schroedinger and Klein-Gordon equations
Energy Technology Data Exchange (ETDEWEB)
Kopylova, Elena A [Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow (Russian Federation)
2010-01-01
This is a survey of results on the long-time asymptotic behaviour of solutions of the Schroedinger and Klein-Gordon equations in weighted energy norms. Results obtained from 1975 to 2001 in the spectral scattering theory of Agmon, Jensen-Kato, Jensen-Nenciu, and Murata are described for the Schroedinger equation, along with the author's recent results obtained jointly with A.I. Komech for the Klein-Gordon equation. The methods used develop the spectral approach as applied to relativistic equations. Bibliography: 40 titles.
Monodromy of the matrix Schroedinger equations and Darboux transformations
Goncharenko, V M
1998-01-01
A Schroedinger operator L=-d sup 2 /dz sup 2 +U(z) with a matrix-valued rational potential U(z) is said to have trivial monodromy if all the solutions of the corresponding Schroedinger equations L psi=lambda psi are single-valued in the complex plane z is an element of C for any lambda. A local criterion of this property in terms of the Laurent coefficients of the potential U near its singularities, which are assumed to be regular, is found. It is proved that any such operator with a potential vanishing at infinity can be obtained by a matrix analogue of the Darboux transformation from the Schroedinger operator L sub o =-d sup 2 /dz sup 2. This generalizes the well known Duistermaat-Gruenbaum result to the matrix case and gives the explicit description of the Schroedinger operators with trivial monodromy in this case. (author)
Mickens, Ronald E.
1989-01-01
A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented.
Generalized fractional Schroedinger equation with space-time fractional derivatives
International Nuclear Information System (INIS)
Wang Shaowei; Xu Mingyu
2007-01-01
In this paper the generalized fractional Schroedinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schroedinger equation and the ones in standard quantum
On the invariant measure for the nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Zhidkov, P.R.
1991-01-01
The invariant measure for the nonlinear Schroedinger equation is constructed. In fact, it is assumed that the nonlinearity in the equation is semilinear. The main aim of the paper is the explanation of the Fermi - Past - Ulam phenomenon. Poincare theorem gives the answer to this question. 17 refs
Exact solutions for the cubic-quintic nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Zhu Jiamin; Ma Zhengyi
2007-01-01
In this paper, the cubic-quintic nonlinear Schroedinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions
Energy Technology Data Exchange (ETDEWEB)
Paul, Wolfgang; Koeppe, Jeanette [Institut fuer Physik, Martin Luther Universitaet, 06099 Halle (Germany); Grecksch, Wilfried [Institut fuer Mathematik, Martin Luther Universitaet, 06099 Halle (Germany)
2016-07-01
The standard approach to solve a non-relativistic quantum problem is through analytical or numerical solution of the Schroedinger equation. We show a way to go around it. This way is based on the derivation of the Schroedinger equation from conservative diffusion processes and the establishment of (several) stochastic variational principles leading to the Schroedinger equation under the assumption of a kinematics described by Nelson's diffusion processes. Mathematically, the variational principle can be considered as a stochastic optimal control problem linked to the forward-backward stochastic differential equations of Nelson's stochastic mechanics. The Hamilton-Jacobi-Bellmann equation of this control problem is the Schroedinger equation. We present the mathematical background and how to turn it into a numerical scheme for analyzing a quantum system without using the Schroedinger equation and exemplify the approach for a simple 1d problem.
Integrable discretization s of derivative nonlinear Schroedinger equations
International Nuclear Information System (INIS)
Tsuchida, Takayuki
2002-01-01
We propose integrable discretizations of derivative nonlinear Schroedinger (DNLS) equations such as the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schroedinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa-Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations. (author)
New method for solving three-dimensional Schroedinger equation
International Nuclear Information System (INIS)
Melezhik, V.S.
1990-01-01
The method derived recently for solving a multidimensional scattering problem is applied to a three-dimensional Schroedinger equation. As compared with direct three-dimensional calculations of finite elements and finite differences, this approach gives sufficiently accurate upper and lower approximations to the helium-atom binding energy, which demonstrates its efficiency. 15 refs.; 1 fig.; 2 tabs
Consequences of the Schroedinger equation for atomic and molecular physics
International Nuclear Information System (INIS)
Thirring, W.E.
1986-01-01
The non-relativistic Schroedinger equation for a system of nuclei and electrons is considered and general properties of Hamiltonian H are treated and commented: self-adjontness of H, the spectrum of H, the discrete spectrum, the continuous spectrum, the limit of infinite nuclear mass, the limit of infinite nuclear charge. (G.Q.)
Quantum osp-invariant non-linear Schroedinger equation
International Nuclear Information System (INIS)
Kulish, P.P.
1985-04-01
The generalizations of the non-linear Schroedinger equation (NS) associated with the orthosymplectic superalgebras are formulated. The simplest osp(1/2)-NS model is solved by the quantum inverse scattering method on a finite interval under periodic boundary conditions as well as on the wholeline in the case of a finite number of excitations. (author)
Travelling solitons in the parametrically driven nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Barashenkov, I.V.; Zemlyanaya, E.V.; Baer, M.
2000-01-01
We show that the parametrically driven nonlinear Schroedinger equation has wide classes of travelling soliton solutions, some of which are stable. For small driving strengths stable nonpropagating and moving solitons co-exist while strongly forced solitons can only be stable when moving sufficiently fast
New method for solving three-dimensional Schroedinger equation
International Nuclear Information System (INIS)
Melezhik, V.S.
1992-01-01
A new method is developed for solving the multidimensional Schroedinger equation without the variable separation. To solve the Schroedinger equation in a multidimensional coordinate space X, a difference grid Ω i (i=1,2,...,N) for some of variables, Ω, from X={R,Ω} is introduced and the initial partial-differential equation is reduced to a system of N differential-difference equations in terms of one of the variables R. The arising multi-channel scattering (or eigenvalue) problem is solved by the algorithm based on a continuous analog of the Newton method. The approach has been successfully tested for several two-dimensional problems (scattering on a nonspherical potential well and 'dipole' scatterer, a hydrogen atom in a homogenous magnetic field) and for a three-dimensional problem of the helium-atom bound states. (author)
Iteration of some discretizations of the nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Ross, K.A.; Thompson, C.J.
1986-01-01
We consider several discretizations of the nonlinear Schroedinger equation which lead naturally to the study of some symmetric difference equations of the form PHIsub(n+1) + PHIsub(n-1) = f(PHIsub(n)). We find a variety of interesting and exotic behaviour from simple closed orbits to intricate patterns of orbits and loops in the (PHIsub(n+1),PHIsub(n)) phase-plane. Some analytical results for a special case are also presented. (orig.)
Asymptotic Value Distribution for Solutions of the Schroedinger Equation
International Nuclear Information System (INIS)
Breimesser, S. V.; Pearson, D. B.
2000-01-01
We consider the Dirichlet Schroedinger operator T=-(d 2 /d x 2 )+V, acting in L 2 (0,∞), where Vis an arbitrary locally integrable potential which gives rise to absolutely continuous spectrum. Without any other restrictive assumptions on the potential V, the description of asymptotics for solutions of the Schroedinger equation is carried out within the context of the theory of value distribution for boundary values of analytic functions. The large x asymptotic behaviour of the solution v(x,λ) of the equation Tf(x,λ)=λf(x,λ), for λ in the support of the absolutely continuous part μ a.c. of the spectral measure μ, is linked to the spectral properties of this measure which are determined by the boundary value of the Weyl-Titchmarsh m-function. Our main result (Theorem 1) shows that the value distribution for v'(N,λ)/v(N,λ) approaches the associated value distribution of the Herglotz function m N (z) in the limit N → ∞, where m N (z) is the Weyl-Titchmarsh m-function for the Schroedinger operator -(d 2 /d x 2 )+Vacting in L 2 (N,∞), with Dirichlet boundary condition at x=N. We will relate the analysis of spectral asymptotics for the absolutely continuous component of Schroedinger operators to geometrical properties of the upper half-plane, viewed as a hyperbolic space
Solution of the Schroedinger equation by a spectral method
International Nuclear Information System (INIS)
Feit, M.D.; Fleck, J.A. Jr.; Steiger, A.
1982-01-01
A new computational method for determining the eigenvalues and eigenfunctions of the Schroedinger equation is described. Conventional methods for solving this problem rely on diagonalization of a Hamiltonian matrix or iterative numerical solutions of a time independent wave equation. The new method, in contrast, is based on the spectral properties of solutions to the time-dependent Schroedinger equation. The method requires the computation of a correlation function from a numerical solution psi(r, t). Fourier analysis of this correlation function reveals a set of resonant peaks that correspond to the stationary states of the system. Analysis of the location of these peaks reveals the eigenvalues with high accuracy. Additional Fourier transforms of psi(r, t) with respect to time generate the eigenfunctions. The effectiveness of the method is demonstrated for a one-dimensional asymmetric double well potential and for the two-dimensional Henon--Heiles potential
The Schroedinger equation as a singular perturbation problem
International Nuclear Information System (INIS)
Jager, E.M. de; Kuepper, T.
1978-01-01
Comparisons are made of the eigenvalues and the corresponding eigenfunctions of the eigenvalue problem connected with the one dimensional Schroedinger equation in Hilbert space. The difference of the eigenvalues is estimated by applying Weyl's monotonicity principle and the minimum maximum principle. The difference of the eigenfunctions is estimated in L 2 norm and in maximum norm obtained by using simple tools from operator theory in Hilbert spaces. An application concerning perturbations of the Planck ideal linear oscillator is given. (author)
Strong phase correlations of solitons of nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Litvak, A.G.; Mironov, V.A.; Protogenov, A.P.
1994-06-01
We discuss the possibility to suppress the collapse in the nonlinear 2+1 D Schroedinger equation by using the gauge theory of strong phase correlations. It is shown that invariance relative to q-deformed Hopf algebra with deformation parameter q being the fourth root of unity makes the values of the Chern-Simons term coefficient, k=2, and of the coupling constant, g=1/2, fixed; no collapsing solutions are present at those values. (author). 21 refs
Inhomogeneous critical nonlinear Schroedinger equations with a harmonic potential
International Nuclear Information System (INIS)
Cao Daomin; Han Pigong
2010-01-01
In this paper, we study the Cauchy problem of the inhomogeneous nonlinear Schroedinger equation with a harmonic potential: i∂ t u=-div(f(x)∇u)+|x| 2 u-k(x)|u| 4/N u, x is an element of R N , N≥1, which models the remarkable Bose-Einstein condensation. We discuss the existence and nonexistence results and investigate the limiting profile of blow-up solutions with critical mass.
Exactly solvable position dependent mass schroedinger equation
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Koc, R.; Tuetuencueler, H.; Koercuek, E.
2002-01-01
Exact solution of the Schrodinger equation with a variable mass is presented. We have derived general expressions for the eigenstates and eigenvalues of the position dependent mass systems. We provide supersymmetric and Lie algebraic methods to discuss the position dependent mass systems
On quantization, the generalised Schroedinger equation and classical mechanics
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Jones, K.R.W.
1991-01-01
A ψ-dependent linear functional operator, was defined, which solves the problem of quantization in non-relativistic quantum mechanics. Weyl ordering is implemented automatically and permits derivation of many of the quantum to classical correspondences. The parameter λ presents a natural C ∞ deformation of the dynamical structure of quantum mechanics via a non-linear integro-differential 'Generalised Schroedinger Equation', admitting an infinite family of soliton solutions. All these solutions are presented and it is shown that this equation gives an exact dynamic and energetic reproduction of classical mechanics with the correct measurement theoretic limit. 23 refs
Numerical solution of the Schroedinger equation with a polynomial potential
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Campoy, G.; Palma, A.
1986-01-01
A numerical method for solving the Schroedinger equation for a potential expressed as a polynomial is proposed. The basic assumption relies on the asymptotic properties of the solution of this equation. It is possible to obtain the energies and the stationary state functions simultaneously. They analyze, in particular, the cases of the quartic anharmonic oscillator and a hydrogen atom perturbed by a quadratic term, obtaining its energy eigenvalues for some values of the perturbation parameter. Together with the Hellmann-Feynman theorem, they use their algorithm to calculate expectation values of x'' for arbitrary positive values of n. 4 tables
Properties of some nonlinear Schroedinger equations motivated through information theory
International Nuclear Information System (INIS)
Yuan, Liew Ding; Parwani, Rajesh R
2009-01-01
We update our understanding of nonlinear Schroedinger equations motivated through information theory. In particular we show that a q-deformation of the basic nonlinear equation leads to a perturbative increase in the energy of a system, thus favouring the simplest q = 1 case. Furthermore the energy minimisation criterion is shown to be equivalent, at leading order, to an uncertainty maximisation argument. The special value η = 1/4 for the interpolation parameter, where leading order energy shifts vanish, implies the preservation of existing supersymmetry in nonlinearised supersymmetric quantum mechanics. Physically, η might be encoding relativistic effects.
Equivalence of two alternative approaches to Schroedinger equations
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Goenuel, B; Koeksal, K
2006-01-01
A recently developed simple approach for the exact/approximate solution of Schroedinger equations with constant/position-dependent mass, in which the potential is considered as in the perturbation theory, is shown to be equivalent to the one leading to the construction of exactly solvable potentials via the solution of second-order differential equations in terms of known special functions. The formalism in the former solves difficulties encountered in the latter in revealing the corrections explicitly to the unperturbed piece of the solutions whereas the other obviates cumbersome procedures used in the calculations of the former
Quantum gravitational corrections to the functional Schroedinger equation
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Kiefer, C.; Singh, T.P.
1990-10-01
We derive corrections to the Schroedinger equation which arise from the quantization of the gravitational field. This is achieved through an expansion of the full functional Wheeler-DeWitt equation with respect to powers of the Planck mass. We demonstrate that the corrections terms are independent of the factor ordering which is chosen for the gravitational kinetic term. Although the corrections are numerically extremely tiny, we show how they lead, at least in principle, to shift in the spectral lines of hydrogen type atoms. We discuss the significance of these corrections for quantum field theory near the Planck scale. (author). 35 refs
Schroedinger equations with indefinite effective mass
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Levai, G.; Znojil, M.
2012-01-01
Complete text of publication follows. The interaction of a particle with the medium around it is usually described by some potential function V (x). It is also often necessary to take into consideration the effects of this medium using a position-dependent effective mass. A wide variety of effective masses m(x) have been used in methodological studies and applications mainly restricted to one dimensional problems, including mass functions that vanish at certain locations or those reaching infinity in some limit. The common feature of these m(x) functions was that they were all non-negative. In our recent study on the PT -symmetric version of the Coulomb potential we found that an asymptotically negative effective mass is necessary for the stability of the energy spectrum. This result inspired us to investigate under which conditions can one apply mass functions that are negative at least in some domains of the coordinate space. For the sake of simplicity we considered the infinitely deep squarewell potential in one dimension V(x) = (+∞, /x/ > L > 1, 0, /x/ 0 , /x/ 0 the energy spectrum becomes unbounded from below. This is not surprising considering that with a negative mass the kinetic energy also becomes negative. In order to stabilize the spectrum we considered energy-dependent effective mass functions that kept the mass finite even for increasing values of the energy. Our first choice was m(x,E) = (1, /x/ ∈ (1,L), -tanh (E), /x/ 2 tanh λ(k) tan k(L - 1) = -1, where λ(k) = k √tanh k 2 . With this choice the energy spectrum was found to be bounded from below. Qualitatively similar results were found for our second example, where we considered a threshold energy E thr by m(x,E) = 1, /x/ ∈ (1,L) , -1, E ≥ E thr , +1, E thr ), /x/ 2 , /x/ 0 and b = b(E) > 0. This lead to the rescaled secular equation tan κa/b x tanh κ(L - a) = b. (3) This setting allowed the investigation of the special limit in which the m(x) turns into the Dirac delta function. We
Energy Technology Data Exchange (ETDEWEB)
Xin, Zhou [Wisconsin Univ., Madison (USA). Dept. of Mathematics
1990-03-01
For the direct-inverse scattering transform of the time dependent Schroedinger equation, rigorous results are obtained based on an operator-triangular-factorization approach. By viewing the equation as a first order operator equation, similar results as for the first order n x n matrix system are obtained. The nonlocal Riemann-Hilbert problem for inverse scattering is shown to have solution. (orig.).
International Nuclear Information System (INIS)
Zhou Xin
1990-01-01
For the direct-inverse scattering transform of the time dependent Schroedinger equation, rigorous results are obtained based on an operator-triangular-factorization approach. By viewing the equation as a first order operator equation, similar results as for the first order n x n matrix system are obtained. The nonlocal Riemann-Hilbert problem for inverse scattering is shown to have solution. (orig.)
Inverse Schroedinger equation and the exact wave function
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Nakatsuji, Hiroshi
2002-01-01
Using the inverse of the Hamiltonian, we introduce the inverse Schroedinger equation (ISE) that is equivalent to the ordinary Schroedinger equation (SE). The ISE has the variational principle and the H-square group of equations as the SE has. When we use a positive Hamiltonian, shifting the energy origin, the inverse energy becomes monotonic and we further have the inverse Ritz variational principle and cross-H-square equations. The concepts of the SE and the ISE are combined to generalize the theory for calculating the exact wave function that is a common eigenfunction of the SE and ISE. The Krylov sequence is extended to include the inverse Hamiltonian, and the complete Krylov sequence is introduced. The iterative configuration interaction (ICI) theory is generalized to cover both the SE and ISE concepts and four different computational methods of calculating the exact wave function are presented in both analytical and matrix representations. The exact wave-function theory based on the inverse Hamiltonian can be applied to systems that have singularities in the Hamiltonian. The generalized ICI theory is applied to the hydrogen atom, giving the exact solution without any singularity problem
The Schroedinger and Dirac free particle equations without quantum mechanics
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Ord, G.N.
1996-01-01
Einstein close-quote s theory of Brownian Movement has provided a well accepted microscopic model of diffusion for many years. Until recently the relationship between this model and Quantum Mechanics has been completely formal. Brownian motion provides a microscopic model for diffusion, but quantum mechanics and diffusion are related by a formal analytic continuation, so the relationship between Brownian motion and Quantum Mechanics has been correspondingly vague. Some recent work has changed this picture somewhat and here we show that a random walk model of Brownian motion produces the diffusion equation or the telegraph equations as a descriptions of particle densities, while at the same time the correlations in the space-time geometry of these same Brownian particles obey the Schroedinger and Dirac equations respectively. This is of interest because the equations of Quantum Mechanics appear here naturally in a classical context without the problems of interpretation they have in the usual context. copyright 1996 Academic Press, Inc
On the solution of the Schroedinger equation through continued fractions
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Mignaco, J.A.
1979-05-01
The domain of interest for the applications of a method to solve the Schroedinger equation through continued fractions is studied. It is argued that the method applies almost equally well to quantum mechanical regimes (lower energy levels, low energy scattering) as well as to semiclassical ones simultaneously; this is illustrated by the example of the central power law potentials r sup(ν)(ν>o). The explanation of this behaviour is given in terms of the mathematical approximations involved and its relationship to physically interesting quantities. (Author) [pt
Perturbative approach to non-Markovian stochastic Schroedinger equations
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Gambetta, Jay; Wiseman, H.M.
2002-01-01
In this paper we present a perturbative procedure that allows one to numerically solve diffusive non-Markovian stochastic Schroedinger equations, for a wide range of memory functions. To illustrate this procedure numerical results are presented for a classically driven two-level atom immersed in an environment with a simple memory function. It is observed that as the order of the perturbation is increased the numerical results for the ensemble average state ρ red (t) approach the exact reduced state found via Imamog-barlu ' s enlarged system method [Phys. Rev. A 50, 3650 (1994)
A new method for the solution of the Schroedinger equation
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Amore, Paolo; Aranda, Alfredo; De Pace, Arturo
2004-01-01
We present a new method for the solution of the Schroedinger equation applicable to problems of a non-perturbative nature. The method works by identifying three different scales in the problem, which then are treated independently: an asymptotic scale, which depends uniquely on the form of the potential at large distances; an intermediate scale, still characterized by an exponential decay of the wavefunction; and, finally, a short distance scale, in which the wavefunction is sizable. The notion of optimized perturbation is then used in the last two regimes. We apply the method to the quantum anharmonic oscillator and find it suitable to treat both energy eigenvalues and wavefunctions, even for strong couplings
Integrability of a system of two nonlinear Schroedinger equations
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Zhukhunashvili, V.Z.
1989-01-01
In recent years the inverse scattering method has achieved significant successes in the integration of nonlinear models that arise in different branches of physics. However, its region of applicability is still restricted, i.e., not all nonlinear models can be integrated. In view of the great mathematical difficulties that arise in integration, it is clearly worth testing a model for integrability before turning to integration. Such a possibility is provided by the Zakharov-Schulman method. The question of the integrability of a system of two nonlinear Schroedinger equations is resolved. It is shown that the previously known cases exhaust all integrable variants
Nonrelativistic Schroedinger equation in quasi-classical theory
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Wignall, J.W.G.
1987-01-01
The author has recently proposed a quasi-classical theory of particles and interactions in which particles are pictured as extended periodic disturbances in a universal field chi(x,t), interacting with each other via nonlinearity in the equation of motion for chi. The present paper explores the relationship of this theory to nonrelativistic quantum mechanics; as a first step, it is shown how it is possible to construct from chi a configuration-space wave function Psi(x 1 , X 2 , t), and that the theory requires that Psi satisfy the two-particle Schroedinger equation in the case where the two particles are well separated from each other. This suggests that the multiparticle Schroedinger equation can be obtained as a direct consequence of the quasi-classical theory without any use of the usual formalism (Hilbert space, quantization rules, etc.) of conventional quantum theory and in particular without using the classical canonical treatment of a system as a crutch theory which has subsequently to be quantized. The quasi-classical theory also suggests the existence of a preferred absolute gauge for the electromagnetic potentials
Soliton-like solutions to the ordinary Schroedinger equation
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Zamboni-Rached, Michel [Universidade Estadual de Campinas (DMO/FEEC/UNICAMP), Campinas, SP (Brazil). Fac. de Engenharia Eletrica e de Computacao. Dept. de Microondas e Optica; Recami, Erasmo, E-mail: recami@mi.infn.i [Universita Statale di Bergamo, Bergamo (Italy). Facolta di Ingegneria
2011-07-01
In recent times it has been paid attention to the fact that (linear) wave equations admit of soliton-like solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G or Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation within standard Quantum Mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we show therefore how to obtain nite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. (author)
Soliton-like solutions to the ordinary Schroedinger equation
International Nuclear Information System (INIS)
Zamboni-Rached, Michel; Recami, Erasmo
2011-01-01
In recent times it has been paid attention to the fact that (linear) wave equations admit of soliton-like solutions, known as Localized Waves or Non-diffracting Waves, which propagate without distortion in one direction. Such Localized Solutions (existing also for K-G or Dirac equations) are a priori suitable, more than Gaussian's, for describing elementary particle motion. In this paper we show that, mutatis mutandis, Localized Solutions exist even for the ordinary Schroedinger equation within standard Quantum Mechanics; and we obtain both approximate and exact solutions, also setting forth for them particular examples. In the ideal case such solutions bear infinite energy, as well as plane or spherical waves: we show therefore how to obtain nite-energy solutions. At last, we briefly consider solutions for a particle moving in the presence of a potential. (author)
Lower bounds for solutions of the Schroedinger equation
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Froese, R.G.
1983-01-01
For a large class of generalized N-body Hamiltonians H = -Δ + V the large absolute value x behavior of solutions to the Schroedinger equation H psi = H psi is studied. If E lies below the essential spectrum of H, then it is proved that lim R -1 log (absolute value psi/sub R/) = -α 0 R → infinity where absolute value psi/sub R/ 2 is the integral of absolute value psi 2 over a sphere of radius R and α 0 2 + E is a threshold or α 0 0. For E not necessarily below the essential spectrum of H, the same equation holds with absolute value psi/sub R/ 2 replaced by an integral of absolute value psi 2 over a spherical shell
Limited-diffraction solutions to Maxwell and Schroedinger equations
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Lu, Jian-yu; Greenleaf, J.F.
1996-10-01
The authors have developed a new family of limited diffraction electromagnetic X-shaped waves based on the scalar X-shaped waves discovered previously. These waves are diffraction-free in theory and particle-like (wave packets), in that they maintain their shape as they propagate to an infinite distance. The 'X waves' possess (theoretically) infinitely extended 'arms' and - at least, the ones studied in this paper - have an infinite total energy: therefore, they are not physically realizable. However, they can be truncated in both space and time and 'approximated' by means of a finite aperture radiator so to get a large enough depth of interest (depth of field). In addition to the Maxwell equations, X wave solutions to the free Schroedinger equation are also obtained. Possible applications of these new waves are discussed. Finally, the authors discuss the appearance of the X-shaped solutions from the purely geometric point of view of the special relativity theory
Nonlinear Schroedinger equation with U(p,q) isotopical group
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Makhankov, V.G.; Pashaev, O.K.
1981-01-01
The properties of the nonlinear Schroedinger equation (NLS) with U(1,1) isogroup are considered in detail. This example illustrates the essential difference between the system and the well-known ''vector'' NLS, i.e. the large set of allowed boundary conditions on the fields that leads to a rich set of solutions of the system. Four types of boundary conditions and related soliton solutions are considered. The Bohr-Sommerfeld quantization allows to interpret them in terms of ''drops'' and ''bubbles'' as bound states of a large number of constituent bosons subject to the thermodynamical relations for gas mixtures. The U(1,1) system under the vanishing boundary conditions may be considered as continuous analog of the Hubbard model and therefore the paper is concluded by studying the inverse scattering equations for this case [ru
Analytical exact solution of the non-linear Schroedinger equation
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Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da
2011-01-01
Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)
The frictional Schroedinger-Newton equation in models of wave function collapse
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Diosi, Lajos [Research Institute for Particle and Nuclear Physics, H-1525 Budapest 114, PO Box 49 (Hungary)
2007-05-15
Replacing the Newtonian coupling G by -iG, the Schroedinger--Newton equation becomes {sup f}rictional{sup .} Instead of the reversible Schroedinger-Newton equation, we advocate its frictional version to generate the set of pointer states for macroscopic quantum bodies.
Derivation of the Schroedinger equation from stochastic mechanics
International Nuclear Information System (INIS)
Wallstrom, T.C.
1988-01-01
The thesis is divided into four largely independent chapters. The first three chapters treat mathematical problems in the theory of stochastic mechanics. The fourth chapter deals with stochastic mechanisms as a physical theory and shows that the Schroedinger equation cannot be derived from existing formulations of stochastic mechanics, as had previously been believed. Since the drift coefficients of stochastic mechanical diffusions are undefined on the nodes, or zeros of the density, an important problem has been to show that the sample paths stay away from the nodes. In Chapter 1, it is shown that for a smooth wavefunction, the closest approach to the nodes can be bounded solely in terms of the time-integrated energy. The ergodic properties of stochastic mechanical diffusions are greatly complicated by the tendency of the particles to avoid the nodes. In Chapter 2, it is shown that a sufficient condition for a stationary process to be ergodic is that there exist positive t and c such that for all x and y, p t (x,y) > cp(y), and this result is applied to show that the set of spin-1/2 diffusions is uniformly ergodic. Nelson has conjectured that in the limit as the particle's moment of inertia I goes to zero, the projections of the Bopp-Haag-Dankel diffusions onto IR 3 converge to a Markovian limit process. This conjecture is proved for the spin-1/2 case in Chapter 3, and the limit process identified as the diffusion naturally associated with the solution to the regular Pauli equation. In Chapter 4 it is shown that the general solution of the stochastic Newton equation does not correspond to a solution of the Schroedinger equation
Soliton interaction in the coupled mixed derivative nonlinear Schroedinger equations
International Nuclear Information System (INIS)
Zhang Haiqiang; Tian Bo; Lue Xing; Li He; Meng Xianghua
2009-01-01
The bright one- and two-soliton solutions of the coupled mixed derivative nonlinear Schroedinger equations in birefringent optical fibers are obtained by using the Hirota's bilinear method. The investigation on the collision dynamics of the bright vector solitons shows that there exists complete or partial energy switching in this coupled model. Such parametric energy exchanges can be effectively controlled and quantificationally measured by analyzing the collision dynamics of the bright vector solitons. The influence of two types of nonlinear coefficient parameters on the energy of each vector soliton, is also discussed. Based on the significant energy transfer between the two components of each vector soliton, it is feasible to exploit the future applications in the design of logical gates, fiber directional couplers and quantum information processors.
Extensions of the auxiliary field method to solve Schroedinger equations
International Nuclear Information System (INIS)
Silvestre-Brac, Bernard; Semay, Claude; Buisseret, Fabien
2008-01-01
It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schroedinger equation. This technique can generate the spectrum associated with an arbitrary potential V(r) starting from the analytically known spectrum of a particular potential P(r). In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of P(r). The method is extended in order to find accurate analytical energy formulae for radial potentials of the form aP(r) + V(r), and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed
Extensions of the auxiliary field method to solve Schroedinger equations
Energy Technology Data Exchange (ETDEWEB)
Silvestre-Brac, Bernard [LPSC Universite Joseph Fourier, Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble, Avenue des Martyrs 53, F-38026 Grenoble-Cedex (France); Semay, Claude; Buisseret, Fabien [Groupe de Physique Nucleaire Theorique, Universite de Mons-Hainaut, Academie universitaire Wallonie-Bruxelles, Place du Parc 20, B-7000 Mons (Belgium)], E-mail: silvestre@lpsc.in2p3.fr, E-mail: claude.semay@umh.ac.be, E-mail: fabien.buisseret@umh.ac.be
2008-10-24
It has recently been shown that the auxiliary field method is an interesting tool to compute approximate analytical solutions of the Schroedinger equation. This technique can generate the spectrum associated with an arbitrary potential V(r) starting from the analytically known spectrum of a particular potential P(r). In the present work, general important properties of the auxiliary field method are proved, such as scaling laws and independence of the results on the choice of P(r). The method is extended in order to find accurate analytical energy formulae for radial potentials of the form aP(r) + V(r), and several explicit examples are studied. Connections existing between the perturbation theory and the auxiliary field method are also discussed.
A global numerical solution of the radial Schroedinger equation by second-order perturbation theory
International Nuclear Information System (INIS)
Adam, G.
1979-01-01
A global numerical method, which uses second-order perturbation theory, is described for the solution of the radial Schroedinger equation. The perturbative numerical (PN) solution is derived in two stages: first, the original potential is approximated by a piecewise continuous parabolic function, and second, the resulting Schroedinger equation is solved on each integration step by second-order perturbation theory, starting with a step function reference approximation for the parabolic potential. We get a manageable PN algorithm, which shows an order of accuracy equal to six in the solution of the original Schroedinger equation, and is very stable against round off errors. (author)
Study of nonlinear waves described by the cubic Schroedinger equation
International Nuclear Information System (INIS)
Walstead, A.E.
1980-01-01
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables
Study of nonlinear waves described by the cubic Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Walstead, A.E.
1980-03-12
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables.
Center manifold for nonintegrable nonlinear Schroedinger equations on the line
International Nuclear Information System (INIS)
Weder, R.
2000-01-01
In this paper we study the following nonlinear Schroedinger equation on the line, where f is real-valued, and it satisfies suitable conditions on regularity, on growth as a function of u and on decay as x → ± ∞. The generic potential, V, is real-valued and it is chosen so that the spectrum of H:= -d 2 /dx 2 +V consists of one simple negative eigenvalue and absolutely-continuous spectrum filling (0,∞). The solutions to this equation have, in general, a localized and a dispersive component. The nonlinear bound states, that bifurcate from the zero solution at the energy of the eigenvalue of H, define an invariant center manifold that consists of the orbits of time-periodic localized solutions. We prove that all small solutions approach a particular periodic orbit in the center manifold as t→ ± ∞. In general, the periodic orbits are different for t→ ± ∞. Our result implies also that the nonlinear bound states are asymptotically stable, in the sense that each solution with initial data near a nonlinear bound state is asymptotic as t→ ± ∞ to the periodic orbits of nearby nonlinear bound states that are, in general, different for t→ ± ∞. (orig.)
International Nuclear Information System (INIS)
Steudel, H.
1980-01-01
It is shown that the two-parameter manifold of Baecklund transformations known for the nonlinear Schroedinger equation can be generated from one Baecklund transformation with specified parameters by use of scale transformation and Galilean transformation. (orig.)
On symmetry reduction and exact solutions of the linear one-dimensional Schroedinger equation
International Nuclear Information System (INIS)
Barannik, L.L.
1996-01-01
Symmetry reduction of the Schroedinger equation with potential is carried out on subalgebras of the Lie algebra which is the direct sum of the special Galilei algebra and one-dimensional algebra. Some new exact solutions are obtained
Wave-packet revival for the Schroedinger equation with position-dependent mass
International Nuclear Information System (INIS)
Schmidt, Alexandre G.M.
2006-01-01
We study the temporal evolution of solutions of 1D Schroedinger equation with position-dependent mass inside an infinite well. Revival of wave-packet is shown to exist and partial revivals are different from the usual ones
International Nuclear Information System (INIS)
Killingbeck, J.
1979-01-01
By using the methods of perturbation theory it is possible to construct simple formulae for the numerical integration of the Schroedinger equation, and also to calculate expectation values solely by means of simple eigenvalue calculations. (Auth.)
Combined solitary-wave solution for coupled higher-order nonlinear Schroedinger equations
International Nuclear Information System (INIS)
Tian Jinping; Tian Huiping; Li Zhonghao; Zhou Guosheng
2004-01-01
Coupled nonlinear Schroedinger equations model several interesting physical phenomena. We used a trigonometric function transform method based on a homogeneous balance to solve the coupled higher-order nonlinear Schroedinger equations. We obtained four pairs of exact solitary-wave solutions including a dark and a bright-soliton pair, a bright- and a dark-soliton pair, a bright- and a bright-soliton pair, and the last pair, a combined bright-dark-soliton pair
The phase space of the focused cubic Schroedinger equation: A numerical study
Energy Technology Data Exchange (ETDEWEB)
Burlakov, Yuri O. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
1998-05-01
In a paper of 1988 [41] on statistical mechanics of the nonlinear Schroedinger equation, it was observed that a Gibbs canonical ensemble associated with the nonlinear Schroedinger equation exhibits behavior reminiscent of a phase transition in classical statistical mechanics. The existence of a phase transition in the canonical ensemble of the nonlinear Schroedinger equation would be very interesting and would have important implications for the role of this equation in modeling physical phenomena; it would also have an important bearing on the theory of weak solutions of nonlinear wave equations. The cubic Schroedinger equation, as will be shown later, is equivalent to the self-induction approximation for vortices, which is a widely used equation of motion for a thin vortex filament in classical and superfluid mechanics. The existence of a phase transition in such a system would be very interesting and actually very surprising for the following reasons: in classical fluid mechanics it is believed that the turbulent regime is dominated by strong vortex stretching, while the vortex system described by the cubic Schroedinger equation does not allow for stretching. In superfluid mechanics the self-induction approximation and its modifications have been used to describe the motion of thin superfluid vortices, which exhibit a phase transition; however, more recently some authors concluded that these equations do not adequately describe superfluid turbulence, and the absence of a phase transition in the cubic Schroedinger equation would strengthen their argument. The self-induction approximation for vortices takes into account only very localized interactions, and the existence of a phase transition in such a simplified system would be very unexpected. In this thesis the authors present a numerical study of the phase transition type phenomena observed in [41]; in particular, they find that these phenomena are strongly related to the splitting of the phase space into
On the equivalence between particular types of Navier-Stokes and non-linear Schroedinger equations
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Dietrich, K.; Vautherin, D.
1985-01-01
We derive a Schroedinger equation equivalent to the Navier-Stokes equation in the special case of constant kinematic viscosities. This equation contains a non-linear term similar to that proposed by Kostin for a quantum description of friction [fr
The exact solutions of the Schroedinger equation with the Morse potential via Laplace transforms
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Chen Gang
2004-01-01
In this Letter, we reduce the second-order differential equation about the one-dimensional Schroedinger equation with the Morse potential reduced to the first-order differential equation in terms of Laplace transforms and then obtain the exact bound state solutions
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Khrennikov, A.Yu.
2005-01-01
One derived the general evolutionary differential equation within the Hilbert space describing dynamics of the wave function. The derived contextual model is more comprehensive in contrast to a quantum one. The contextual equation may be a nonlinear one. Paper presents the conditions ensuring linearity of the evolution and derivation of the Schroedinger equation [ru
On reduction and exact solutions of nonlinear many-dimensional Schroedinger equations
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Barannik, A.F.; Marchenko, V.A.; Fushchich, V.I.
1991-01-01
With the help of the canonical decomposition of an arbitrary subalgebra of the orthogonal algebra AO(n) the rank n and n-1 maximal subalgebras of the extended isochronous Galileo algebra, the rank n maximal subalgebras of the generalized extended classical Galileo algebra AG(a,n) the extended special Galileo algebra AG(2,n) and the extended whole Galileo algebra AG(3,n) are described. By using the rank n subalgebras, ansatze reducing the many dimensional Schroedinger equations to ordinary differential equations is found. With the help of the reduced equation solutions exact solutions of the Schroedinger equation are considered
Solving the Schroedinger equation using the finite difference time domain method
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Sudiarta, I Wayan; Geldart, D J Wallace
2007-01-01
In this paper, we solve the Schroedinger equation using the finite difference time domain (FDTD) method to determine energies and eigenfunctions. In order to apply the FDTD method, the Schroedinger equation is first transformed into a diffusion equation by the imaginary time transformation. The resulting time-domain diffusion equation is then solved numerically by the FDTD method. The theory and an algorithm are provided for the procedure. Numerical results are given for illustrative examples in one, two and three dimensions. It is shown that the FDTD method accurately determines eigenfunctions and energies of these systems
Numerical Clifford Analysis for the Non-stationary Schroedinger Equation
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Faustino, N.; Vieira, N.
2007-01-01
We construct a discrete fundamental solution for the parabolic Dirac operator which factorizes the non-stationary Schroedinger operator. With such fundamental solution we construct a discrete counterpart for the Teodorescu and Cauchy-Bitsadze operators and the Bergman projectors. We finalize this paper with convergence results regarding the operators and a concrete numerical example
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Senthilvelan, M; Torrisi, M; Valenti, A
2006-01-01
By using Lie's invariance infinitesimal criterion, we obtain the continuous equivalence transformations of a class of nonlinear Schroedinger equations with variable coefficients. We construct the differential invariants of order 1 starting from a special equivalence subalgebra E χ o . We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schr?dinger equations which can be mapped, by means of an equivalence transformation of E χ o , to the well-known cubic Schroedinger equation. We also provide the explicit form of the transformation
Exact solitary and periodic wave solutions for a generalized nonlinear Schroedinger equation
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Sun Chengfeng; Gao Hongjun
2009-01-01
The generalized nonlinear Schroedinger equation (GNLS) iu t + u xx + β | u | 2 u + γ | u | 4 u + iα (| u | 2 u) x + iτ(| u | 2 ) x u = 0 is studied. Using the bifurcation of travelling waves of this equation, some exact solitary wave solutions were obtained in [Wang W, Sun J,Chen G, Bifurcation, Exact solutions and nonsmooth behavior of solitary waves in the generalized nonlinear Schroedinger equation. Int J Bifucat Chaos 2005:3295-305.]. In this paper, more explicit exact solitary wave solutions and some new smooth periodic wave solutions are obtained.
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Hoffmann, D.
1984-01-01
Erwin Schroedinger (1887-1961) belongs without doubt to the most outstanding physicists of our century. His name is inseparably connected with the development of quantum theory with the formulation of his famous wave equation being his greatest achievement. This relation became generally known as the Schroedinger equation and its understanding was fundamental to the progress of modern quantum theory. In 1933 Schroedinger's work was honoured by the award of the Nobel Prize in physics. In the booklet Schroedinger's life, work and philosophical views are outlined against the social and physico-historical background of his time
Exponential and Bessel fitting methods for the numerical solution of the Schroedinger equation
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Raptis, A.D.; Cash, J.R.
1987-01-01
A new method is developed for the numerical integration of the one dimensional radial Schroedinger equation. This method involves using different integration formulae in different parts of the range of integration rather than using the same integration formula throughout. Two new integration formulae are derived, one which integrates Bessel and Neumann functions exactly and another which exactly integrates certain exponential functions. It is shown that, for large r, these new formulae are much more accurate than standard integration methods for the Schroedinger equation. The benefit of using this new approach is demonstrated by considering some numerical examples based on the Lennard-Jones potential. (orig.)
Non self-similar collapses described by the non-linear Schroedinger equation
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Berge, L.; Pesme, D.
1992-01-01
We develop a rapid method in order to find the contraction rates of the radially symmetric collapsing solutions of the nonlinear Schroedinger equation defined for space dimensions exceeding a threshold value. We explicitly determine the asymptotic behaviour of these latter solutions by solving the non stationary linear problem relative to the nonlinear Schroedinger equation. We show that the self-similar states associated with the collapsing solutions are characterized by a spatial extent which is bounded from the top by a cut-off radius
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Ding Zhonghai; Chen, Goong; Lin, Chang-Shou
2010-01-01
The dimensional scaling (D-scaling) technique is an innovative asymptotic expansion approach to study the multiparticle systems in molecular quantum mechanics. It enables the calculation of ground and excited state energies of quantum systems without having to solve the Schroedinger equation. In this paper, we present a mathematical analysis of the D-scaling technique for the Schroedinger equation with power-law potentials. By casting the D-scaling technique in an appropriate variational setting and studying the corresponding minimization problem, the D-scaling technique is justified rigorously. A new asymptotic dimensional expansion scheme is introduced to compute asymptotic expansions for ground state energies.
Three-Step Predictor-Corrector of Exponential Fitting Method for Nonlinear Schroedinger Equations
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Tang Chen; Zhang Fang; Yan Haiqing; Luo Tao; Chen Zhanqing
2005-01-01
We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three-step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.
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Silvestre-Brac, Bernard [LPSC Universite Joseph Fourier, Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble, Avenue des Martyrs 53, F-38026 Grenoble-Cedex (France); Semay, Claude; Buisseret, Fabien [Groupe de Physique Nucleaire Theorique, Universite de Mons-Hainaut, Academie universitaire Wallonie-Bruxelles, Place du Parc 20, B-7000 Mons (Belgium)], E-mail: silvestre@lpsc.in2p3.fr, E-mail: claude.semay@umh.ac.be, E-mail: fabien.buisseret@umh.ac.be
2009-06-19
The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies of the Schroedinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schroedinger equation with exponential potentials of the form -{alpha}r{sup {lambda}}exp(-{beta}r) can also be analytically solved by using the auxiliary field method. Closed formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn to the Yukawa potential and the pure exponential potential.
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Silvestre-Brac, Bernard; Semay, Claude; Buisseret, Fabien
2009-01-01
The auxiliary field method is a new and efficient way to compute approximate analytical eigenenergies of the Schroedinger equation. This method has already been successfully applied to the case of central potentials of power-law and logarithmic forms. In the present work, we show that the Schroedinger equation with exponential potentials of the form -αr λ exp(-βr) can also be analytically solved by using the auxiliary field method. Closed formulae giving the critical heights and the energy levels of these potentials are presented. Special attention is drawn to the Yukawa potential and the pure exponential potential
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Chithiika Ruby, V.; Senthilvelan, M.
2010-01-01
In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schroedinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schroedinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new.
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Amirkhanov, I.V.; Zhidkov, E.P.; Konnova, S.V.
2000-01-01
For the case of spherical-symmetrical potential we have considered the convergence of the solution of singular-perturbated Schroedinger equation of the 4th order to the solution of the corresponding standard nonrelativistic Schroedinger equation by numerical and analytical methods. The questions of existence of the solutions are explored. Numerical results are given. (author)
Fendzi-Donfack, Emmanuel; Nguenang, Jean Pierre; Nana, Laurent
2018-02-01
We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. We determine the solutions of fractional nonlinear electrical transmission lines (NETL) and the perturbed nonlinear Schroedinger (NLS) equation with the Kerr law nonlinearity term. The solutions are obtained for the parameters in the range (0<α≤1) of the derivative operator and we found the traditional solutions for the limiting case of α =1. We show that according to the modified Riemann-Liouville derivative, the solutions found can describe physical systems with memory effect, transient effects in electrical systems and nonlinear transmission lines, and other systems such as optical fiber.
Travelling solitons in the damped driven nonlinear Schroedinger equation
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Barashenkov, I.V.; Zemlyanaya, E.V.
2003-01-01
The well known effect of the linear damping on the moving nonlinear Schroedinger soliton (even when there is energy supply via the spatially homogeneous driving) is to quench its momentum to zero. Surprisingly, the zero momentum does not necessarily mean zero velocity. We show that two or more parametrically driven damped solitons can form a complex travelling with zero momentum at a nonzero constant speed. All travelling complexes we have found so far, turned out to be unstable. Thus, the parametric driving is capable of sustaining the uniform motion of damped solitons, but some additional agent is required to make this motion stable
Discrete coupled derivative nonlinear Schroedinger equations and their quasi-periodic solutions
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Geng Xianguo; Su Ting
2007-01-01
A hierarchy of nonlinear differential-difference equations associated with a discrete isospectral problem is proposed, in which a typical differential-difference equation is a discrete coupled derivative nonlinear Schroedinger equation. With the help of the nonlinearization of the Lax pairs, the hierarchy of nonlinear differential-difference equations is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems. Based on the theory of algebraic curve, the Abel-Jacobi coordinates are introduced to straighten out the corresponding flows, from which quasi-periodic solutions for these differential-difference equations are obtained resorting to the Riemann-theta functions. Moreover, a (2+1)-dimensional discrete coupled derivative nonlinear Schroedinger equation is proposed and its quasi-periodic solutions are derived
Intertwining relations and Darboux transformations for Schroedinger equations in (n+1) dimensions
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Schulze-Halberg, Axel
2010-01-01
We evaluate the intertwining relation for Schroedinger equations in (n+1) dimensions. The conditions for the existence of a Darboux transformation are analyzed and compared to their (1+1) dimensional counterparts. A complete solution of the conditions is given for (2+1) dimensions, and a Darboux transformation is constructed.
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Belmonte-Beitia, Juan; Calvo, Gabriel F.
2009-01-01
In this Letter, by means of similarity transformations, we construct explicit solutions to the quintic nonlinear Schroedinger equation with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general approach and use it to study some examples and find nontrivial explicit solutions such as periodic (breathers), quasiperiodic and bright and dark soliton solutions
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Palacios, Sergio L.
2004-01-01
We propose two simple ansaetze that allow us to obtain different analytical solutions of the high dispersive cubic and cubic-quintic nonlinear Schroedinger equations. Among these solutions we can find solitary wave and periodic wave solutions representing the propagation of different waveforms in nonlinear media
New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schroedinger Equation
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Yang Qin; Dai Chaoqing; Zhang Jiefang
2005-01-01
Some new exact travelling wave and period solutions of discrete nonlinear Schroedinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differential-different models.
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Rezende, J.
1983-01-01
We give a simple proof of Feynman's formula for the Green's function of the n-dimensional harmonic oscillator valid for every time t with Im t<=0. As a consequence the Schroedinger equation for the anharmonic oscillator is integrated and expressed by the Feynman path integral on Hilbert space. (orig.)
Interrelation of alternative sets of Lax-pairs for a generalized nonlinear Schroedinger equation
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Iino, Kazuhiro; Ichikawa, Yoshihiko; Wadati, Miki.
1982-05-01
Examination of the inverse scattering transformation schemes for a generalized nonlinear Schroedinger equation reveals the fact that the algorithm of Chen-Lee-Liu gives rise to the Lax-pairs for the squared eigenfunctions of the Wadati-Konno-Ichikawa scheme, which has been formulated as superposition of the Ablowitz-Kaup-Newell-Segur scheme and the Kaup-Newell scheme. (author)
Filamentary structures of the cosmic web and the nonlinear Schroedinger type equation
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Tigrak, E; Weygaert, R van de; Jones, B J T
2011-01-01
We show that the filamentary type structures of the cosmic web can be modeled as solitonic waves by solving the reaction diffusion system which is the hydrodynamical analogous of the nonlinear Schroedinger type equation. We find the analytical solution of this system by applying the Hirota direct method which produces the dissipative soliton solutions to formulate the dynamical evolution of the nonlinear structure formation.
Continuous-time random walk as a guide to fractional Schroedinger equation
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Lenzi, E. K.; Ribeiro, H. V.; Mukai, H.; Mendes, R. S.
2010-01-01
We argue that the continuous-time random walk approach may be a useful guide to extend the Schroedinger equation in order to incorporate nonlocal effects, avoiding the inconsistencies raised by Jeng et al. [J. Math. Phys. 51, 062102 (2010)]. As an application, we work out a free particle in a half space, obtaining the time dependent solution by considering an arbitrary initial condition.
Fractional integration, Morrey spaces and a Schroedinger equation
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Olsen, P.A.
1995-01-01
Let V : R 3 → R be the potential for the 3-dimensional Schroedinger operator -Δ + V. It was shown by Cwikel, Lieb and Rosenblum, [8], that the number of bound states, N(V), of -Δ + V is bounded by N(V) ≤ C ∫ R3 |V(x)| 3/2 dx. Later Fefferman and phong, [4], improved on this inequality. Make a dyadic decomposition of R 3 into cubes. Define a dyadic cube Q to be minimal with respect to ε > 0 if ∫ q |V(x)| p dx ≥ ε p |Q| 1-2p/3 and ∫ Q ' |V(x)| p dx p |Q'| 1-2p/3 for all dyadic cubes Q' contained-in Q. 10 refs., 4 figs., 1 tab
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Yang Xiao; Du Dianlou
2010-01-01
The Poisson structure on C N xR N is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schroedinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.
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Sen, S.; Roy Chowdhury, A.
1989-06-01
The nonlinear Alfven waves are governed by the Vector Derivative nonlinear Schroedinger (VDNLS) equation, which for parallel or quasi parallel propagation reduces to the Derivative Nonlinear Schroedinger (DNLS) equation for the circularly polarized waves. We have formulated the Quantum Inverse problem for a new type of Nonlinear Schroedinger Equation which has many properties similar to the usual NLS problem but the structure of classical and quantum R matrix are distinctly different. The commutation rules of the scattering data are obtained and the Algebraic Bethe Ansatz is formulated to derive the eigenvalue equation for the energy of the excited states. 10 refs
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Dobrev, V. K.; Stoimenov, S.
2010-01-01
The singular vectors in Verma modules over the Schroedinger algebra s(n) in (n + 1)-dimensional space-time are found for the case of general representations. Using the singular vectors, hierarchies of equations invariant under Schroedinger algebras are constructed.
On exact solitary wave solutions of the nonlinear Schroedinger equation with a source
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Raju, T Solomon; Kumar, C Nagaraja; Panigrahi, Prasanta K
2005-01-01
We use a fractional transformation to connect the travelling wave solutions of the nonlinear Schroedinger equation (NLSE), phase locked with a source, to the elliptic equations satisfying, f-Prime ± af ± λf 3 = 0. The solutions are necessarily of rational form, containing both trigonometric and hyperbolic types as special cases. Bright and dark solitons, as well as singular solitons, are obtained in a suitable range of parameter values. (letter to the editor)
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Todorov, N S [Low Temperature Department of the Institute of Solid State Physics of the Bulgarian Academy of Sciences, Sofia
1981-04-01
It is shown that the nonstationary Schroedinger equation does not satisfy a well-known adiabatical principle in thermodynamics. A ''renormalization procedure'' based on the possible existence of a time-irreversible basic evolution equation is proposed with the help of which one comes to agreement in a variety of specific cases of an adiabatic inclusion of a perturbing potential. The ideology of the present article rests essentially on the ideology of the preceding articles, in particular article I.
Energy Technology Data Exchange (ETDEWEB)
Todorov, N S
1981-04-01
It is shown that the nonstationary Schroedinger equation does not satisfy a well-known adiabatical principle in thermodynamics. A ''renormalization procedure'' based on the possible existence of a time-irreversible basic evolution equation is proposed with the help of which one comes to agreement in a variety of specific cases of an adiabatic inclusion of a perturbing potential. The ideology of the present article IV rests essentially on the ideology of the preceding articles, in particular article I.
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Pelinovsky, D. E.; Stefanov, A.
2008-01-01
Based on the recent work [Komech et al., 'Dispersive estimates for 1D discrete Schroedinger and Klein-Gordon equations', Appl. Anal. 85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schroedinger operator, Hφ=(-Δ+V)φ=-(φ n+1 +φ n-1 -2φ n )+V n φ n . We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of H consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates parallel e itH P a.c. (H) parallel l σ 2 →l -σ 2 -3/2 for any fixed σ>(5/2) and any t>0, where P a.c. (H) denotes the spectral projection to the absolutely continuous spectrum of H. In addition, based on the scattering theory for the discrete Jost solutions and the previous results by Stefanov and Kevrekidis [''Asymptotic behaviour of small solutions for the discrete nonlinear Schroedinger and Klein-Gordon equations,'' Nonlinearity 18, 1841 (2005)], we find new dispersive estimates parallel e itH P a.c. (H) parallel l 1 →l ∞ -1/3 , which are sharp for the discrete Schroedinger operators even for V=0
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Li Biao; Chen Yong
2007-01-01
In this paper, the inhomogeneous nonlinear Schroedinger equation with the loss/gain and the frequency chirping is investigated. With the help of symbolic computation, three families of exact analytical solutions are presented by employing the extended projective Riccati equation method. From our results, many previous known results of nonlinear Schroedinger equation obtained by some authors can be recovered by means of some suitable selections of the arbitrary functions and arbitrary constants. Of optical and physical interests, soliton propagation and soliton interaction are discussed and simulated by computer, which include snake-soliton propagation and snake-solitons interaction, boomerang-like soliton propagation and boomerang-like solitons interaction, dispersion managed (DM) bright (dark) soliton propagation and DM solitons interaction
Global existence of small solutions to semilinear Schroedinger equations
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Chihara, Hiroyuki
1996-01-01
We present global existence theorem for semilinear Schrodinger equations. In general, Schrodinger-type equations do not admit the classical energy estimates. To avoid this difficulty, we use S. Doi's method for linear Schrodinger-type equations. Combining his method and L p -L q estimates, we prove the global existence of solutions with small initial data
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Morales, J.; Ovando, G.; Pena, J. J.
2010-01-01
One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis a vis the standard Schroedinger equation with constant mass [1]. However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so-called effective mass that is in general variable and dependent on position. Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position-dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi-spaced spectrum solutions, i.e. harmonic oscillator isospectral partners. To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm-Liouville type, into a Schroedinger-like standard equation is applied to the PDMSE. In that case, the former potential associated to the PDMSE and the potential involved in the Schroedinger-like standard equation are related through a Riccati-type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions-dependent mass distribution (PDMD)m(x). Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.
Embedded solitons in the third-order nonlinear Schroedinger equation
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Pal, Debabrata; Ali, Sk Golam; Talukdar, B
2008-01-01
We work with a sech trial function with space-dependent soliton parameters and envisage a variational study for the nonlinear Schoedinger (NLS) equation in the presence of third-order dispersion. We demonstrate that the variational equations for pulse evolution in this NLS equation provide a natural basis to derive a potential model which can account for the existence of a continuous family of embedded solitons supported by the third-order NLS equation. Each member of the family is parameterized by the propagation velocity and co-efficient of the third-order dispersion
The discretized Schroedinger equation and simple models for semiconductor quantum wells
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Boykin, Timothy B; Klimeck, Gerhard
2004-01-01
The discretized Schroedinger equation is one of the most commonly employed methods for solving one-dimensional quantum mechanics problems on the computer, yet many of its characteristics remain poorly understood. The differences with the continuous Schroedinger equation are generally viewed as shortcomings of the discrete model and are typically described in purely mathematical terms. This is unfortunate since the discretized equation is more productively viewed from the perspective of solid-state physics, which naturally links the discrete model to realistic semiconductor quantum wells and nanoelectronic devices. While the relationship between the discrete model and a one-dimensional tight-binding model has been known for some time, the fact that the discrete Schroedinger equation admits analytic solutions for quantum wells has gone unnoted. Here we present a solution to this new analytically solvable problem. We show that the differences between the discrete and continuous models are due to their fundamentally different bandstructures, and present evidence for our belief that the discrete model is the more physically reasonable one
Solution of Schroedinger equation for particle moving in two-well potential
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Ivanova, O.I.; Sabirov, R.Kh.
2000-01-01
The solution of the Schroedinger equation for the particle, moving in the two-well potential is given on the basis of a single variational method. This potential constitutes the sum of the harmonic potential and the Gaussian addition. The analytical expression for the wave function of the particle basic state is obtained. The dependence of the obtained solutions on the potential barrier height and width is studied. It is shown that the better separation of the potential barrier provides for higher accuracy of the calculations. The values of the two-well potential, whereby good agreement between the calculations and exact numerical solution of the Schroedinger equation may be expected, are presented [ru
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Dobrev, V.K.; Doebner, H.D.; Mrugalla, C.
1995-12-01
We give a q-deformation S-perpendicular q of the centrally extended Schroedinger algebra. We construct the lowest weight representations of S-perpendicular q , starting from the Verma modules over S-perpendicular q , finding their singular vectors and factoring the Verma submodules built on the singular vectors. We also give a vector-field realization of S-perpendicular q which provides polynomial realization of the lowest weight representations and an infinite hierarchy of q-difference equations which may be called generalized q-deformed heat equations. We also apply our methods to the on-shell q-Schroedinger algebra proposed by Floreanini and Vinet. (author). 12 refs
Form-preserving Transformations for the Time-dependent Schroedinger Equation in (n + 1) Dimensions
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Schulze-Halberg, Axel
2006-01-01
We define a form-preserving transformation (also called point canonical transformation) for the time-dependent Schroedinger equation (TDSE) in (n+1) dimensions. The form-preserving transformation is shown to be invertible and to preserve L 2 -normalizability. We give a class of time-dependent TDSEs that can be mapped onto stationary Schroedinger equations by our form-preserving transformation. As an example, we generate a solvable, time-dependent potential of Coulombic ring-shaped type together with the corresponding exact solution of the TDSE in (3+1) dimensions. We further consider TDSEs with position-dependent (effective) masses and show that there is no form-preserving transformation between them and the conventional TDSEs, if the spatial dimension of the system is higher than one
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Ponomarev, L.I.; Puzynin, I.V.; Puzynina, T.P.
1975-01-01
The paper is a part of further development of investigations in which a numerical solution method of the Schroedinger equation for the case of a discrete spectrum has been developed and applied. The suggested algorithm (CAMEN scheme) is generalized and applied to quasistationary solutions of the Schroedinger equation system. Some specific features of the CAMEN scheme realization (such as establishing boundary conditions are observed while calculating quasistationary levels of hydrogen mesic molecules. The calculations have been carried out for energies and wave functions of quasistationary states of hydrogen mesic molecules. The choice of the initial approximation, the accuracy of calculations and characteristics of the convergence of the method have been investigated. The CAMEN algorithm has been realized in the form of the FORTRAN program packet. The CAMEN scheme can be also used for solving scatering problems
Quantum Gelfand-Levitan equations for nonlinear Schroedinger model of spin-1/2 particles
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Pu, F.; Zhao, B.
1984-01-01
The quantum Gelfand-Levitan equations for the nonlinear Schroedinger model of spin-(1/2) particles are obtained. Two Izergin-Korepin relations are used in the derivation. A new type commutation relation of L operators is introduced to get the commutation relations which are needed for the study of S matrices and Green's functions. As examples, the four-point Green's functions and the two-body S matrices are given
International Nuclear Information System (INIS)
Makhan'kov, V.G.; Slavov, S.I.
1989-01-01
Vector nonlinear Schroedinger equations (VS3) is investigated under quasi-constant boundary conditions. New two-soliton solutions are obtained with such non-trivial dynamics that they may be called the breather solutions. A version of the basic Novikov-Dubrovin-Krichever algebro-geometrical approach is applied to obtain breather like solutions existing for all types of internal symmetry is specified are formulated in terms of the soliton velocity expressed via the parameters of the problem. 4 refs
Exact solutions of fractional Schroedinger-like equation with a nonlocal term
International Nuclear Information System (INIS)
Jiang Xiaoyun; Xu Mingyu; Qi Haitao
2011-01-01
We study the time-space fractional Schroedinger equation with a nonlocal potential. By the method of Fourier transform and Laplace transform, the Green function, and hence the wave function, is expressed in terms of H-functions. Graphical analysis demonstrates that the influence of both the space-fractal parameter α and the nonlocal parameter ν on the fractional quantum system is strong. Indeed, the nonlocal potential may act similar to a fractional spatial derivative as well as fractional time derivative.
International Nuclear Information System (INIS)
Carow-Watamura, U.; Schlieker, M.; Watamura, S.
1991-01-01
We construct a differential calculus on the N-dimensional non-commutative Euclidean space, i.e., the space on which the quantum group SO q (N) is acting. The differential calculus is required to be manifestly covariant under SO q (N) transformations. Using this calculus, we consider the Schroedinger equation corresponding to the harmonic oscillator in the limit of q→1. The solution of it is given by q-deformed functions. (orig.)
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Bayramoglu, Mehmet; Tasci, Fatih; Zeynalov, Djafar
2004-01-01
We study the discrete part of spectrum of a singular non-self-adjoint second-order differential equation on a semiaxis with an operator coefficient. Its boundedness is proved. The result is applied to the Schroedinger boundary value problem -Δu+q(x)u=λ 2 u, u vertical bar ∂D =0, with a complex potential q(x) in an angular domain
Regularity of the Rotation Number for the One-Dimensional Time-Continuous Schroedinger Equation
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Amor, Sana Hadj, E-mail: sana_hadjamor@yahoo.fr [Ecole Nationale d' Ingenieurs de Monastir (Tunisia)
2012-12-15
Starting from results already obtained for quasi-periodic co-cycles in SL(2, R), we show that the rotation number of the one-dimensional time-continuous Schroedinger equation with Diophantine frequencies and a small analytic potential has the behavior of a 1/2-Hoelder function. We give also a sub-exponential estimate of the length of the gaps which depends on its label given by the gap-labeling theorem.
Quasiseparation of variables in the Schroedinger equation with a magnetic field
International Nuclear Information System (INIS)
Charest, F.; Hudon, C.; Winternitz, P.
2007-01-01
We consider a two-dimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the separation of variables in the Schroedinger equation. We introduce the concept of 'quasiseparation of variables' and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra
International Nuclear Information System (INIS)
Yasuk, F.; Tekin, S.; Boztosun, I.
2010-01-01
In this study, the exact solutions of the d-dimensional Schroedinger equation with a position-dependent mass m(r)=1/(1+ζ 2 r 2 ) is presented for a free particle, V(r)=0, by using the method of point canonical transformations. The energy eigenvalues and corresponding wavefunctions for the effective potential which is to be a generalized Poeschl-Teller potential are obtained within the framework of the asymptotic iteration method.
Oscillatory integrals on Hilbert spaces and Schroedinger equation with magnetic fields
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Albeverio, S.; Brzezniak, Z.
1994-01-01
We extend the theory of oscillatory integrals on Hilbert spaces (the mathematical version of ''Feynman path integrals'') to cover more general integrable functions, preserving the property of the integrals to have converging finite dimensional approximations. We give an application to the representation of solutions of the time dependent Schroedinger equation with a scalar and a magnetic potential by oscillatory integrals on Hilbert spaces. A relation with Ramer's functional in the corresponding probabilistic setting is found. (orig.)
A method of solving simple harmonic oscillator Schroedinger equation
Maury, Juan Carlos F.
1995-01-01
A usual step in solving totally Schrodinger equation is to try first the case when dimensionless position independent variable w is large. In this case the Harmonic Oscillator equation takes the form (d(exp 2)/dw(exp 2) - w(exp 2))F = 0, and following W.K.B. method, it gives the intermediate corresponding solution F = exp(-w(exp 2)/2), which actually satisfies exactly another equation, (d(exp 2)/dw(exp 2) + 1 - w(exp 2))F = 0. We apply a different method, useful in anharmonic oscillator equations, similar to that of Rampal and Datta, and although it is slightly more complicated however it is also more general and systematic.
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Ding Qing
2007-01-01
We prove that the integrable-nonintegrable discrete nonlinear Schroedinger equation (AL-DNLS) introduced by Cai, Bishop and Gronbech-Jensen (Phys. Rev. Lett. 72 591(1994)) is the discrete gauge equivalent to an integrable-nonintegrable discrete Heisenberg model from the geometric point of view. Then we study whether the transmission and bifurcation properties of the AL-DNLS equation are preserved under the action of discrete gauge transformations. Our results reveal that the transmission property of the AL-DNLS equation is completely preserved and the bifurcation property is conditionally preserved to those of the integrable-nonintegrable discrete Heisenberg model
A Greenian approach to the solution of the Schroedinger equation for periodic lattice potentials
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Minelli, T.A.
1976-01-01
A modified structural Green's function (MSGF), exploiting all the information contained in the previously solved Schroedinger equation for the electron interacting with a single lattice site, has been introduced and used in order to obtain, from a Dyson-type equation, a kernel whose poles and residues give the E-vs.-k relation and, respectively, the Bloch functions. Such a formulation suggests an alternative technique for the approximate solution of the KKR equations. The MSGF formalism has been also used in order to determine the structure constants of a one-dimensional lattice in a general representation
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El-Jaick, Lea Jaccoud; Figueiredo, Bartolomeu D.B.
2009-01-01
We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schroedinger equation with an inverted quasi-exactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole. For the first problem we find finite and infinite-series solutions which are convergent and bounded for any value of the independent variable. For the angular equation, we also find expansions in series of Jacobi polynomials. (author)
The Schroedinger-Newton equation as model of self-gravitating quantum systems
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Grossardt, Andre
2013-01-01
The Schroedinger-Newton equation (SN equation) describes a quantummechanical one-particle-system with gravitational self-interaction and might play a role answering the question if gravity must be quantised. As non-relativistic limit of semi-classical gravity, it provides testable predictions of the effects that classical gravity has on genuinely quantum mechanical systems in the mass regime between a few thousand proton masses and the Planck mass, which is experimentally unexplored. In this thesis I subsume the mathematical properties of the SN equation and justify it as a physical model. I will give a short outline of the controversial debate around semi-classical gravity as a fundamental theory, along with the idea of the SN equation as a model of quantum state reduction. Subsequently, I will respond to frequent objections against nonlinear Schrodinger equations. I will show how the SN equation can be obtained from Einstein's General Relativity coupled to either a KleinGordon or a Dirac equation, in the same sense as the linear Schroedinger equation can be derived in flat Minkowski space-time. The equation is, to this effect, a non-relativistic approximation of the semi-classical Einstein equations. Additionally, I will discuss, first by means of analytic estimations and later numerically, in which parameter range effects of gravitational selfinteraction - e.g. in molecular-interferometry experiments - should be expected. Besides the one-particle SN equation I will provide justification for a modified equation describing the centre-of-mass wave-function of a many-particle system. Furthermore, for this modified equation, I will examine, numerically, the consequences for experiments. Although one arrives at the conclusion that no effects of the SN equation can be expected for masses up to six or seven orders of magnitude above those considered in contemporary molecular interferometry experiments, tests of the equation, for example in satellite experiments, seem
KAM for the non-linear Schroedinger equation
Eliasson, L H
2006-01-01
We consider the $d$-dimensional nonlinear Schr\\"o\\-dinger equation under periodic boundary conditions:-i\\dot u=\\Delta u+V(x)*u+\\ep|u|^2u;\\quad u=u(t,x),\\;x\\in\\T^dwhere $V(x)=\\sum \\hat V(a)e^{i\\sc{a,x}}$ is an analytic function with $\\hat V$ real. (This equation is a popular model for the `real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\\ep=0$ the equation is linear and has time--quasi-periodic solutions $u$,u(t,x)=\\sum_{s\\in \\AA}\\hat u_0(a)e^{i(|a|^2+\\hat V(a))t}e^{i\\sc{a,x}}, \\quad 0<|\\hat u_0(a)|\\le1,where $\\AA$ is any finite subset of $\\Z^d$. We shall treat $\\omega_a=|a|^2+\\hat V(a)$, $a\\in\\AA$, as free parameters in some domain $U\\subset\\R^{\\AA}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, in particular, will have the following consequence: \\smallskip {\\it If $|\\ep|$ is sufficiently small, then there is a large subset $U'$ of $U$ such that for all $...
Structure and properties of Hughston's stochastic extension of the Schroedinger equation
International Nuclear Information System (INIS)
Adler, Stephen L.; Horwitz, Lawrence P.
2000-01-01
Hughston has recently proposed a stochastic extension of the Schroedinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics. (c) 2000 American Institute of Physics
Quasi-classical derivation of the Dirac and one-particle Schroedinger equations
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Wignall, J.W.G.
1990-08-01
The quasi-classical approach, in which particles are regarded as extended periodic excitations of a classical nonlinear field, is for the first time applied quantitatively in the quantum domain. It is shown that the twofold intrinsic 'spin' degree of freedom possessed by an electron can be interpreted in a purely classical way, and that the Lorentz covariant incorporation of this degree of freedom requires that the spacetime evolution of an electron excitation in a prescribed external field be given by the Dirac equation and hence, in the nonrelativistic limit, by the Pauli or Schroedinger one-particle equations. 17 refs
Dynamical theory of neutron diffraction. [One-body Schroedinger equation, review
Energy Technology Data Exchange (ETDEWEB)
Sears, V F [Atomic Energy of Canada Ltd., Chalk River, Ontario. Chalk River Nuclear Labs.
1978-10-01
We present a review of the dynamical theory of neutron diffraction by macroscopic bodies which provides the theoretical basis for the study of neutron optics. We consider both the theory of dispersion, in which it is shown that the coherent wave in the medium satisfies a macroscopic one-body Schroedinger equation, and the theory of reflection, refraction, and diffraction in which the above equation is solved for a number of special cases of interest. The theory is illustrated with the help of experimental results obtained over the past 10 years by a number of new techniques such as neutron gravity refractometry. Pendelloesung interference, and neutron interferometry.
Path space measures for Dirac and Schroedinger equations: Nonstandard analytical approach
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Nakamura, T.
1997-01-01
A nonstandard path space *-measure is constructed to justify the path integral formula for the Dirac equation in two-dimensional space endash time. A standard measure as well as a standard path integral is obtained from it. We also show that, even for the Schroedinger equation, for which there is no standard measure appropriate for a path integral, there exists a nonstandard measure to define a *-path integral whose standard part agrees with the ordinary path integral as defined by a limit from time-slice approximant. copyright 1997 American Institute of Physics
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Yan, Z.; Zhang, H.
2001-01-01
In this paper, an isospectral problem and one associated with a new hierarchy of nonlinear evolution equations are presented. As a reduction, a representative system of new generalized derivative nonlinear Schroedinger equations in the hierarchy is given. It is shown that the hierarchy possesses bi-Hamiltonian structures by using the trace identity method and is Liouville integrable. The spectral problem is non linearized as a finite-dimensional completely integrable Hamiltonian system under a constraint between the potentials and spectral functions. Finally, the involutive solutions of the hierarchy of equations are obtained. In particular, the involutive solutions of the system of new generalized derivative nonlinear Schroedinger equations are developed
Energy Technology Data Exchange (ETDEWEB)
Kravchenko, Vladislav V [Departmento de Telecomunicaciones, SEPI, Escuela Superior de IngenierIa Mecanica y Electrica, Instituto Politecnico Nacional, CP 07738 Mexico DF (Mexico)
2005-01-28
Given a particular solution of a one-dimensional stationary Schroedinger equation this equation of second order can be reduced to a first-order linear ordinary differential equation. This is done with the aid of an auxiliary Riccati differential equation. In the present work we show that the same fact is true in a multidimensional situation also. For simplicity we consider the case of two or three independent variables. One particular solution of the stationary Schroedinger equation allows us to reduce this second-order equation to a linear first-order quaternionic differential equation. As in the one-dimensional case this is done with the aid of an auxiliary quaternionic Riccati equation. The resulting first-order quaternionic equation is equivalent to the static Maxwell system and is closely related to the Dirac equation. In the case of two independent variables it is the well-known Vekua equation from theory of pseudoanalytic (or generalized analytic) functions. Nevertheless, we show that even in this case it is very useful to consider not only complex valued functions, solutions of the Vekua equation, but complete quaternionic functions. In this way the first-order quaternionic equation represents two separate Vekua equations, one of which gives us solutions of the Schroedinger equation and the other one can be considered as an auxiliary equation of a simpler structure. Moreover for the auxiliary equation we always have the corresponding Bers generating pair (F, G), the base of the Bers theory of pseudoanalytic functions, and what is very important, the Bers derivatives of solutions of the auxiliary equation give us solutions of the main Vekua equation and as a consequence of the Schroedinger equation. Based on this fact we obtain an analogue of the Cauchy integral theorem for solutions of the stationary Schroedinger equation. Other results from theory of pseudoanalytic functions can be written for solutions of the Schroedinger equation. Moreover, for an ample
Genus two finite gap solutions to the vector nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Woodcock, Thomas; Warren, Oliver H; Elgin, John N
2007-01-01
A recently published article presents a technique used to derive explicit formulae for odd genus solutions to the vector nonlinear Schroedinger equation. In another article solutions of genus two are derived using a different approach which assumes a separable ansatz. In this communication, the extension of the first technique to the even genus case is discussed, and this extension is carried out explicitly for genus two. Furthermore, a birational mapping is found between the spectral curves that arise in the two approaches. (fast track communication)
Chirped self-similar solutions of a generalized nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Fei Jin-Xi [Lishui Univ., Zhejiang (China). College of Mathematics and Physics; Zheng Chun-Long [Shaoguan Univ., Guangdong (China). School of Physics and Electromechanical Engineering; Shanghai Univ. (China). Shanghai Inst. of Applied Mathematics and Mechanics
2011-01-15
An improved homogeneous balance principle and an F-expansion technique are used to construct exact chirped self-similar solutions to the generalized nonlinear Schroedinger equation with distributed dispersion, nonlinearity, and gain coefficients. Such solutions exist under certain conditions and impose constraints on the functions describing dispersion, nonlinearity, and distributed gain function. The results show that the chirp function is related only to the dispersion coefficient, however, it affects all of the system parameters, which influence the form of the wave amplitude. As few characteristic examples and some simple chirped self-similar waves are presented. (orig.)
Minimal gravitational coupling in the Newtonian theory and the covariant Schroedinger equation
International Nuclear Information System (INIS)
Duval, C.; Kuenzle, H.P.
1983-02-01
The role of the Bargmann group (11-dimensional extended Galilei group) in non relativistic gravitation theory is investigated. The generalized Newtonian gravitation theory (Newton-Cartan theory) achieves the status of a gauge theory about as much as General Relativity and couples minimally to a complex scalar field leading to a fourdimensionally covariant Schroedinger equation. Matter current and stress-energy tensor follow correctly from the Lagrangian. This theory on curved Newtonian space-time is also shown to be a limit of the Einstein-Klein-Gordon theory
Quantum scattering via the discretisation of Schroedinger's equation
Energy Technology Data Exchange (ETDEWEB)
Alexopoulos, A. [Electronic Warfare and Radar Division, Defence Science and Technology Organisation (DSTO), PO Box 1500, Edinburgh 5111 (Australia)]. E-mail: aris.alexopoulos@dsto.defence.gov.au
2007-03-19
We obtain the scattering matrix for particles that encounter a quantum potential by discretising Schroedinger's time independent differential equation without the need to resort to the manipulation of the eigenfunctions directly. The singularities that arise in some solutions by other methods are handled with ease including the effects of resonances while convergence is excellent in all limits with only a small number of orders required to give accurate results. Our method compares the tunnelling probability with that of the WKB theory, exact numerical solutions and the modified Airy function method.
Arbitrary l-wave solutions of the Schroedinger equation for the screen Coulomb potential
International Nuclear Information System (INIS)
Dong, Shishan; Sun, Guohua; Dong, Shihai
2013-01-01
Using improved approximate schemes for centrifugal term and the singular factor 1/r appearing in potential itself, we solve the Schroedinger equation with the screen Coulomb potential for arbitrary angular momentum state l. The bound state energy levels are obtained. A closed form of normalization constant of the wave functions is also found. The numerical results show that our results are in good agreement with those obtained by other methods. The key issue is how to treat two singular points in this quantum system. (author)
Solution of Schroedinger Equation for Two-Dimensional Complex Quartic Potentials
International Nuclear Information System (INIS)
Singh, Ram Mehar; Chand, Fakir; Mishra, S. C.
2009-01-01
We investigate the quasi-exact solutions of the Schroedinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x 1 + ip 3 , y = x 2 + ip 4 , p x = p 1 + ix 3 , p y = p 2 + ix 4 . Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetric one, are also worked out. (general)
Auxiliary fields as a tool for computing analytical solutions of the Schroedinger equation
International Nuclear Information System (INIS)
Silvestre-Brac, Bernard; Semay, Claude; Buisseret, Fabien
2008-01-01
We propose a new method to obtain approximate solutions for the Schroedinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows to find in many cases, analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which greatly improves the corresponding formulae that can be found in the literature
Auxiliary fields as a tool for computing analytical solutions of the Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Silvestre-Brac, Bernard [LPSC Universite Joseph Fourier, Grenoble 1, CNRS/IN2P3, Institut Polytechnique de Grenoble, Avenue des Martyrs 53, F-38026 Grenoble-Cedex (France); Semay, Claude; Buisseret, Fabien [Groupe de Physique Nucleaire Theorique, Universite de Mons-Hainaut, Academie universitaire Wallonie-Bruxelles, Place du Parc 20, B-7000 Mons (Belgium)], E-mail: silvestre@lpsc.in2p3.fr, E-mail: claude.semay@umh.ac.be, E-mail: fabien.buisseret@umh.ac.be
2008-07-11
We propose a new method to obtain approximate solutions for the Schroedinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows to find in many cases, analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which greatly improves the corresponding formulae that can be found in the literature.
Exact solution of the Schroedinger equation with the spin-boson Hamiltonian
International Nuclear Information System (INIS)
Gardas, Bartlomiej
2011-01-01
We address the problem of obtaining the exact reduced dynamics of the spin-half (qubit) immersed within the bosonic bath (environment). An exact solution of the Schroedinger equation with the paradigmatic spin-boson Hamiltonian is obtained. We believe that this result is a major step ahead and may ultimately contribute to the complete resolution of the problem in question. We also construct the constant of motion for the spin-boson system. In contrast to the standard techniques available within the framework of the open quantum systems theory, our analysis is based on the theory of block operator matrices.
International Nuclear Information System (INIS)
Ravi Kanth, A.S.V.; Aruna, K.
2009-01-01
In this paper, we propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schroedinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.
Minimal gravitational coupling in the Newtonian theory and the covariant Schroedinger equation
International Nuclear Information System (INIS)
Duval, C.; Kuenzle, H.P.
1984-01-01
The role of the Bargmann group (11-dimensional extended Galilei group) in nonrelativistic gravitation theory is investigated. The generalized Newtonian gravitation theory (Newton-Cartan theory) achieves the status of a gauge theory about as much as general relativity and couples minimally to a complex scalar field leading to a four-dimensionally covariant Schroedinger equation. Matter current and stress-energy tensor follow correctly from the Lagrangian. This theory on curved Newtonian space-time is also shown to be a limit of the Einstein-Klein-Gordon theory. (author)
International Nuclear Information System (INIS)
Feizi, H.; Rajabi, A.A.; Shojaei, M.R.
2011-01-01
In this work, the three dimensional Woods-Saxon potential is studied within the context of Supersymmetry Quantum Mechanics. We have applied the SUSY method by using the Pekeris approximation to the centrifugal potential l ≠ 0 states. By application of this method, it is possible to solve the Schroedinger equation for this potential. We obtain exactly bound state spectrum and wave function of Woods-Saxon potential for nonzero angular momentum. Hamiltonian hierarchy method and the shape invariance property are used in the calculations. (authors)
Energy Technology Data Exchange (ETDEWEB)
Arevalo, Edward, E-mail: arevalo@temf.tu-darmstadt.d [Technische Universitaet Darmstadt, Institut fuer Theorie elektromagnetischer Felder, TEMF, Schlossgartenstr. 8, D-64289 Darmstadt (Germany)
2009-09-21
The effect of instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schroedinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves in discrete systems is a signature of the modulation instability is used. With the help of this notion we conjecture that instability effects on moving solitons can be qualitative estimated from the analytical solutions. Results from numerical simulations are presented to support this conjecture.
International Nuclear Information System (INIS)
Zhao Dun; Zhang Yujuan; Lou Weiwei; Luo Honggang
2011-01-01
By constructing nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, we investigate the nonautonomous nonlinear Schroedinger (NLS) equations which have been used to describe the Feshbach resonance management in matter-wave solitons in Bose-Einstein condensate and the dispersion and nonlinearity managements for optical solitons. It is found that these equations are some special cases of a new integrable model of nonlocal nonautonomous NLS equations. Based on the Lax pairs, the Darboux transformation and conservation laws are explored. It is shown that the local external potentials would break down the classical infinite number of conservation laws. The result indicates that the integrability of the nonautonomous NLS systems may be nontrivial in comparison to the conventional concept of integrability in the canonical case.
International Nuclear Information System (INIS)
Weiland, J.; Ichikawa, Y.H.; Wilhelmsson, H.
1977-12-01
The Bogoliubov-Mitropolsky perturbation method has been applied to the study of a perturbation on soliton solutions to the nonlinear Schroedinger equation. The results are compared to those of Karpman and Maslov using the inverse scattering method and to those by Ott and Sudan on the KdV equation. (auth.)
International Nuclear Information System (INIS)
Gosson, Maurice A de
2008-01-01
The nearby orbit method is a powerful tool for constructing semi-classical solutions of Schroedinger's equation when the initial datum is a coherent state. In this paper, we first extend this method to arbitrary squeezed states and thereafter apply our results to the Schroedinger equation in phase space. This adaptation requires the phase-space Weyl calculus developed in previous work of ours. We also study the regularity of the semi-classical solutions from the point of view of the Feichtinger algebra familiar from the theory of modulation spaces
International Nuclear Information System (INIS)
Rasolofoson, N.G.
2014-01-01
The properties of a physical system may vary significantly due to the presence of matter or energy. This change can be defined by the deformation of the space which is described as the variation of its curvature. In order to describe this law of physics, we have used differential geometry and studied especially a Schroedinger equation which describes a system evolving with time on a Riemannian manifold of constant curvature. Therefore, we have established and solved the Schroedinger equation using appropriate mathematics tools. As perspective, the study of string theory may be considered. [fr
International Nuclear Information System (INIS)
Zhang Zaiyun; Liu Zhenhai; Miao Xiujin; Chen Yuezhong
2011-01-01
In this Letter, we investigate the perturbed nonlinear Schroedinger's equation (NLSE) with Kerr law nonlinearity. All explicit expressions of the bounded traveling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded traveling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.
Schroedinger equation from 0 (h/2π) to o(h/2πinfinity)
International Nuclear Information System (INIS)
Voros, A.
1985-08-01
The Balian and Bloch idea, that the semiclassical treatment of the Schroedinger equation can be carried out exactly to all orders, o(h/2πinfinity), has been explicitly confirmed upon the time-independent equation with a polynomial potential V(q) in one degree of freedom. The global analytic structure of certain functions, which encode the full eigenvalue distribution, has indeed been computed in great detail with the complex WKB method, yielding a structure called a resurgence algebra. In the special case V(q) = q 2 sub(M), this leads to sum rules for the eigenvalues, which have been verified numerically. Inasmuch as the leading order 0(h/2π) of the WKB expansion amounts to the stationary phase evaluation of the Feynman path integral, it is a yet unsolved challenge to reproduce our results by an exact analysis of this path integral using a generalized saddle-point treatment
International Nuclear Information System (INIS)
Nordbrock, U.; Kienzler, R.
2007-01-01
Conservation laws are a recognized tool in physical and engineering sciences. The classical procedure to construct conservation laws is to apply Noether's Theorem. It requires the existence of a Lagrange-function for the system under consideration. Two unknown sets of functions have to be found. A broader class of such laws is obtainable, if Noether's Theorem is used together with the Bessel-Hagen extension, raising the number of sets of unknown functions to three. By using the recently developed Neutral-Action Method, the same conservation laws can be obtained by calculating only one unknown set of functions. Moreover the Neutral Action Method can also be applied in the absence of a Lagrangian, since only the governing differential equations are required for this procedure. In the paper, an application of this method to the Schroedinger equation is presented. (authors)
Inverse periodic problem for the discrete approximation of the Schroedinger nonlinear equation
International Nuclear Information System (INIS)
Bogolyubov, N.N.; Prikarpatskij, A.K.; AN Ukrainskoj SSR, Lvov. Inst. Prikladnykh Problem Mekhaniki i Matematiki)
1982-01-01
The problem of numerical solution of the Schroedinger nonlinear equation (1) iPSIsub(t) = PSIsub(xx)+-2(PSI)sup(2)PSI. The numerical solution of nonlinear differential equation supposes its discrete approximation is required for the realization of the computer calculation process. Tor the equation (1) there exists the following discrete approximation by variable x(2) iPSIsub(n, t) = (PSIsub(n+1)-2PSIsub(n)+PSIsub(n-1))/(Δx)sup(2)+-(PSIsub(n))sup(2)(PSIsub(n+1)+PSIsub(n-1)), n=0, +-1, +-2... where PSIsub(n)(+) is the corresponding value of PSI(x, t) function in the node and divisions with the equilibrium step Δx. The main problem is obtaining analytically exact solutions of the equations (2). The analysis of the equation system (2) is performed on the base of the discrete analogue of the periodic variant of the inverse scattering problem method developed with the aid of nonlinear equations of the Korteweg-de Vries type. Obtained in explicit form are analytical solutions of the equations system (2). The solutions are expressed through the Riemann THETA-function [ru
The two-fermion relativistic wave equations of Constraint Theory in the Pauli-Schroedinger form
International Nuclear Information System (INIS)
Mourad, J.; Sazdjian, H.
1994-01-01
The two-fermion relativistic wave equations of Constraint Theory are reduced, after expressing the components of the 4x4 matrix wave function in terms of one of the 2x2 components, to a single equation of the Pauli-Schroedinger type, valid for all sectors of quantum numbers. The potentials that are present belong to the general classes of scalar, pseudoscalar and vector interactions and are calculable in perturbation theory from Feynman diagrams. In the limit when one of the masses becomes infinite, the equation reduces to the two-component form of the one-particle Dirac equation with external static potentials. The Hamiltonian, to order 1/c 2 , reproduces most of the known theoretical results obtained by other methods. The gauge invariance of the wave equation is checked, to that order, in the case of QED. The role of the c.m. energy dependence of the relativistic interquark confining potential is emphasized and the structure of the Hamiltonian, to order 1/c 2 , corresponding to confining scalar potentials, is displayed. (authors). 32 refs., 2 figs
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Wu Hongyu; Fei Jinxi; Zheng Chunlong
2010-01-01
An improved homogeneous balance principle and an F-expansion technique are used to construct exact self-similar solutions to the cubic-quintic nonlinear Schroedinger equation. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and the external potential. Some simple self-similar waves are presented. (general)
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Cobian, Hector; Schulze-Halberg, Axel
2011-01-01
We construct Darboux transformations for time-dependent Schroedinger equations with position-dependent mass in (2 + 1) dimensions. Several examples illustrate our results, which complement and generalize former findings for the constant mass case in two spatial variables (Schulze-Halberg 2010 J. Math. Phys. 51 033521).
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Leung Shingyu; Qian Jianliang
2010-01-01
We propose the backward phase flow method to implement the Fourier-Bros-Iagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schroedinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in . In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying the FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to compute beam ingredients rapidly. Numerical examples demonstrate the efficiency and accuracy of the proposed algorithms.
Stability of plane wave solutions of the two-space-dimensional nonlinear Schroedinger equation
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Martin, D.U.; Yuen, H.C.; Saffman, P.G.
1980-01-01
The stability of plane, periodic solutions of the two-dimensional nonlinear Schroedinger equation to infinitesimal, two-dimensional perturbation has been calculated and verified numerically. For standing wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains but that associated with the even mode disappears, which is consistent with the results of Zakharov and Rubenchik, Saffman and Yuen and Ablowitz and Segur on the stability of solitons. In addition, we have identified travelling wave instabilities for the even mode perturbations which are absent in the long-wave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities may also be present for the soliton. In other words, the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, travelling-wave perturbations. (orig.)
Universal Critical Power for Nonlinear Schroedinger Equations with a Symmetric Double Well Potential
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Sacchetti, Andrea
2009-01-01
Here we consider stationary states for nonlinear Schroedinger equations in any spatial dimension n with symmetric double well potentials. These states may bifurcate as the strength of the nonlinear term increases and we observe two different pictures depending on the value of the nonlinearity power: a supercritical pitchfork bifurcation, and a subcritical pitchfork bifurcation with two asymmetric branches occurring as the result of saddle-node bifurcations. We show that in the semiclassical limit, or for a large barrier between the two wells, the first kind of bifurcation always occurs when the nonlinearity power is less than a critical value; in contrast, when the nonlinearity power is larger than such a critical value then we always observe the second scenario. The remarkable fact is that such a critical value is a universal constant in the sense that it does not depend on the shape of the double well potential and on the dimension n.
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Pelinovsky, Dmitry E.; Yang Jianke
2005-01-01
We study the generalized third-order nonlinear Schroedinger (NLS) equation which admits a one-parameter family of single-hump embedded solitons. Analyzing the spectrum of the linearization operator near the embedded soliton, we show that there exists a resonance pole in the left half-plane of the spectral parameter, which explains linear stability, rather than nonlinear semistability, of embedded solitons. Using exponentially weighted spaces, we approximate the resonance pole both analytically and numerically. We confirm in a near-integrable asymptotic limit that the resonance pole gives precisely the linear decay rate of parameters of the embedded soliton. Using conserved quantities, we qualitatively characterize the stable dynamics of embedded solitons
Soliton on a cnoidal wave background in the coupled nonlinear Schroedinger equation
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Shin, H J
2004-01-01
An application of the Darboux transformation on a cnoidal wave background in the coupled nonlinear Schroedinger equation gives a new solution which describes a soliton moving on a cnoidal wave. This is a generalized version of the previously known soliton solutions of dark-bright pair. Here a dark soliton resides on a cnoidal wave instead of on a constant background. It also exhibits a new type of soliton solution in a self-focusing medium, which describes a breakup of a generalized dark-bright pair into another generalized dark-bright pair and an 'oscillating' soliton. We calculate the shift of the crest of the cnoidal wave along a soliton and the moving direction of the soliton on a cnoidal wave
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Yomba, Emmanuel
2008-01-01
With the aid of symbolic computation, we demonstrate that the known method which is based on the new generalized hyperbolic functions and the new kinds of generalized hyperbolic function transformations, generates classes of exact solutions to a system of coupled nonlinear Schroedinger equations. This system includes the modified Hubbard model and the system of coupled nonlinear Schroedinger derived by Lazarides and Tsironis. Four types of solutions for this system are given explicitly, namely: new bright-bright, new dark-dark, new bright-dark and new dark-bright solitons
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Lima, M.L.; Mignaco, J.A.
1983-01-01
The power law potentials in the Schroedinger equation solved recently are shown to come from the classical treatment of the singularities of a linear, second order differential equation. This allows to enlarge the class of solvable power law potentials. (Author) [pt
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Helal, M A [Mathematics Department, Faculty of Science, Cairo University (Egypt); Seadawy, A R [Mathematics Department, Faculty of Science, Beni-Suef University (Egypt)], E-mail: mahelal@yahoo.com, E-mail: aly742001@yahoo.com
2009-09-15
The derivative nonlinear Schroedinger equation (DNLSE) arises as a physical model for ultra-short pulse propagation. In this paper, the existence of a Lagrangian and the invariant variational principle (i.e. in the sense of the inverse problem of calculus of variations through deriving the functional integral corresponding to a given coupled nonlinear partial differential equations) for two-coupled equations describing the nonlinear evolution of the Alfven wave with magnetosonic waves at a much larger scale are given and the functional integral corresponding to those equations is derived. We found the solutions of DNLSE by choice of a trial function in a region of a rectangular box in two cases, and using this trial function, we find the functional integral and the Lagrangian of the system without loss. Solution of the general case for the two-box potential can be obtained on the basis of a different ansatz where we approximate the Jost function using polynomials of order n instead of the piecewise linear function. An example for the third order is given for illustrating the general case.
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Theodorakis, Stavros
2003-01-01
We emulate the cubic term Ψ 3 in the nonlinear Schroedinger equation by a piecewise linear term, thus reducing the problem to a set of uncoupled linear inhomogeneous differential equations. The resulting analytic expressions constitute an excellent approximation to the exact solutions, as is explicitly shown in the case of the kink, the vortex, and a δ function trap. Such a piecewise linear emulation can be used for any differential equation where the only nonlinearity is a Ψ 3 one. In particular, it can be used for the nonlinear Schroedinger equation in the presence of harmonic traps, giving analytic Bose-Einstein condensate solutions that reproduce very accurately the numerically calculated ones in one, two, and three dimensions
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Gambetta, Jay; Wiseman, H.M.
2002-01-01
Do stochastic Schroedinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schroedinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schroedinger equation introduced by Strunz, Diosi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction
The effective Schroedinger equation of the optical model of composite nuclei elastic collisions
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Mondragon, A.; Hernandez, E.
1980-01-01
An effective hamiltonian for elastic collisions between composite nuclei is obtained from the Schroedinger equation of the complete many-body system and its fully antisymmetric wave functions by means of a projection operator technique. This effective hamiltonian, defined in such a way that it has to reproduce the scattering amplitude in full detail, including exchange effects, is explicitly Galilean invariant. The effective interaction operator is a function of the relative distance between the centers of mass of the colliding nuclei and the constants of the motion of the whole system. The interaction operator of the optical model is obtained next, requiring as usual, that it reproduces the average over the energy of the scattering amplitude and keeping the Galilean invariance. The resulting optical potential operator has some terms identical to those obtained in the Resonating Group Method, and others coming from the elimination of all non elastic channels and the delayed elastic scattering. This result makes the relation existing among the projection operator method to the Feshbach and the cluster model equations of motion for positive energies (RGM) explicit. The additional interaction terms due to the flux loss in the elastic channel are non-local, and non-hermitean operators expressed in terms of the transition amplitudes and the density of states of the compound nucleus in such a way that an approximate evaluation, in a systematic fashion, seems possible. Theangular momentum dependence of the optical potential operator is discussed in some detail. (author)
Phase integral approximation for coupled ordinary differential equations of the Schroedinger type
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Skorupski, Andrzej A.
2008-01-01
Four generalizations of the phase integral approximation (PIA) to sets of ordinary differential equations of Schroedinger type [u j '' (x)+Σ k=1 N R jk (x)u k (x)=0, j=1,2,...,N] are described. The recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix R(x)[≡(R jk (x))] either Hermitian or non-Hermitian. For Hermitian and negative definite R(x) matrices, a Wronskian conserving PIA theory is formulated, which generalizes Fulling's current conserving theory pertinent to positive definite R(x) matrices. The idea of a modification of the PIA, which is well known for one equation [u '' (x)+R(x)u(x)=0], is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a nondegenerate eigenvalue of the R(x) matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-Hermitian R(x) matrices. The general theory is illustrated by a few examples automatically generated by using the author's program in MATHEMATICA published in e-print arXiv:0710.5406 [math-ph
Reduction of the equation for lower hybrid waves in a plasma to a nonlinear Schroedinger equation
Karney, C. F. F.
1977-01-01
Equations describing the nonlinear propagation of waves in an anisotropic plasma are rarely exactly soluble. However it is often possible to make approximations that reduce the exact equations into a simpler equation. The use of MACSYMA to make such approximations, and so reduce the equation describing lower hybrid waves into the nonlinear Schrodinger equation which is soluble by the inverse scattering method is demonstrated. MACSYMA is used at several stages in the calculation only because there is a natural division between calculations that are easiest done by hand, and those that are easiest done by machine.
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Kostadinov, S.I.; Petrov, G.
1992-01-01
From a special class of systems has been used a Schroedinger equation with impulse effect in Minkowski space field theory with time dependent boundary conditions, i.e. those of moving mirrors. The field theoretical approach for studying the properties of the vacuum starts from an analysis of the behaviour of local field quantities in Minkowski space with uniformly moving mirrors. For the impulsive moving mirror model is the real process of interaction between the quantum field and the external mirror a subject to disturbances in its evolution acting in time very short compared with the entire duration of the process. So the stability of the solution of the Schroedinger evolution equation for the process in the stability of the vacuum of Casimir. 8 refs
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Al Khawaja, U.
2010-01-01
We derive the integrability conditions of nonautonomous nonlinear Schroedinger equations using the Lax pair and similarity transformation methods. We present a comparative analysis of these integrability conditions with those of the Painleve method. We show that while the Painleve integrability conditions restrict the dispersion, nonlinearity, and dissipation/gain coefficients to be space independent and the external potential to be only a quadratic function of position, the Lax Pair and the similarity transformation methods allow for space-dependent coefficients and an external potential that is not restricted to the quadratic form. The integrability conditions of the Painleve method are retrieved as a special case of our general integrability conditions. We also derive the integrability conditions of nonautonomous nonlinear Schroedinger equations for two- and three-spacial dimensions.
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Pavlus, M.
1997-01-01
The entire potential and the rest of wave functions are determined in parallelepiped domain if the entire discrete spectrum and the apriori information about the wave functions on one side of parallelepiped are given. Formulation for solving the Schroedinger discrete equation in two and higher dimensions is proposed and new formulas are derived for their solution. Two examples for a 2D case and one example for a 3D case are demonstrated
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Xu Guiqiong; Li Zhibin
2005-01-01
It is proven that generalized coupled higher-order nonlinear Schroedinger equations possess the Painleve property for two particular choices of parameters, using the Weiss-Tabor-Carnevale method and Kruskal's simplification. Abundant families of periodic wave solutions are obtained by using the Jacobi elliptic function expansion method with the assistance of symbolic manipulation system, Maple. It is also shown that these solutions exactly degenerate to bright soliton, dark soliton and mixed dark and bright soliton solutions with physical interests
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Cash, J.R.; Raptis, A.D.; Simos, T.E.
1990-01-01
An efficient algorithm is described for the accurate numerical integration of the one-dimensional Schroedinger equation. This algorithm uses a high-order, variable step Runge-Kutta like method in the region where the potential term dominates, and an exponential or Bessel fitted method in the asymptotic region. This approach can be used to compute scattering phase shifts in an efficient and reliable manner. A Fortran program which implements this algorithm is provided and some test results are given. (orig.)
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Belmonte-Beitia, Juan; Konotop, Vladimir V.; Perez-Garcia, Victor M.; Vekslerchik, Vadym E.
2009-01-01
Using similarity transformations we construct explicit solutions of the nonlinear Schroedinger equation with linear and nonlinear periodic potentials. We present explicit forms of spatially localized and periodic solutions, and study their properties. We put our results in the framework of the exploited perturbation techniques and discuss their implications on the properties of associated linear periodic potentials and on the possibilities of stabilization of gap solitons using polychromatic lattices.
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Anastassi, Z. A.; Simos, T. E.
2010-01-01
We develop a new family of explicit symmetric linear multistep methods for the efficient numerical solution of the Schroedinger equation and related problems with oscillatory solution. The new methods are trigonometrically fitted and have improved intervals of periodicity as compared to the corresponding classical method with constant coefficients and other methods from the literature. We also apply the methods along with other known methods to real periodic problems, in order to measure their efficiency.
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Lobanov, Yu.Yu.; Shahbagian, R.R.; Zhidkov, E.P.
1991-01-01
A new method for numerical solution of the boundary problem for Schroedinger-like partial differential equations in R n is elaborated. The method is based on representation of multidimensional Green function in the form of multiple functional integral and on the use of approximation formulas which are constructed for such integrals. The convergence of approximations to the exact value is proved, the remainder of the formulas is estimated. Method reduces the initial differential problem to quadratures. 16 refs.; 7 tabs
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Prinari, Barbara; Ablowitz, Mark J.; Biondini, Gino
2006-01-01
The inverse scattering transform for the vector defocusing nonlinear Schroedinger (NLS) equation with nonvanishing boundary values at infinity is constructed. The direct scattering problem is formulated on a two-sheeted covering of the complex plane. Two out of the six Jost eigenfunctions, however, do not admit an analytic extension on either sheet of the Riemann surface. Therefore, a suitable modification of both the direct and the inverse problem formulations is necessary. On the direct side, this is accomplished by constructing two additional analytic eigenfunctions which are expressed in terms of the adjoint eigenfunctions. The discrete spectrum, bound states and symmetries of the direct problem are then discussed. In the most general situation, a discrete eigenvalue corresponds to a quartet of zeros (poles) of certain scattering data. The inverse scattering problem is formulated in terms of a generalized Riemann-Hilbert (RH) problem in the upper/lower half planes of a suitable uniformization variable. Special soliton solutions are constructed from the poles in the RH problem, and include dark-dark soliton solutions, which have dark solitonic behavior in both components, as well as dark-bright soliton solutions, which have one dark and one bright component. The linear limit is obtained from the RH problem and is shown to correspond to the Fourier transform solution obtained from the linearized vector NLS system
Interpretation of non-Markovian stochastic Schroedinger equations as a hidden-variable theory
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Gambetta, Jay; Wiseman, H.M.
2003-01-01
Do diffusive non-Markovian stochastic Schroedinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system 'conditioned' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit
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Marumori, Toshio; Hayashi, Akihisa; Tomoda, Toshiaki; Kuriyama, Atsushi; Maskawa, Toshihide
1980-01-01
The aim of this series of papers is to propose a microscopic theory to go beyond the situations where collective motions are described by the random phase approximation, i.e., by small amplitude harmonic oscillations about equilibrium. The theory is thus appropriate for the microscopic description of the large amplitude collective motion of soft nuclei. The essential idea is to develop a method to determine the collective subspace (or submanifold) in the many-particle Hilbert space in an optimal way, on the basis of a fundamental principle called the invariance principle of the Schroedinger equation. By using the principle within the framework of the Hartree-Fock theory, it is shown that the theory can clarify the structure of the so-called ''phonon-bands'' by self-consistently deriving the collective Hamiltonian where the number of the ''physical phonon'' is conserved. The purpose of this paper is not to go into detailed quantitative discussion, but rather to develop the basic idea. (author)
Excitation of multiphase waves of the nonlinear Schroedinger equation by capture into resonances
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Friedland, L.; Shagalov, A.G.
2005-01-01
A method for adiabatic excitation and control of multiphase (N-band) waves of the periodic nonlinear Schroedinger (NLS) equation is developed. The approach is based on capturing the system into successive resonances with external, small amplitude plane waves having slowly varying frequencies. The excitation proceeds from zero and develops in stages, as an (N+1)-band (N=0,1,2,...), growing amplitude wave is formed in the (N+1)th stage from an N-band solution excited in the preceding stage. The method is illustrated in simulations, where the excited multiphase waves are analyzed via the spectral approach of the inverse scattering transform method. The theory of excitation of 0- and 1-band NLS solutions by capture into resonances is developed on the basis of a weakly nonlinear version of Whitham's averaged variational principle. The phenomenon of thresholds on the driving amplitudes for capture into successive resonances and the stability of driven, phase-locked solutions in these cases are discussed
Stationary localized modes of the quintic nonlinear Schroedinger equation with a periodic potential
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Alfimov, G. L.; Konotop, V. V.; Pacciani, P.
2007-01-01
We consider localized modes (bright solitons) of the one-dimensional quintic nonlinear Schroedinger equation with a periodic potential, describing several mean-field models of low-dimensional condensed gases. In the case of attractive nonlinearity we deduce sufficient conditions for collapse. We show that there exist spatially localized modes with arbitrarily large numbers of particles. We study such solutions in the semi-infinite gap (attractive case) and in the first gap (attractive and repulsive cases), and show that a nonzero minimum value of the number of particles is necessary for a localized mode to be created. In the limit of large negative frequencies (attractive case) we observe quantization of the number of particles of the stationary modes. Such solutions can be interpreted as coupled Townes solitons and appear to be stable. The modes in the first gap have numbers of particles infinitely growing with frequencies approaching one of the gap edges, which is explained by the power decay of the modes. Stability of the localized modes is discussed
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Faisal, F H.M. [Bielefeld Univ. (Germany, F.R.). Fakultaet fuer Physik
1976-06-11
In this work the perturbation theory for multiphoton processes at high intensities is investigated and it is described an analytical method of summing the perturbation series to extract the contribution from all terms that give rise to the absorption of N photons by an atomic system. The method is first applied to the solution of a simple model problem and the result is confirmed by direct integration of the model Schroedinger equation. The usual lowest (nonvanishing)-order perturbation-theoretical calculation is also carried out for this model to demonstrate explicitly that the full result correctly reproduces that of the lowest-order theory in the limit of low intensity. The method is then extended to the case of an atomic system with well-developed spectrum (e.g. H atom) and the N-photon T-matrix is derived in terms of a ''photon matrix'' asub(N), for which a three-term recurrence relation is established. Next, from the vantage point of the general result obtained here, A probe is made into the nature of several approximate nonperturbative solutions that have appeared in the literature in the past. It is shown here that their applicability is severely restricted by the requirement of the essential spectral degeneracy of the atomic system. Finally, appendix A outlines a prescription of computing the photon matrix asub(N), which (as in the usual lowest-order perturbation-theoretical calculation)requires a knowledge of the eigenfunctions and eigenvalues of the atomic Hamiltonian only.
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Klein, A.; Tanabe, K.
1984-01-01
The invariance principle of the Schroedinger equation provides a basis for theories of collective motion with the help of the time-dependent variational principle. It is formulated here with maximum generality, requiring only the motion of intrinsic state in the collective space. Special cases arise when the trial vector is a generalized coherent state and when it is a uniform superposition of collective eigenstates. The latter example yields variational principles uncovered previously only within the framework of the equations of motion method. (orig.)
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Dubrovsky, V.G.; Formusatik, I.B.
2003-01-01
The scheme for calculating via Zakharov-Manakov ∂-macron-dressing method of new rational solutions with constant asymptotic values at infinity of the famous two-dimensional Veselov-Novikov (VN) integrable nonlinear evolution equation and new exact rational potentials of two-dimensional stationary Schroedinger (2DSchr) equation with multiple pole wave functions is developed. As examples new lumps of VN nonlinear equation and new exact rational potentials of 2DSchr equation with multiple pole of order two wave functions are calculated. Among the constructed rational solutions are as nonsingular and also singular
Frozen and broken color: a matrix Schroedinger equation in the semiclassical limit
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Orbach, H.S.
1981-01-01
We consider the case of frozen color, i.e, where global color symmetry remains exact, but where colored states have a mass large compared to color-singlet mesons. Using semiclassical WKB formalism, we construct the spectrum of bound states. In order to determine the charge of the constituents, we then consider deep-inelastic scattering of an external probe (e.g., lepton) from our one-dimensional meson. We calculate explicitly the structure function, W, in the WKB limit and show how Lipkin's mechanism is manifested, as well as how scaling behavior comes. We derive the WKB formalism as a special case of a method of obtaining WKB type solutions for generalized Schroedinger equations for which the Hamiltonian is an arbitrary matrix function of any number of pairs of canonical operators. We generalize these considerations to the case of broken color symmetry - but where the breaking is not so strong as to allow low-lying states to have a large amount of mixing with the colored states. In this case, the degeneracy of excited colored states can be broken. We find that local excitation of color guarantees global excitation of color; i.e., if at a given energy colored semiclassical states can be constructed with size comparable to that of the ground state wave function, colored states of that energy will also exist in the spectrum of the full theory and will be observed. However, global existence of color does not guarantee the excitation of colored states via deep-inelastic processes
The paradox of Schroedinger's waves
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Gribben, John.
1987-01-01
The paper examines the contribution of the work of Erwin Schroedinger in quantum physics. The Schroedinger equation was developed to explain the behavior of electrons within an atom in terms of waves, and it has proved one of the most useful tools in quantum physics. The Schroedinger 'Cat' experiment is also described and discussed. Finally Schroedinger's ideas on chromosomes in molecular biology are briefly outlined. (U.K.)
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Qiao Haoxue; Cai Qingyu; Rao Jianguo; Li Baiwen
2002-01-01
A spectral fitting method for solving the time-dependent Schroedinger equation has been developed and applied to the atom in intense laser fields. This method allows us to obtain a highly accurate time-dependent wave function with a contribution from the high-order term of Δt. Moreover, the time-dependent wave function is determined on a small number of discrete mesh points, thus making calculations simple and accurate. This method is illustrated by computing wave functions and harmonic generation spectra of a model atom in laser fields
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Boiti, M.; Pempinelli, F.; Pogrebkov, A.K.; Polivanov, M.C.
1993-01-01
The resolvent operator of the linear problem is determined as the full Green function continued in the complex domain in two variables. An analog of the known Hilbert identity is derived. The authors demonstrate the role of this identity in the study of two-dimensional scattering. Considering the nonstationary Schroedinger equation as an example, it is shown that all types of solutions of the linear problem, as well as spectral data known in the literature, are given as specific values of this unique function - the resolvent function. A new form of the inverse problem is formulated. 7 refs
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Hu Xianquan; Luo Guang; Cui Lipeng; Niu Lianbin; Li Fangyu
2009-01-01
The analytic solution of the radial Schroedinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schroedinger equation is V(r) = α 1 r 8 + α 2 r 3 + α 3 r 2 + β 3 r -1 + β 2 r -3 + β 1 r -4 . Generally speaking, there is only an approximate solution, but not analytic solution for Schroedinger equation with several potentials' superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schroedinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r → and r → 0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions of the stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial Schroedinger equation; and lastly, they discuss the solutions and make conclusions. (general)
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Pali, R.; Coss, R. de; Mustre de Leon, J.
1999-01-01
The adimensionalization of equations which govern the dynamics of a physical system can be very useful when studying the qualitative behavior of any variable involved in those equations. In a dynamic system like a particle moving in an effective potential, the isotopic coefficient measure the degree of anharmonicity of the potential. In general each eigenstate has a different coefficient. In this work, we determined the isotopic coefficients for potentials of the form V(x) ∝ x 2N (N=1,2,3,...) through the adimensionalization process of the Schroedinger equation. We found an analytic expression for the isotopic coefficient which depends only of N but not on the eigenstate. The isotopic coefficient value starts at 1/2 for N=1 (harmonic potential) and gradually converges to 1.0 when N increments. This reflects the fact that the potential is more anharmonic for increasing N. (Author)
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Kutzelnigg, Werner; Mukherjee, Debashis
2004-01-01
The k-particle irreducible Brillouin conditions IBC k and the k-particle irreducible contracted Schroedinger equations ICSE k for a closed-shell state are analyzed in terms of a Moeller-Plesset-type perturbation expansion. The zeroth order is Hartree-Fock. From the IBC 2 (1) , i.e., from the two-particle IBC to first order in the perturbation parameter μ, one gets the leading correction λ 2 (1) to the two-particle cumulant λ 2 correctly. However, in order to construct the second-order energy E 2 , one also needs the second-order diagonal correction γ D (2) to the one-particle density matrix γ. This can be obtained: (i) from the idempotency of the n-particle density matrix, i.e., essentially from the requirement of n-representability; (ii) from the ICSE 1 (2) ; or (iii) by means of perturbation theory via a unitary transformation in Fock space. Method (ii) is very unsatisfactory, because one must first solve the ICSE 3 (2) to get λ 3 (2) , which is needed in the ICSE 2 (2) to get λ 2 (2) , which, in turn, is needed in the ICSE 1 (2) to get γ (2) . Generally the (k+1)-particle approximation is needed to obtain E k correctly. One gains something, if one replaces the standard hierarchy, in which one solves the ICSE k , ignoring λ k+1 and λ k+2 , by a renormalized hierarchy, in which only λ k+2 is ignored, and λ k+1 is expressed in terms of the λ p of lower particle rank via the partial trace relation for λ k+2 . Then the k-particle approximation is needed to obtain E k correctly. This is still poorer than coupled-cluster theory, where the k-particle approximation yields E k+1 . We also study the possibility to use some simple necessary n-representability conditions, based on the non-negativity of γ (2) and two related matrices, in order to get estimates for γ D (2) in terms of λ 2 (1) . In general these estimates are rather weak, but they can become close to the best possible bounds in special situations characterized by a very sparse structure of λ 2
International Nuclear Information System (INIS)
Boykin, Timothy B; Klimeck, Gerhard
2005-01-01
The discretized Schroedinger equation is most often used to solve one-dimensional quantum mechanics problems numerically. While it has been recognized for some time that this equation is equivalent to a simple tight-binding model and that the discretization imposes an underlying bandstructure unlike free-space quantum mechanics on the problem, the physical implications of this equivalence largely have been unappreciated and the pedagogical advantages accruing from presenting the problem as one of solid-state physics (and not numerics) remain generally unexplored. This is especially true for the analytically solvable discretized finite square well presented here. There are profound differences in the physics of this model and its continuous-space counterpart which are direct consequences of the imposed bandstructure. For example, in the discrete model the number of bound states plus transmission resonances equals the number of atoms in the quantum well
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Fewo, S I; Kenfack-Jiotsa, A; Kofane, T C
2006-01-01
With the help of the one-dimensional quintic complex Ginzburg-Landau equation (CGLE) as perturbations of the nonlinear Schroedinger equation (NLSE), we derive the equations of motion of pulse parameters called collective variables (CVs), of a pulse propagating in dispersion-managed (DM) fibre optic links. The equations obtained are investigated numerically in order to view the evolution of pulse parameters along the propagation distance, and also to analyse effects of initial amplitude and width on the propagating pulse. Nonlinear gain is shown to be beneficial in stabilizing DM solitons. A fully numerical simulation of the one-dimensional quintic CGLE as perturbations of NLSE finally tests the results of the CV theory. A good agreement is observed between both methods
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Malenfant, J.
1988-01-01
The Breit equation for two equal-mass spin-1/2 particles interacting through an attractive Coulomb potential is separated into its angular and radial parts, obtaining coupled sets of first-order differential equations for the radial wave functions. The radial equations for the 1 J/sub J/, 3 J/sub J/, and 3 P 0 states are further reduced to a single, one-dimensional Schroedinger equation with a relatively simple effective potential. No approximations, other than the initial one of an instantaneous Coulomb interaction, are made in deriving this equation; it accounts for all relativistic effects, as well as for mixing between different components of the wave function. Approximate solutions are derived for this Schroedinger equation, which gives the correct O(α 4 ) term for the 1 1 S 0 energy and for the n 1 J/sub J/ energies, for J>0. The radial equations for the 3 (J +- 1)/sub J/ states are reduced to two second-order coupled equations. At small r, the Breit Coulomb wave functions behave as r/sup ν//sup -1/, where ν is either √J(J+1)+1-α 2 /4 or √J(J+1)-α 2 /4 . The 1 S 0 and 3 P 0 wave functions therefore diverge at the origin as r/sup //sup √//sup 1-//sup α//sup <2//4 -1$. This divergence of the J = 0 states, however, does not occur when the spin-spin interaction, -(α/r)αxα, is added to the Coulomb potential
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Wurm, A.; LaChapelle, J.
1997-01-01
The authors comment on the paper by J. LaChapelle, J. Math. Phys. 37, 4310 (1996), and give explicit expressions for the parametrization, its solution, and the Lie derivatives of the Schroedinger equation for the case of n-dimensional spherical coordinates
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Zagrebaev, V.I.; ); Samarin, V.V.
2006-01-01
Fusion of heavy nuclei was analyzed on the basis of the numerical solution of the Schroedinger three-body and three-dimensional nonstationary equations. One revealed the increase of fuss ion probability in 66 He + 2O Pb reaction caused by transfer and collectivization of valent neutrons [ru
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Hoang-Do, Ngoc-Tram; Hoang, Van-Hung; Le, Van-Hoang [Department of Physics, Ho Chi Minh City University of Pedagogy, 280 An Duong Vuong Street, District 5, Ho Chi Minh City (Viet Nam)
2013-05-15
The Feranchuk-Komarov operator method is developed by combining with the Levi-Civita transformation in order to construct analytical solutions of the Schroedinger equation for a two-dimensional exciton in a uniform magnetic field of arbitrary strength. As a result, analytical expressions for the energy of the ground and excited states are obtained with a very high precision of up to four decimal places. Especially, the precision is uniformly stable for the whole range of the magnetic field. This advantage appears due to the consideration of the asymptotic behaviour of the wave-functions in strong magnetic field. The results could be used for various physical analyses and the method used here could also be applied to other atomic systems.
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Dyakin, V.V.; Petrukhnovskii, S.I.
1988-01-01
Three-dimensional periodic Schroedinger operators with potentials that are square integrable on the unit cell (single-electron model of a crystal) are considered. A description is given of the class of rational curves that do not have more than a finite number of common points with any isoenergy surface (in particular, the Fermi surface) of an arbitrary operator of the considered form. A consequence of a theorem proved in the paper is the absence on the isoenergy surfaces of elements of planes, cones, and cylinders with straight generators, and all possible paraboloids and hyperboloids. Another interesting consequence is the following assertion: The topological dimension of an isoenergy manifold does not exceed two, which justifies the use of the word surface. The results generalize the assertion of Thomas's theorem on the absence on isoenergy surfaces of straight edges
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Lue Xing; Zhu Hongwu; Yao Zhenzhi; Meng Xianghua; Zhang Cheng; Zhang Chunyi; Tian Bo
2008-01-01
In this paper, the multisoliton solutions in terms of double Wronskian determinant are presented for a generalized variable-coefficient nonlinear Schroedinger equation, which appears in space and laboratory plasmas, arterial mechanics, fluid dynamics, optical communications and so on. By means of the particularly nice properties of Wronskian determinant, the solutions are testified through direct substitution into the bilinear equations. Furthermore, it can be proved that the bilinear Baecklund transformation transforms between (N - 1)- and N-soliton solutions
Rauscher, Elizabeth A
2011-01-01
The Maxwell, Einstein, Schrödinger and Dirac equations are considered the most important equations in all of physics. This volume aims to provide new eight- and twelve-dimensional complex solutions to these equations for the first time in order to reveal
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Kist, Tarso B.L.; Orszag, M.; Davidovich, L.
1997-01-01
The dynamics of open system is frequently modeled in terms of a small system S coupled to a reservoir R, the last having an infinitely larger number of degree of freedom than S. Usually the dynamics of the S variables may be of interest, which can be studied using either Langevin equations, or master equations, or yet the path integral formulation. Useful alternatives for the master equation method are the Monte Carlo Wave-function method (MCWF), and Stochastic Schroedinger Equations (SSE's). The methods MCWF and SSE's recently experienced a fast development both in their theoretical background and applications to the study of the dissipative quantum systems dynamics in quantum optics. Even though these alternatives can be shown to be formally equivalent to the master equation approach, they are often regarded as mathematical tricks, with no relation to a concrete physical evolution of the system. The advantage of using them is that one has to deal with state vectors, instead of density matrices, thus reducing the total amount of matrix elements to be calculated. In this work, we consider the possibility of giving a physical interpretation to these methods, in terms of continuous measurements made on the evolving system. We show that physical realizations of the two methods are indeed possible, for a mode of the electromagnetic field in a cavity interacting with a continuum of modes corresponding to the field outside the cavity. Two schemes are proposed, consisting of a mode of the electromagnetic field interacting with a beam of Rydberg two-level atoms. In these schemes, the field mode plays the role of a small system and the atomic beam plays the role of a reservoir (infinitely larger number of degrees of freedom at finite temperature, the interaction between them being given by the Jaynes-Cummings model
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Jaulent, M.; Jean, C.
1976-01-01
The one-dimensional Schroedinger equation y + ''+ ) 7k 2 -V + (k,x){y + =0, x belonging to R, was previously considered when the potential V + (k,x) depends on the energy k 2 in the following way: V + (k,x)=U(x)+2kQ(x), (U(x), Q(x)) belonging to a large class of pairs of real potentials admitting no bound state). The two systems of differential and integral equations then introduced are solved. Then, investigating the inverse scattering problem it is found that a necessary and sufficient condition for one of the functions S + (k) and Ssub(-1)sup(+)(k) to be the scattering matrix associated with a pair (U(x), Q(x)) is that S + (k) (or equivalently Ssub(-1)sup(+)(k) belongs to the class S introduced. This pair is the only one admitting this function as its scattering matrix. Investigating the inverse reflection problem, it is found that a necessary and sufficient condition for a function S 21 + (k) to be the reflection coefficient to the right associated with a pair (U(x), Q(x)) is that S 21 + (k) belongs to the class R introduced. This pair is the only one admitting this function as
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Mahlab, M.S.
1975-01-01
All the presently available techniques for solving Schroedinger's differential equation for helium-like atoms display poor convergence of the wave function in the neighborhood of the singularities of the Hamiltonian operator. In general most of the methods of solving this equation will converge in the appropriate limit to the exact wave function; however, convergence is slow, especially near the singularities of this differential equation. These difficulties become readily apparent from local energy studies. A technique is presented that avoids these difficulties. The wave function it produces is specifically most accurate at the singularities of the Hamiltonian. The novel aspect of this treatment is the subdivision of the space spanned by the wave function. Different expansions are picked such that they converge rapidly in each of the different subdivisions. These expansions may be constructed in such a way that they obey the boundary conditions in their respective subdivision. Most importantly, all the information available from the recursion relations associated with the differential equation may be incorporated into these expansions. A systematic procedure is presented such that these expansions may be brought together to form a wave function that satisfies all the continuity requirements. An S-state helium wave function was constructed to demonstrate that this method of treatment is feasible, and capable of indefinite systematic improvement. A discussion of several new asymptotic expansions that were constructed for the helium wave function, as well as an improved functional form for the small electron-nucleus wave function, is included in this presentation
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Abdou, M.A.
2008-01-01
The generalized F-expansion method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for the generalized nonlinear Schrodinger equation with a source. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics
A new fourth-order Fourier-Bessel split-step method for the extended nonlinear Schroedinger equation
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Nash, Patrick L.
2008-01-01
Fourier split-step techniques are often used to compute soliton-like numerical solutions of the nonlinear Schroedinger equation. Here, a new fourth-order implementation of the Fourier split-step algorithm is described for problems possessing azimuthal symmetry in 3 + 1-dimensions. This implementation is based, in part, on a finite difference approximation Δ perpendicular FDA of 1/r (∂)/(∂r) r(∂)/(∂r) that possesses an associated exact unitary representation of e i/2λΔ perpendicular FDA . The matrix elements of this unitary matrix are given by special functions known as the associated Bessel functions. Hence the attribute Fourier-Bessel for the method. The Fourier-Bessel algorithm is shown to be unitary and unconditionally stable. The Fourier-Bessel algorithm is employed to simulate the propagation of a periodic series of short laser pulses through a nonlinear medium. This numerical simulation calculates waveform intensity profiles in a sequence of planes that are transverse to the general propagation direction, and labeled by the cylindrical coordinate z. These profiles exhibit a series of isolated pulses that are offset from the time origin by characteristic times, and provide evidence for a physical effect that may be loosely termed normal mode condensation. Normal mode condensation is consistent with experimentally observed pulse filamentation into a packet of short bursts, which may occur as a result of short, intense irradiation of a medium
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Chudnovsky, David; Chudnovsky, G.V.
1978-01-01
The relations between many particle problem with inverse square potential on the line and meromorphic eigenfunctions of Schroedinger operator are presented. This gives new type of Backlund transformations for many particle problem [fr
Cummings, Patrick
We consider the approximation of solutions of two complicated, physical systems via the nonlinear Schrodinger equation (NLS). In particular, we discuss the evolution of wave packets and long waves in two physical models. Due to the complicated nature of the equations governing many physical systems and the in-depth knowledge we have for solutions of the nonlinear Schrodinger equation, it is advantageous to use approximation results of this kind to model these physical systems. The approximations are simple enough that we can use them to understand the qualitative and quantitative behavior of the solutions, and by justifying them we can show that the behavior of the approximation captures the behavior of solutions to the original equation, at least for long, but finite time. We first consider a model of the water wave equations which can be approximated by wave packets using the NLS equation. We discuss a new proof that both simplifies and strengthens previous justification results of Schneider and Wayne. Rather than using analytic norms, as was done by Schneider and Wayne, we construct a modified energy functional so that the approximation holds for the full interval of existence of the approximate NLS solution as opposed to a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et al. We then consider the Klein-Gordon-Zakharov system and prove a long wave approximation result. In this case there is a non-trivial resonance that cannot be eliminated via a normal form transform. By combining the normal form transform for small Fourier modes and using analytic norms elsewhere, we can get a justification result on the order 1 over epsilon squared time scale.
Chaotic synchronization of symbolic information in the discrete nonlinear Schroedinger equation
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Pando L, C.L.
2003-08-01
We have studied the discrete nonlinear Schrodinger equation (DNLSE) with on-site defects and periodic boundary conditions. When the array dynamics becomes chaotic, the otherwise quasiperiodic amplitude correlations between the oscillators are destroyed. However, we show that synchronization of symbolic information of suitable amplitude signals is possible in this hamiltonian system. (author)
KAM for the non-linear Schroedinger equation a short presentation
Eliasson, H L
2006-01-01
We consider the $d$-dimensional nonlinear Schr\\"o\\-dinger equation under periodic boundary conditions:-i\\dot u=\\Delta u+V(x)*u+\\ep \\frac{\\p F}{\\p \\bar u}(x,u,\\bar u) ;\\quad u=u(t,x),\\;x\\in\\T^dwhere $V(x)=\\sum \\hat V(a)e^{i\\sc{a,x}}$ is an analytic function with $\\hat V$ real and $F$ is a real analytic function in $\\Re u$, $\\Im u$ and $x$. (This equation is a popular model for the `real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\\ep=0$ the equation is linear and has time--quasi-periodic solutions $u$,u(t,x)=\\sum_{s\\in \\AA}\\hat u_0(a)e^{i(|a|^2+\\hat V(a))t}e^{i\\sc{a,x}}, \\quad 0<|\\hat u_0(a)|\\le1,where $\\AA$ is any finite subset of $\\Z^d$. We shall treat $\\omega_a=|a|^2+\\hat V(a)$, $a\\in\\AA$, as free parameters in some domain $U\\subset\\R^{\\AA}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, in particular, will have the following consequence: \\smallskip {\\it ...
Born approximation to a perturbative numerical method for the solution of the Schroedinger equation
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Adam, Gh.
1978-01-01
A step function perturbative numerical method (SF-PN method) is developed for the solution of the Cauchy problem for the second order liniar differential equation in normal form. An important point stressed in the present paper, which seems to have been previously ignored in the literature devoted to the PN methods, is the close connection between the first order perturbation theory of the PN approach and the wellknown Born approximation, and, in general, the connection between the varjous orders of the PN corrections and the Neumann series. (author)
A new fundamental model of moving particle for reinterpreting Schroedinger equation
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Umar, Muhamad Darwis
2012-01-01
The study of Schrödinger equation based on a hypothesis that every particle must move randomly in a quantum-sized volume has been done. In addition to random motion, every particle can do relative motion through the movement of its quantum-sized volume. On the other way these motions can coincide. In this proposed model, the random motion is one kind of intrinsic properties of the particle. The every change of both speed of randomly intrinsic motion and or the velocity of translational motion of a quantum-sized volume will represent a transition between two states, and the change of speed of randomly intrinsic motion will generate diffusion process or Brownian motion perspectives. Diffusion process can take place in backward and forward processes and will represent a dissipative system. To derive Schrödinger equation from our hypothesis we use time operator introduced by Nelson. From a fundamental analysis, we find out that, naturally, we should view the means of Newton’s Law F(vector sign) = ma(vector sign) as no an external force, but it is just to describe both the presence of intrinsic random motion and the change of the particle energy.
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Skoczen, A.; Machowski, W.; Kaprzyk, S.
1990-07-01
Computer program aiming at application in quantum mechanics didactics has been proposed. This program can generate the moving pictures of one-dimensional quantum mechanics scattering phenomena. Constructions of this program provide two options. In the first option the wave packet is generated in infinite one-dimensional well which has walls on the borders of graphic window. In the second option the square potential barrier is located in this well and transmission and reflection of wave packet are shown. We have selected a Gaussian wave packet to represent the initial state of the particle. The wave equation is solved numerically by a method discussed in detail. Solutions for the succesive time moments are graphically presented on the monitor screen. In this way observer can watch whole time-development of physical system. Graphically presented results are physically realistic when program parameters satisfy conditions discussed in this paper. (author)
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Kan, K.K.
1983-01-01
The relationship of nuclear internal flow and collective inertia, the difference of this flow from that of a classical fluid, and the approach of this flow to rigid flow in independent-particle model rotation are elucidated by reviewing the theory of Schroedinger fluid and its implications for collective vibration and rotation. (author)
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Arvieu, R.; Carbonell, J.; Gignoux, C.; Mangin-Brinet, M.; Rozmej, P.
1997-01-01
The time evolution of coherent rotational wave packets associated to a diatomic molecule or to a deformed nucleus has been studied. Assuming a rigid body dynamics the J(J+1) law leads to a mechanism of cloning: the way function is divided into wave packets identical to the initial one at specific time. Applications are studied for a nuclear wave packed formed by Coulomb excitation. Exact boundary conditions at finite distance for the solution of the time-dependent Schroedinger equation are derived. A numerical scheme based on Crank-Nicholson method is proposed to illustrate its applicability in several examples. (authors)
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Sakmann, Kaspar
2010-07-21
In this thesis, the physics of trapped, interacting Bose-Einstein condensates is analyzed by solving the many-body Schroedinger equation. Particular emphasis is put on coherence, fragmentation and reduced density matrices. First, the ground state of a trapped Bose-Einstein condensate and its correlation functions are obtained. Then the dynamics of a bosonic Josephson junction is investigated by solving the time-dependent many-body Schroedinger equation numerically exactly. These are the first exact results in literature in this context. It is shown that the standard approximations of the field, Gross-Pitaevskii theory and the Bose-Hubbard model fail at weak interaction strength and within their range of expected validity. For stronger interactions the dynamics becomes strongly correlated and a new equilibration phenomenon is discovered. By comparison with exact results it is shown that a symmetry of the Bose- Hubbard model between attractive and repulsive interactions must be considered an artefact of the model. A conceptual innovation of this thesis are time-dependent Wannier functions. Equations of motion for time-dependent Wannier functions are derived from the variational principle. By comparison with exact results it is shown that lattice models can be greatly improved at little computational cost by letting the Wannier functions of a lattice model become time-dependent. (orig.)
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Ixaru, G.L.
1978-03-01
The method developed in the previous paper (preprint, C.I.Ph. (Bucharest), MC-2-78, 1978) is here investigated from computational point of view. Special emphasis is paid to the two basic descriptors of the efficiency: the volume of memory required and the computational effort (timing). Next, two experimental cases are reported. They (i) confirm the theoretical estimates for the rate cf convergence of each version of the present method and (ii) show that the present method is substantially faster than the others. Specifically, it is found that for typical physical problems it is faster by a factor of ten up to twenty than the methods commonly used, viz. Numerov and de Vogelaere. The data reported also allow an inUirect comparison with the method of Gordon. I l/ allow an indirect comparison with the method of Gordon. It is shown that, while this exhibits the same rate as our basic, lowest order version, the computational effort for the latter is, in case of systems with nine equations, only half than for the method of Gordon. At the end of the paper some types of physical problems are suggested which should be the most benefitting if solved numerically with the present method. (author)
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Keanini, R.G.
2011-01-01
Research highlights: → Systematic approach for physically probing nonlinear and random evolution problems. → Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. → Organization of near-molecular scale vorticity mediated by hydrodynamic modes. → Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the
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Li Juan; Zhang Haiqiang; Xu Tao; Zhang, Ya-Xing; Tian Bo
2007-01-01
For the long-distance communication and manufacturing problems in optical fibers, the propagation of subpicosecond or femtosecond optical pulses can be governed by the variable-coefficient nonlinear Schroedinger equation with higher order effects, such as the third-order dispersion, self-steepening and self-frequency shift. In this paper, we firstly determine the general conditions for this equation to be integrable by employing the Painleve analysis. Based on the obtained 3 x 3 Lax pair, we construct the Darboux transformation for such a model under the corresponding constraints, and then derive the nth-iterated potential transformation formula by the iterative process of Darboux transformation. Through the one- and two-soliton-like solutions, we graphically discuss the features of femtosecond solitons in inhomogeneous optical fibers
Correspondence passed between Einstein and Schroedinger
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Balibar, F.
1992-01-01
The main points of the 26 year long correspondence between Einstein and Schroedinger are reviewed: from the de Broglie thesis and the Bose-Einstein statistics to the Schroedinger equation (1925-1926); from the EPR paradox to the cat parable (1935); a complete collaboration on unitary theories
Energy Technology Data Exchange (ETDEWEB)
Briscese, Fabio [Northumbria University, Department of Mathematics, Physics and Electrical Engineering, Newcastle upon Tyne (United Kingdom); Citta Universitaria, Istituto Nazionale di Alta Matematica Francesco Severi, Gruppo Nazionale di Fisica Matematica, Rome (Italy)
2017-09-15
In this paper it is argued how the dynamics of the classical Newtonian N-body system can be described in terms of the Schroedinger-Poisson equations in the large N limit. This result is based on the stochastic quantization introduced by Nelson, and on the Calogero conjecture. According to the Calogero conjecture, the emerging effective Planck constant is computed in terms of the parameters of the N-body system as ℎ ∝ M{sup 5/3}G{sup 1/2}(N/ left angle ρ right angle){sup 1/6}, where is G the gravitational constant, N and M are the number and the mass of the bodies, and left angle ρ right angle is their average density. The relevance of this result in the context of large scale structure formation is discussed. In particular, this finding gives a further argument in support of the validity of the Schroedinger method as numerical double of the N-body simulations of dark matter dynamics at large cosmological scales. (orig.)
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Lubkin, E [Wisconsin Univ., Madison (USA). Dept. of Physics
1979-08-01
The issue is to seek quantum interference effects in an arbitrary field, in particular in psychology. For this a digest of quantum mechanics over finite-n-dimensional Hilbert space is invented. In order to match crude data not only von Neumann's mixed states are used but also a parallel notion of unsharp tests. The mathematically styled text (and earlier work on multibin tests, designated MB) deals largely with these new tests. Quantum psychology itself is only given a foundation. It readily engenders objections; its plausibility is developed gradually, in interlocking essays. There is also the empirically definite proposal that (state, test, outcome)-indexed counts be gathered to record data, then fed to a matrix format (MF) search for quantum models. A previously proposed experiment in visual perception which has since failed to find significant quantum correlations, is discussed. The suspicion that quantum mechanics is all around goes beyond MF, and Schroedinger's cat symbolizes this broader perspective.
International Nuclear Information System (INIS)
Lubkin, E.
1979-01-01
The issue is to seek quantum interference effects in an arbitrary field, in particular in psychology. For this a digest of quantum mechanics over finite-n-dimensional Hilbert space is invented. In order to match crude data not only von Neumann's mixed states are used but also a parallel notion of unsharp tests. The mathematically styled text (and earlier work on multibin tests, designated MB) deals largely with these new tests. Quantum psychology itself is only given a foundation. It readily engenders objections; its plausibility is developed gradually, in interlocking essays. There is also the empirically definite proposal that (state, test, outcome)-indexed counts be gathered to record data, then fed to a 'matrix format' (MF) search for quantum models. A previously proposed experiment in visual perception which has since failed to find significant quantum correlations, is discussed. The suspicion that quantum mechanics is all around goes beyond MF, and 'Schroedinger's cat' symbolizes this broader perspective. (author)
International Nuclear Information System (INIS)
Ducomet, Bernard; Zlotnik, Alexander; Zlotnik, Ilya
2014-01-01
We consider an initial-boundary value problem for a generalized 2D time-dependent Schroedinger equation (with variable coefficients) on a semi-infinite strip. For the Crank-Nicolson-type finite-difference scheme with approximate or discrete transparent boundary conditions (TBCs), the Strang-type splitting with respect to the potential is applied. For the resulting method, the unconditional uniform in time L2-stability is proved. Due to the splitting, an effective direct algorithm using FFT is developed now to implement the method with the discrete TBC for general potential. Numerical results on the tunnel effect for rectangular barriers are included together with the detailed practical error analysis confirming nice properties of the method. (authors)
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Kovarik, M.D.; Barnes, T.; Tennessee Univ., Knoxville, TN
1993-01-01
We describe a Monte Carlo simulation of a dynamical fermion problem in two spatial dimensions on an Intel iPSC/860 hypercube. The problem studied is the determination of the dispersion relation of a dynamical hole in the t-J model of the high temperature superconductors. Since this problem involves the motion of many fermions in more than one spatial dimensions, it is representative of the class of systems that suffer from the ''minus sign problem'' of dynamical fermions which has made Monte Carlo simulation very difficult. We demonstrate that for small values of the hole hopping parameter one can extract the entire hole dispersion relation using the GRW Monte Carlo algorithm, which is a simulation of the Euclidean time Schroedinger equation, and present results on 4 x 4 and 6 x 6 lattices. Generalization to physical hopping parameter values wig only require use of an improved trial wavefunction for importance sampling
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Balibar, F. [Paris-7 Univ., 75 (France)
1992-12-31
The main points of the 26 year long correspondence between Einstein and Schroedinger are reviewed: from the de Broglie thesis and the Bose-Einstein statistics to the Schroedinger equation (1925-1926); from the EPR paradox to the cat parable (1935); a complete collaboration on unitary theories.
Schroedinger operators and evolutionary strategies
International Nuclear Information System (INIS)
Asselmeyer, T.
1997-01-01
First we introduce a simple model for the description of evolutionary algorithms, which is based on 2nd order partial differential equations for the distribution function of the individuals. Then we turn to the properties of Boltzmann's and Darwin's strategy. the next chapter is dedicated to the mathematical properties of Schroedinger operators. Both statements on the spectral density and their reproducibility during the simulation are summarized. The remaining of this chapter are dedicated to the analysis of the kernel as well as the dependence of the Schroedinger operator on the potential. As conclusion from the results of this chapter we obtain the classification of the strategies in dependence of the fitness. We obtain the classification of the evolutionary strategies, which are described by a 2nd order partial differential equation, in relation to their solution behaviour. Thereafter we are employed with the variation of the mutation distribution
The Schroedinger operator as a generalized Laplacian
International Nuclear Information System (INIS)
Grabowska, Katarzyna; Urbanski, Pawel; Grabowski, Janusz
2008-01-01
The Schroedinger operators on the Newtonian spacetime are defined in a way which make them independent of the class of inertial observers. In this picture the Schroedinger operators act not on functions on the spacetime but on sections of a certain one-dimensional complex vector bundle-the Schroedinger line bundle. This line bundle has trivializations indexed by inertial observers and is associated with an U(1)-principal bundle with an analogous list of trivializations-the Schroedinger principal bundle. If an inertial frame is fixed, the Schroedinger bundle can be identified with the trivial bundle over spacetime, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wavefunctions' cannot be viewed as actual functions on the spacetime. In this approach, the change of an observer results not only in the change of actual coordinates in the spacetime but also in a change of the phase of wavefunctions. For the Schroedinger principal bundle, a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of the Laplace-Beltrami operator associated with a pseudo-Riemannian metric. The free Schroedinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schroedinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schroedinger operator in a given reference frame-which is supported by producing right physics predictions-and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton-Jacobi equations that makes a bridge between the classical and quantum theory
International Nuclear Information System (INIS)
Lima, M.L.; Mignaco, J.A.
1985-01-01
It is shown that the rational power law potentials in the two-body radial Schrodinger equations admit a systematic treatment available from the classical theory of ordinary linear differential equations of the second order. The resulting potentials come into families evolved from equations having a fixed number of elementary regular singularities. As a consequence, relations are found and discussed among the several potentials in a family. (Author) [pt
International Nuclear Information System (INIS)
Lima, M.L.; Mignaco, J.A.
1985-01-01
It is shown that the rational power law potentials in the two-body radial Schoedinger equation admit a systematic treatment available from the classical theory of ordinary linear differential equations of the second order. The admissible potentials come into families evolved from equations having a fixed number of elementary singularities. As a consequence, relations are found and discussed among the several potentials in a family. (Author) [pt
Unifying quanta and relativity. Schroedinger`s attitude to relativistic quantum mechanics
Energy Technology Data Exchange (ETDEWEB)
Kragh, H. [Roskilde Universitetscenter (Denmark)
1992-12-31
A considerable part of Schroedinger`s scientific work focused on the relationship between quantum theory and the theory of relativity. This paper provides a historical analysis of his occupation on this subject in the period 1925-1934. The first section surveys the role played by relativity in Schroedinger`s formation of wave mechanics in 1925-1926; the second section analyzes his attempt to make sense of Dirac`s theory of the electron by proposing a relativistic wave equation with positive energies only. In this work, which took place in 1930-1931, Schroedinger discovered the Zitterbewegung that Dirac electrons will exhibit even in a field-free case. Schroedinger`s failed attempt to introduce an alternative to the Dirac theory was part of his general dissatisfaction with the current state of quantum mechanics. It is argued that, to a large extent, his work on the Dirac theory was philosophically motivated and that it contributed to his alienation from mainstream quantum physics in the 1930s. (author). 54 refs.
Energy Technology Data Exchange (ETDEWEB)
Arvieu, R.; Carbonell, J.; Gignoux, C.; Mangin-Brinet, M. [Inst. des Sciences Nucleaires, Grenoble-1 Univ., 38 (France); Rozmej, P. [Uniwersytet Marii Curie-Sklodowskiej, Lublin (Poland)
1997-12-31
The time evolution of coherent rotational wave packets associated to a diatomic molecule or to a deformed nucleus has been studied. Assuming a rigid body dynamics the J(J+1) law leads to a mechanism of cloning: the way function is divided into wave packets identical to the initial one at specific time. Applications are studied for a nuclear wave packed formed by Coulomb excitation. Exact boundary conditions at finite distance for the solution of the time-dependent Schroedinger equation are derived. A numerical scheme based on Crank-Nicholson method is proposed to illustrate its applicability in several examples. (authors) 3 refs.
Some spectral equivalences between Schroedinger operators
International Nuclear Information System (INIS)
Dunning, C; Hibberd, K E; Links, J
2008-01-01
Spectral equivalences of the quasi-exactly solvable sectors of two classes of Schroedinger operators are established, using Gaudin-type Bethe ansatz equations. In some instances the results can be extended leading to full isospectrality. In this manner we obtain equivalences between PT-symmetric problems and Hermitian problems. We also find equivalences between some classes of Hermitian operators
The Universe according to Schroedinger and Milo
Wolff, Milo
2009-10-01
The puzzling electron is due to the belief that it is a discrete particle. Schroedinger, (1937) eliminated discrete particles writing: What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). Thus he rejected wave-particle duality. Schroedinger's concept was developed by Milo Wolff using a Scalar Wave Equation in 3D quantum space to find wave solutions. The resulting Wave Structure of Matter (WSM) contains all the electron's properties including the Schroedinger Equation. Further, Newton's Law F=ma is no longer a puzzle; It originates from Mach's principle of inertia (1883) that depends on the space medium and the WSM. These the origin of all the Natural Laws. Carver Mead (1999) at CalTech used the WSM to design Intel micro-chips and to correct errors of Maxwell's Equations. Applications of the WSM describe matter at molecular dimensions: Industrial alloys, catalysts, biology and medicine, molecular computers and memories. See book ``Schroedinger's Universe'' - at Amazon.com. Pioneers of the WSM are growing rapidly. Some are: SpaceAndMotion.com, QuantumMatter.com, treeincarnation.com/audio/milowolff.htm, daugerresearch.com/orbitals/index.shtml, glafreniere.com/matter.html =A new Universe.
Schroedinger's Wave Structure of Matter (WSM)
Wolff, Milo; Haselhurst, Geoff
2009-10-01
The puzzling electron is due to the belief that it is a discrete particle. Einstein deduced this structure was impossible since Nature does not allow the discrete particle. Clifford (1876) rejected discrete matter and suggested structures in `space'. Schroedinger, (1937) also eliminated discrete particles writing: What we observe as material bodies and forces are nothing but shapes and variations in the structure of space. Particles are just schaumkommen (appearances). He rejected wave-particle duality. Schroedinger's concept was developed by Milo Wolff and Geoff Haselhurst (SpaceAndMotion.com) using the Scalar Wave Equation to find spherical wave solutions in a 3D quantum space. This WSM, the origin of all the Natural Laws, contains all the electron's properties including the Schroedinger Equation. The origin of Newton's Law F=ma is no longer a puzzle; It originates from Mach's principle of inertia (1883) that depends on the space medium and the WSM. Carver Mead (1999) at CalTech used the WSM to design Intel micro-chips correcting errors of Maxwell's magnetic Equations. Applications of the WSM also describe matter at molecular dimensions: alloys, catalysts, biology and medicine, molecular computers and memories. See ``Schroedinger's Universe'' - at Amazon.com
Crystallized Schroedinger cat states
International Nuclear Information System (INIS)
Castanos, O.; Lopez-Pena, R.; Man'ko, V.I.
1995-01-01
Crystallized Schroedinger cat states (male and female) are introduced on the base of extension of group construction for the even and odd coherent states of the electromagnetic field oscillator. The Wigner and Q functions are calculated and some are plotted for C 2 , C 3 , C 4 , C 5 , C 3v Schroedinger cat states. Quadrature means and dispersions for these states are calculated and squeezing and correlation phenomena are studied. Photon distribution functions for these states are given explicitly and are plotted for several examples. A strong oscillatory behavior of the photon distribution function for some field amplitudes is found in the new type of states
Almost periodic Schroedinger operators
International Nuclear Information System (INIS)
Bellissard, J.; Lima, R.
1984-01-01
These lectures are devoted to recent developments in the theory of almost-periodic Schroedinger Operators. We specially describe the algebraic point of view, with applications to gap-labelling theorems. Particular models are also presented which exhibit various spectral properties. (orig.)
Improved algorithm for solving nonlinear parabolized stability equations
Zhao, Lei; Zhang, Cun-bo; Liu, Jian-xin; Luo, Ji-sheng
2016-08-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. Project supported by the National Natural Science Foundation of China (Grant Nos. 11332007 and 11402167).
Improved algorithm for solving nonlinear parabolized stability equations
International Nuclear Information System (INIS)
Zhao Lei; Zhang Cun-bo; Liu Jian-xin; Luo Ji-sheng
2016-01-01
Due to its high computational efficiency and ability to consider nonparallel and nonlinear effects, nonlinear parabolized stability equations (NPSE) approach has been widely used to study the stability and transition mechanisms. However, it often diverges in hypersonic boundary layers when the amplitude of disturbance reaches a certain level. In this study, an improved algorithm for solving NPSE is developed. In this algorithm, the mean flow distortion is included into the linear operator instead of into the nonlinear forcing terms in NPSE. An under-relaxation factor for computing the nonlinear terms is introduced during the iteration process to guarantee the robustness of the algorithm. Two case studies, the nonlinear development of stationary crossflow vortices and the fundamental resonance of the second mode disturbance in hypersonic boundary layers, are presented to validate the proposed algorithm for NPSE. Results from direct numerical simulation (DNS) are regarded as the baseline for comparison. Good agreement can be found between the proposed algorithm and DNS, which indicates the great potential of the proposed method on studying the crossflow and streamwise instability in hypersonic boundary layers. (paper)
On the recovering of a coupled nonlinear Schroedinger potential
Energy Technology Data Exchange (ETDEWEB)
Corona, Gulmaro Corona [Area de Analisis Matematico y sus Aplicaciones, Universidad Autonoma Metropolitana, Atzcapotzalco, DF (Mexico)]. E-mail: ccg@hp9000a1.uam.mx
2000-04-28
We establish a priori conditions for a Gel'fand-Levitan (GL) integral using some results of the Fredholm theory. As consequence, we obtain a recovering formula for the potential of the coupled nonlinear Schroedinger equations. The remarkable fact is that the recovering formula is given in terms of the solutions of a classical GL-integral equation. (author)
An introduction to the self-adjointness and spectral analysis of Schroedinger operators
International Nuclear Information System (INIS)
Simon, B.
1977-01-01
The author first explains the basic results about self adjointness, from a point of view which emphasizes the connection with solvability of the Schroedinger equation. He then describes four methods that define self ajoint Hamiltonians, for most Schroedinger operators and discusses types of spectra, closing by considering the essential spectrum in the two body case. (P.D.)
Energy Technology Data Exchange (ETDEWEB)
Asselmeyer, T.
1997-12-22
First we introduce a simple model for the description of evolutionary algorithms, which is based on 2nd order partial differential equations for the distribution function of the individuals. Then we turn to the properties of Boltzmann's and Darwin's strategy. the next chapter is dedicated to the mathematical properties of Schroedinger operators. Both statements on the spectral density and their reproducibility during the simulation are summarized. The remaining of this chapter are dedicated to the analysis of the kernel as well as the dependence of the Schroedinger operator on the potential. As conclusion from the results of this chapter we obtain the classification of the strategies in dependence of the fitness. We obtain the classification of the evolutionary strategies, which are described by a 2nd order partial differential equation, in relation to their solution behaviour. Thereafter we are employed with the variation of the mutation distribution.
Philosophy of Erwin Schroedinger: a diachronic view of Schroedinger's thoughts
International Nuclear Information System (INIS)
Melgar, M.F.
1988-01-01
There is no agreement within the scientific community about the philosophy of Schroedinger. Some people think that he was a realist, while others defend him as an idealist. In this paper we study a number of Schroedinger's works and we show that the epithets of realist and idealist do not do him justice. Toward the end we conclude that it would be more adequate to place him in the trend known as the philosophy of immanence
Mobile localization in nonlinear Schroedinger lattices
International Nuclear Information System (INIS)
Gomez-Gardenes, J.; Falo, F.; Floria, L.M.
2004-01-01
Using continuation methods from the integrable Ablowitz-Ladik lattice, we have studied the structure of numerically exact mobile discrete breathers in the standard discrete nonlinear Schroedinger equation. We show that, away from that integrable limit, the mobile pulse is dressed by a background of resonant plane waves with wavevectors given by a certain selection rule. This background is seen to be essential for supporting mobile localization in the absence of integrability. We show how the variations of the localized pulse energy during its motion are balanced by the interaction with this background, allowing the localization mobility along the lattice
Properties of squeezed Schroedinger cats
International Nuclear Information System (INIS)
Obada, A.S.F.; Omar, Z.M.
1995-09-01
In this article we investigate some statistical properties of the even and odd squeezed (squeezed Schroedinger cat) states. The quasi-probability distribution functions especially W(α) and Q(α) are calculated and discussed for these states. The phase distribution function is discussed. A generation scheme is proposed for either the squeezed generalized Schroedinger cat, or the squeezed number state. (author). 35 refs, 5 figs
Localization for off-diagonal disorder and for continuous Schroedinger operators
International Nuclear Information System (INIS)
Delyon, F.; Souillard, B.; Simon, B.
1987-01-01
We extend the proof of localization by Delyon, Levy, and Souillard to accommodate the Anderson model with off-diagonal disorder and the continuous Schroedinger equation with a random potential. (orig.)
Generalized non-linear Schroedinger hierarchy
International Nuclear Information System (INIS)
Aratyn, H.; Gomes, J.F.; Zimerman, A.H.
1994-01-01
The importance in studying the completely integrable models have became evident in the last years due to the fact that those models present an algebraic structure extremely rich, providing the natural scenery for solitons description. Those models can be described through non-linear differential equations, pseudo-linear operators (Lax formulation), or a matrix formulation. The integrability implies in the existence of a conservation law associated to each of degree of freedom. Each conserved charge Q i can be associated to a Hamiltonian, defining a time evolution related to to a time t i through the Hamilton equation ∂A/∂t i =[A,Q i ]. Particularly, for a two-dimensions field theory, infinite degree of freedom exist, and consequently infinite conservation laws describing the time evolution in space of infinite times. The Hamilton equation defines a hierarchy of models which present a infinite set of conservation laws. This paper studies the generalized non-linear Schroedinger hierarchy
International Nuclear Information System (INIS)
Da Costa, N.C.A.; Krause, D.; French, S.
1992-01-01
Schroedinger introduced discussions about the inconsistency between the classical conception of particles as individual entities and the way in which modern physics treats such particles. In particular, it is noted that quantal particles apparently appear to lack individuality, and that certain suppositions of quantum theory imply that permutations of 'identical' particles are not regarded as observable, hence implying that they must be taken as 'non-individuals' of some kind. An overview is presented in this paper of some results obtained by the authors in the field of non-reflexive logics, which have some bearings on these problems and which can perhaps provide an adequate mathematical tool for dealing with some of the fundamental features of elementary particles, such as for instance the fact that identity apparently lacks sense with respect to them, that particle permutations are not regarded as observable and that a collection of these entities cannot be considered as a set in the sense of the usual theories of sets. The main objective of the paper is to show that the nature of elementary particles can be described in terms of certain non-classical logics, despite the problems regarding their individuality. (authors). 28 refs
A life of Erwin Schroedinger; Erwin Schroedinger. Eine Biographie
Energy Technology Data Exchange (ETDEWEB)
Moore, Walter J.
2012-07-01
Erwin Schroedinger (1887-1961) was a pioneer of quantum physics, one of the most important scientists of the 20th century at all and - a charming Austrian. He was a man with a passionate interest in people and ideas. Mostly known he became by his representation of quantum theory in the form of wave mechanics, for which he got the Nobel prize for physics and naturally by the famous thought experiment ''Schroedinger's cat''. Walter Moore's biography is very close to the person of Schroedinger and presents his scientific work in the context of his private friendships, his interest in mysticism, and in front of the moving background of the political events in Germany and Austria.
International Nuclear Information System (INIS)
Pando L, C.L.; Doedel, E.J.
2006-08-01
We investigate the nonlinear dynamics in a trimer, described by the one-dimensional discrete nonlinear Schrodinger equation (DNLSE), with periodic boundary conditions in the presence of a single on-site defect. We make use of numerical continuation to study different families of stationary and periodic solutions, which allows us to consider suitable perturbations. Taking into account a Poincare section, we are able to study the dynamics in both a thin stochastic layer solution and a global stochasticity solution. We find that the time series of the transit times, the time intervals to traverse some suitable sets in phase space, generate 1/f noise for both stochastic solutions. In the case of the thin stochastic layer solution, we find that transport between two almost invariant sets along with intermittency in small and large time scales are relevant features of the dynamics. These results are reflected in the behaviour of the standard map with suitable parameters. In both chaotic solutions, the distribution of transit times has a maximum and a tail with exponential decay in spite of the presence of long-range correlations in the time series. We motivate our study by considering a ring of weakly-coupled Bose-Einstein condensates (BEC) with attractive interactions, where inversion of populations between two spatially symmetric sites and phase locking take place in both chaotic solutions. (author)
Schroedinger's variational method of quantization revisited
International Nuclear Information System (INIS)
Yasue, K.
1980-01-01
Schroedinger's original quantization procedure is revisited in the light of Nelson's stochastic framework of quantum mechanics. It is clarified why Schroedinger's proposal of a variational problem led us to a true description of quantum mechanics. (orig.)
International Nuclear Information System (INIS)
Moore, Walter J.
2012-01-01
Erwin Schroedinger (1887-1961) was a pioneer of quantum physics, one of the most important scientists of the 20th century at all and - a charming Austrian. He was a man with a passionate interest in people and ideas. Mostly known he became by his representation of quantum theory in the form of wave mechanics, for which he got the Nobel prize for physics and naturally by the famous thought experiment ''Schroedinger's cat''. Walter Moore's biography is very close to the person of Schroedinger and presents his scientific work in the context of his private friendships, his interest in mysticism, and in front of the moving background of the political events in Germany and Austria.
Schroedinger and the wave mechanics
International Nuclear Information System (INIS)
Bassalo, J.M.F.
1987-01-01
In commemoration of the centennial of Schroedinger's birth, in 1987, we show in this paper some aspects of his academic life, and his philosophical and scientific work. Among Schroedinger's innumerable contributions to almost all areas of philosophy and science, we choose here the creation of quantum mechanics (1926), considered one of the pillars of Modern quantum theory, and the importance of his philosophical essay What is life (1944). This publication was responsible for a great in the studies of biology, culminating in the discovery of the DNA molecular structure, in 1953, by Crick and Watson, thanks to the X-rays diffraction technique of the DNA developed by Wilkens. (author) [pt
Approximated solutions to the Schroedinger equation
International Nuclear Information System (INIS)
Rico, J.F.; Fernandez-Alonso, J.I.
1977-01-01
The authors are currently working on a couple of the well-known deficiencies of the variation method and present here some of the results that have been obtained so far. The variation method does not give information a priori on the trial functions best suited for a particular problem nor does it give information a posteriori on the degree of precision attained. In order to clarify the origin of both difficulties, a geometric interpretation of the variation method is presented. This geometric interpretation is the starting point for the exact formal solution to the fundamental state and for the step-by-step approximations to the exact solution which are also given. Some comments on these results are included. (Auth.)
Derivation of an applied nonlinear Schroedinger equation
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Pitts, Todd Alan [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Laine, Mark Richard [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Schwarz, Jens [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Rambo, Patrick K. [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Karelitz, David B. [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
2015-01-01
We derive from first principles a mathematical physics model useful for understanding nonlinear optical propagation (including filamentation). All assumptions necessary for the development are clearly explained. We include the Kerr effect, Raman scattering, and ionization (as well as linear and nonlinear shock, diffraction and dispersion). We explain the phenomenological sub-models and each assumption required to arrive at a complete and consistent theoretical description. The development includes the relationship between shock and ionization and demonstrates why inclusion of Drude model impedance effects alters the nature of the shock operator. Unclassified Unlimited Release
The recursive solution of the Schroedinger equation
International Nuclear Information System (INIS)
Haydock, R.
The transformation of an arbitrary quantum model and its subsequent analysis is proposed. The chain expresses mathematically the physical concept of local environment. The recursive transformation yields analytic chains for some systems, but it is also convenient and efficient for constructing numerical chain models enabling the solution of problems which are too big for numerical matrix methods. The chain model sugests new approach to quantum mechanical models. Because of the simple solution of chain models, the qualitative behaviour of different physical properties can be determined. Unlike many methods for solving quantum models, one has rigorous results about the convergence of approximation. Because they are defined recursively, the approsimations are suited to computation. (Ha)
Integrability in the theory of Schroedinger operator and harmonic analysis
International Nuclear Information System (INIS)
Chalykh, O.A.; Veselov, A.P.
1993-01-01
The algebraic integrability for the Schroedinger equation in R n and the role of the quantum Calogero-Sutherland problem and root systems in this context are discussed. For the special values of the parameters in the potential the explicit formula for the eigenfunction of the corresponding Sutherland operator is found. As an application the explicit formula for the zonal spherical functions on the symmetric spaces SU 2 * n /Sp n (type A II in Cartan notations) is presented. (orig.)
Nonlinear von Neumann equations for quantum dissipative systems
International Nuclear Information System (INIS)
Messer, J.; Baumgartner, B.
1978-01-01
For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Auth.)
Nonlinear von Neumann equations for quantum dissipative systems
International Nuclear Information System (INIS)
Messer, J.; Baumgartner, B.
For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Author)
In search of Schroedinger's cat
International Nuclear Information System (INIS)
Gribbin, John.
1984-01-01
The book explains how the paradox of Schroedinger's cat led to an understanding of reality in quantum physics. The contents of the book is divided into three parts. Part one concerns light, atoms and Bohr's atom. Quantum mechanics is discussed in Part Two, including photons and electrons, matrices and waves, and applications of quanta. The last part deals with chance and uncertainty, paradoxes and possibilities, the experimental proof of the paradoxical reality of the quantum world, and the many-worlds interpretation of quantum mechanics. (U.K.)
Introduction to Schroedinger inverse scattering
International Nuclear Information System (INIS)
Roberts, T.M.
1991-01-01
Schroedinger inverse scattering uses scattering coefficients and bound state data to compute underlying potentials. Inverse scattering has been studied extensively for isolated potentials q(x), which tend to zero as vertical strokexvertical stroke→∞. Inverse scattering for isolated impurities in backgrounds p(x) that are periodic, are Heaviside steps, are constant for x>0 and periodic for x<0, or that tend to zero as x→∞ and tend to ∞ as x→-∞, have also been studied. This paper identifies literature for the five inverse problems just mentioned, and for four other inverse problems. Heaviside-step backgrounds are discussed at length. (orig.)
Connection between Dirac and matrix Schroedinger inverse-scattering transforms
International Nuclear Information System (INIS)
Jaulent, M.; Leon, J.J.P.
1978-01-01
The connection between two applications of the inverse scattering method for solving nonlinear equations is established. The inverse method associated with the massive Dirac system (D) : (iσ 3 d/dx - i q 3 σ 1 - q 1 σ 2 + mσ 2 )Y = epsilonY is rediscovered from the inverse method associated with the 2 x 2 matrix Schroedinger equation (S) : Ysub(xx) + (k 2 -Q)Y = 0. Here Q obeys a nonlinear constraint equivalent to a linear constraint on the reflection coefficient for (S). (author)
Continuity relations and quantum wave equations
International Nuclear Information System (INIS)
Goedecke, G.H.; Davis, B.T.
2010-01-01
We investigate the mathematical synthesis of the Schroedinger, Klein-Gordon, Pauli-Schroedinger, and Dirac equations starting from probability continuity relations. We utilize methods similar to those employed by R. E. Collins (Lett. Nuovo Cimento, 18 (1977) 581) in his construction of the Schroedinger equation from the position probability continuity relation for a single particle. Our new results include the mathematical construction of the Pauli-Schroedinger and Dirac equations from the position probability continuity relations for a particle that can transition between two states or among four states, respectively.
Some threshold spectral problems of Schroedinger operators
International Nuclear Information System (INIS)
Jia, X.
2009-01-01
This Ph.D. thesis deals with some spectral problems of the Schroedinger operators. We first consider the semi-classical limit of the number of bound states of unique two-cluster N-body Schroedinger operator. Then we use Dirichlet-Neumann bracket to get semi-classical limit of Riesz means of the discrete eigenvalues of N-body Schroedinger operator. The effective potential of N-body Schroedinger operator with Coulomb potential is also considered and we find that the effective potential has critical decay at infinity. Thus, the Schroedinger operator with critical potential is studied in this thesis. We study the coupling constant threshold of Schroedinger operator with critical potential and the asymptotic expansion of resolvent of Schroedinger operator with critical potential. We use that expansion to study low-energy asymptotics of derivative of spectral shift function for perturbation with critical decay. After that, we use this result and the known result for high-energy asymptotic expansion of spectral shift function to obtain the Levinson theorem. (author)
Schroedinger representation in quantum field theory
International Nuclear Information System (INIS)
Luescher, M.
1985-01-01
Until recently, the Schroedinger representation in quantum field theory had not received much attention, even more so because there were reasons to believe that in the presence of interactions it did not exist in a mathematically well-defined sense. When Symanzik set out to solve this problem, he was motivated by a special 2-dimensional case, the relativistic string model, in which the Schroedinger wave functionals are the primary objects of physical interest. Also, he knew that if it were possible to demonstrate the existence of the Schroedinger representation, the (then unproven) ultraviolet finiteness of the Casimir force in renormalizable quantum field theories would probably follow. (orig./HSI)
Erwin Schroedinger, Francis Crick and epigenetic stability
Directory of Open Access Journals (Sweden)
Ogryzko Vasily V
2008-04-01
Full Text Available Abstract Schroedinger's book 'What is Life?' is widely credited for having played a crucial role in development of molecular and cellular biology. My essay revisits the issues raised by this book from the modern perspective of epigenetics and systems biology. I contrast two classes of potential mechanisms of epigenetic stability: 'epigenetic templating' and 'systems biology' approaches, and consider them from the point of view expressed by Schroedinger. I also discuss how quantum entanglement, a nonclassical feature of quantum mechanics, can help to address the 'problem of small numbers' that led Schroedinger to promote the idea of a molecular code-script for explaining the stability of biological order.
Erwin Schroedinger, Francis Crick and epigenetic stability.
Ogryzko, Vasily V
2008-04-17
Schroedinger's book 'What is Life?' is widely credited for having played a crucial role in development of molecular and cellular biology. My essay revisits the issues raised by this book from the modern perspective of epigenetics and systems biology. I contrast two classes of potential mechanisms of epigenetic stability: 'epigenetic templating' and 'systems biology' approaches, and consider them from the point of view expressed by Schroedinger. I also discuss how quantum entanglement, a nonclassical feature of quantum mechanics, can help to address the 'problem of small numbers' that led Schroedinger to promote the idea of a molecular code-script for explaining the stability of biological order.
International Nuclear Information System (INIS)
Kravchenko, Viktor G; Kravchenko, Vladislav V
2003-01-01
We show that an ample class of physically meaningful partial differential systems of first order such as the Dirac equation with different one-component potentials, static Maxwell's system and the system describing the force-free magnetic fields are equivalent to a single quaternionic equation which in its turn reduces in general to a Schroedinger equation with quaternionic potential, and in some situations this last can be diagonalized. The rich variety of methods developed for different problems corresponding to the Schroedinger equation can be applied to the systems considered in the present work
Energy Technology Data Exchange (ETDEWEB)
Kravchenko, Viktor G [Faculdade de Ciencias y Tecnologia, Universidade do Algarve, Campus de Gambelas, 8000 Faro (Portugal); Kravchenko, Vladislav V [Depto de Telecomunicaciones, SEPI ESIME Zacatenco, Instituto Politecnico Nacional, Av. IPN S/N, Edif. 1 CP 07738, DF (Mexico)
2003-11-07
We show that an ample class of physically meaningful partial differential systems of first order such as the Dirac equation with different one-component potentials, static Maxwell's system and the system describing the force-free magnetic fields are equivalent to a single quaternionic equation which in its turn reduces in general to a Schroedinger equation with quaternionic potential, and in some situations this last can be diagonalized. The rich variety of methods developed for different problems corresponding to the Schroedinger equation can be applied to the systems considered in the present work.
Accurate high-lying eigenvalues of Schroedinger and Sturm-Liouville problems
International Nuclear Information System (INIS)
Vanden Berghe, G.; Van Daele, M.; De Meyer, H.
1994-01-01
A modified difference and a Numerov-like scheme have been introduced in a shooting algorithm for the determination of the (higher-lying) eigenvalues of Schroedinger equations and Sturm-Liouville problems. Some numerical experiments are introduced. Time measurements have been performed. The proposed algorithms are compared with other previously introduced shooting schemes. The structure of the eigenvalue error is discussed. ((orig.))
International Nuclear Information System (INIS)
Barut, A.O.
1990-06-01
For an arbitrary potential V with classical trajectories x-vector=g-vector(t) we construct localized oscillating three-dimensional wave lumps ψ(x-vector,t,g-vector) representing a single quantum particle. The crest of the envelope of the ripple follows the classical orbit g-vector(t) slightly modified due to potential V and ψ(x-vector,t;g-vector) satisfies the Schroedinger equation. The field energy, momentum and angular momentum calculated as integrals over all space are equal to particle energy, momentum and angular momentum. The relation to coherent states and to Schroedinger waves are also discussed. (author). 6 refs
Some studies of Schroedinger operators
International Nuclear Information System (INIS)
Liu Yang.
1993-09-01
This thesis consists of three papers. Paper 1 deals with the semiclassical approximation for a Schroedinger operator in one dimension with an arbitrary continuous potential. The basic result is that function in the range of a spectral projection associated with an interval are exponentially small (with respect to Plancks parameter h) in places where the potential exceeds the interval. As an application of this result, it is shown that the eigenvalues of the operator localized to the wells appear as resonances of the global operator. This is true also when the potential is not bounded from below. Such results were proved before for analytic potentials by analytic dilatation. In Paper 2, the potential is assumed to have the form of χ + V(χ) (the Stark Hamiltonian) with a well-behaved V(χ), an explicit spectral and scattering theory for such an operator was presented using the time-independent approach. In particular, we derive an eigenfunction expansion theorem which, combined with a construction of an intertwining operator, gives a solution of the inverse scattering problem according to L. Daddeev and A. Melin. The direct part of the second paper has a generalization to higher dimensions, and this was done in the third paper. Also in that paper, the condition on the potentials for doing the inverse scattering theory was relaxed, and an explicit formula for the potentials involving the first approximation of the scattering data was given
Group-theoretical interpretation of the Korteweg-de Vries type equations
International Nuclear Information System (INIS)
Berezin, F.A.; Perelomov, A.M.
1978-01-01
The Korteweg-de Vries equation is studied within the group-theoretical framework. Analogous equations are obtained for which the many-dimensional Schroedinger equation (with nonlocal potential) plays the same role as the one-dimensional Schroedinger equation does in the theory of the Korteweg-de Vries equation
On the Schroedinger representation of the Euclidean quantum field theory
International Nuclear Information System (INIS)
Semmler, U.
1987-04-01
The theme of the present thesis is the Schroedinger representation of the Euclidean quantum field theory: We define the time development of the quantum field states as functional integral in a novel, mathematically precise way. In the following we discuss the consequences which result from this approach to the Euclidean quantum field theory. Chapter 1 introduces the theory of abstract Wiener spaces which is here proved as suitable mathematical tool for the treatment of the physical problems. In chapter 2 the diffusion theory is formulated in the framework of abstract Wiener spaces. In chapter 3 we define the field functional ψ 5 u, t 7 as functional integral, determine the functional differential equation which ψ satisfies (Schroedinger equation), and summarize the consequences resulting from this. Chapter 4 is dedicated to the attempt to determine the kernel of the time-development operator, by the knowledge of which the time development of each initial state is fixed. In chapter 5 the consequences of the theory presented in chapter 3 and 4 are discussed by means of simple examples. In chapter 6 the renormalization which results for the φ 4 potential from the definition of the functional integral in chapter 3 is calculated up to the first-order perturbation theory, and it is shown that the problems in the Symanzik renormalization procedure can be removed. (orig./HSI) [de
On the connection between Schroedinger- and Dirichlet forms
International Nuclear Information System (INIS)
Albeverio, S.; Bochum Univ.; Gesztesy, F.; Karwowski, W.; Streit, L.; Bielefeld Univ.
Relations between Schroedinger forms associated with Schroedinger operators in L 2 (Ω;dsup(n)x), Ω is contained in Rsup(n) open, n >= 1 and the corresponding Dirichlet forms are investigated. Various concrete examples are presented. (orig.)
Single-particle Schroedinger fluid. I. Formulation
International Nuclear Information System (INIS)
Kan, K.K.; Griffin, J.J.
1976-01-01
The problem of a single quantal particle moving in a time-dependent external potential well is formulated specifically to emphasize and develop the fluid dynamical aspects of the matter flow. This idealized problem, the single-particle Schroedinger fluid, is shown to exhibit already a remarkably rich variety of fluid dynamical features, including compressible flow and line vortices. It provides also a sufficient framework to encompass simultaneously various simplified fluidic models for nuclei which have earlier been postulated on an ad hoc basis, and to illuminate their underlying restrictions. Explicit solutions of the single-particle Schroedinger fluid problem are studied in the adiabatic limit for their mathematical and physical implications (especially regarding the collective kinetic energy). The basic generalizations for extension of the treatment to the many-body Schroedinger fluid are set forth
A life of Erwin Schroedinger. 2. ed.
International Nuclear Information System (INIS)
Moore, Walter J.
2015-01-01
Erwin Schroedinger (1887-1961) was a pioneer of quantum physics, one of the most important scientist of the 20th century at all and a charming Austrian. He was a man with a passionate interest for men and ideas. Mostly known he became by his representation of quantum theory in the form of wave mechanics, for which he obtained the Nobel prize for physics and naturally by the famous thought experiment ''Schroedingers cat''. Walter Moore's biography is quite near to the person of Schroedinger and presents his scientific work in the context of his friendships, his interset for mysticism, and in front of the moving background of the political events in Germany and Austria.
Abelian Higgs mechanism in the Schroedinger picture
International Nuclear Information System (INIS)
Kim, S.K.; Namgung, W.; Soh, K.S.; Yee, J.H.
1990-01-01
We have studied symmetry-breaking phenomena in scalar electrodynamics by evaluating the effective potential at one-loop order in the Schroedinger picture. Contributions to the effective potential by the Higgs particle and the transverse and longitudinal components of a photon are compared with other previous works, and they are found to be consistent
Bound states for non-symmetric evolution Schroedinger potentials
Energy Technology Data Exchange (ETDEWEB)
Corona, Gulmaro Corona [Area de Analisis Matematico y sus Aplicaciones, Universidad Autonoma Metropolitana-Azcapotalco, Atzcapotzalco, DF (Mexico)). E-mail: ccg@correo.azc.uam.mx
2001-09-14
We consider the spectral problem associated with the evolution Schroedinger equation, (D{sup 2}+ k{sup 2}){phi}=u{phi}, where u is a matrix-square-valued function, with entries in the Schwartz class defined on the real line. The solution {phi}, called the wavefunction, consists of a function of one real variable, matrix-square-valued with entries in the Schwartz class. This problem has been dealt for symmetric potentials u. We found for the present case that the bound states are localized similarly to the scalar and symmetric cases, but by the zeroes of an analytic matrix-valued function. If we add an extra condition to the potential u, we can determine these states by an analytic scalar function. We do this by generalizing the scalar and symmetric cases but without using the fact that the Wronskian of a pair of wavefunction is constant. (author)
Collective states of externally driven, damped nonlinear Schroedinger solitons
International Nuclear Information System (INIS)
Barashenkov, I.V.; Smirnov, Yu.S.
1997-01-01
We study bifurcations of localized stationary solitons of the externally driven, damped nonlinear Schroedinger equation iΨ t + Ψ xx + 2|Ψ| 2 Ψ=-iγΨ-h e iΩt , in the region of large γ (γ>1/2). For each pair of h and γ, there are two coexisting solitons, Ψ + and Ψ - . As the driver's strength h increases for the fixed γ, the Ψ + soliton merges with the flat background while the Ψ - forms a stationary collective state with two 'psi-pluses': Ψ - → Ψ (+ - +) . We obtain other stationary solutions and identify them as multisoliton complexes Ψ (++) , Ψ (--) , Ψ (-+) , Ψ (---) , Ψ (-+- ) etc. The corresponding intersoliton separations are compared to predictions of a variational approximation
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Amos, K.; Allen, L.J.; Steward, C.; Hodgson, P.E.; Sofianos, S.A.
1995-01-01
Direct solution of the Schroedinger equation and inversion methods of analysis of elastic scattering data are considered to evaluate the information that they can provide about the physical interaction between colliding nuclear particles. It was found that both optical model and inversion methods based upon inverse scattering theories are subject to ambiguities. Therefore, it is essential that elastic scattering data analyses are consistent with microscopic calculations of the potential. 25 refs
Energy Technology Data Exchange (ETDEWEB)
Amos, K.; Allen, L.J.; Steward, C. [Melbourne Univ., Parkville, VIC (Australia). School of Physics; Hodgson, P.E. [Oxford Univ. (United Kingdom). Dept. of Physics; Sofianos, S.A. [University of South Africa (UNISA), Pretoria (South Africa). Dept. of Physics
1995-10-01
Direct solution of the Schroedinger equation and inversion methods of analysis of elastic scattering data are considered to evaluate the information that they can provide about the physical interaction between colliding nuclear particles. It was found that both optical model and inversion methods based upon inverse scattering theories are subject to ambiguities. Therefore, it is essential that elastic scattering data analyses are consistent with microscopic calculations of the potential. 25 refs.
Considerations on Bohr's, Heisenberg's and Schroedinger's philosophy
International Nuclear Information System (INIS)
Shimony, A.
1981-01-01
In denying that the words 'physical reality' are meaningful without reference to an experimental arrangement, Bohr renounces any knowledge of the 'thing-in-itself'. However, the relation of his epistemology to both idealism and positivism remains obscure. Heisenberg departs from Bohr in enunciating a metaphysical implication of quantum mechanics. Heisenberg asserts that there is an intermediate modality -potentiality- between logical possibility and existence. His attempts to explain the transition from potentiality to existence are not convincing. Schroedinger rejects Bohr's interpretation of quantum mechanics as a positivist exercise and seeks instead a realist interpretation. Nevertheless, the metaphysics of Schroedinger is fundamentally idealistic, maintaining that the material aspect of the world is composed of the same elements as mind, but in a different order [fr
Schroedinger and the interpretation of quantum mechanics
International Nuclear Information System (INIS)
Rohrlich, F.
1987-01-01
On the occasion of the centennial of his birth, Schroedinger's life and views are sketched and his critique of the interpretation of quantum mechanics accepted at his time is examined. His own interpretation, which he had to abandon after a short time, provides a prime example of the way in which the tentative meaning of central theoretical terms in a new and revolutionary theory often fails. Schroedinger's strong philosophical convictions have played a key role in his refusal to break with many of the notions of classical physics. At the same time, they made him into a keen and incisive critic of the Copenhagen interpretation. His criticism is compared with present views on quantum mechanics
Completely integrable operator evolutionary equations
International Nuclear Information System (INIS)
Chudnovsky, D.V.
1979-01-01
The authors present natural generalizations of classical completely integrable equations where the functions are replaced by arbitrary operators. Among these equations are the non-linear Schroedinger, the Korteweg-de Vries, and the modified KdV equations. The Lax representation and the Baecklund transformations are presented. (Auth.)
Exchange effects in Relativistic Schroedinger Theory
International Nuclear Information System (INIS)
Sigg, T.; Sorg, M.
1998-01-01
The Relativistic Schroedinger Theory predicts the occurrence of exchange and overlap effects in many-particle systems. For a 2-particle system, the interaction energy of the two particles consists of two contributions: Coulomb energy and exchange energy, where the first one is revealed to be the same as in standard quantum theory. However the exchange energy is mediated by an exchange potential, contrary to the kinematical origin of the exchange term in the standard theory
Schroedinger covariance states in anisotropic waveguides
International Nuclear Information System (INIS)
Angelow, A.; Trifonov, D.
1995-03-01
In this paper Squeezed and Covariance States based on Schroedinger inequality and their connection with other nonclassical states are considered for particular case of anisotropic waveguide in LiNiO 3 . Here, the problem of photon creation and generation of squeezed and Schroedinger covariance states in optical waveguides is solved in two steps: 1. Quantization of electromagnetic field is provided in the presence of dielectric waveguide using normal-mode expansion. The photon creation and annihilation operators are introduced, expanding the solution A-vector(r-vector,t) in a series in terms of the Sturm - Liouville mode-functions. 2. In terms of these operators the Hamiltonian of the field in a nonlinear waveguide is derived. For such Hamiltonian we construct the covariance states as stable (with nonzero covariance), which minimize the Schroedinger uncertainty relation. The evolutions of the three second momenta of q-circumflex j and p-circumflex j are calculated. For this Hamiltonian all three momenta are expressed in terms of one real parameters s only. It is found out how covariance, via this parameter s, depends on the waveguide profile n(x,y), on the mode-distributions u-vector j (x,y), and on the waveguide phase mismatching Δβ. (author). 37 refs
A life of Erwin Schroedinger. 2. ed.; Erwin Schroedinger. Eine Biographie
Energy Technology Data Exchange (ETDEWEB)
Moore, Walter J.
2015-07-01
Erwin Schroedinger (1887-1961) was a pioneer of quantum physics, one of the most important scientist of the 20th century at all and a charming Austrian. He was a man with a passionate interest for men and ideas. Mostly known he became by his representation of quantum theory in the form of wave mechanics, for which he obtained the Nobel prize for physics and naturally by the famous thought experiment ''Schroedingers cat''. Walter Moore's biography is quite near to the person of Schroedinger and presents his scientific work in the context of his friendships, his interset for mysticism, and in front of the moving background of the political events in Germany and Austria.
Exact solutions to the Lienard equation and its applications
International Nuclear Information System (INIS)
Feng Zhaosheng
2004-01-01
In this paper, a kind of explicit exact solutions to the Lienard equation is obtained, and the applications of the result in seeking traveling solitary wave solution of the nonlinear Schroedinger equation are presented
Relativistic supersymmetric quantum mechanics based on Klein-Gordon equation
International Nuclear Information System (INIS)
Znojil, Miloslav
2004-01-01
Witten's the non-relativistic formalism of supersymmetric quantum mechanics was based on a factorization and partnership between Schroedinger equations. We show how it accommodates a transition to the partnership between relativistic Klein-Gordon equations
International Nuclear Information System (INIS)
Myrheim, J.
1993-06-01
The thesis deals with the application of different methods to the quantization problem for system of identical particles in one and two dimensions. The standard method is the analytic quantization method due to Schroedinger, which leads to the concept of fractional statistics in one and two dimensions. Two-dimensional particles with fractional statistics are well known by the name of anyons. Two alternative quantization methods are shown by the author, the algebraic method of Heisenberg and the Feynman path integral method. The Feynman method is closely related to the Schroedinger method, whereas the Heisenberg and Schroedinger methods may give different results. The relation between the Heisenberg and Schroedinger methods is discussed. The Heisenberg method is applied to the equations of motion of vortices in superfluid helium, which have the form of Hamiltonian equations for a one-dimensional system. The same method is also discussed more generally for systems of identical particles in one and two dimensions. An application of the Feynman method to the problem of computing the equation of state for a gas of anyons is presented. 104 refs., 4 figs
International Nuclear Information System (INIS)
Meyenn, Karl von
2011-01-01
After Schroedinger has in the beginning of 1926 published his wave mechanics, he has by this opened many new physical views and perspectives, which have decidingly influenced the further development of quantum theory. Also today the Schroedinger equations forms the foundation of the whole microphysics and their far reaching applications. Therefore it is both for the scientist and for the interested layman very attractive to be informed by first hand about the more direct conditions and the problems of their origin. Letters of famous scientists and researchers have also in the past attracted the interest of the public, and many a scientist has been excited to the study by the lecture of such primary sources. The selection of about 300 letters presented here illuminates especially the origin of wave mechanics and their still controverse interpretation. An extensive introduction, comments, remarks, illustrations, and lists establish the physical and historical relations.
One-dimensional Schroedinger equation as a classical dinamical problem
International Nuclear Information System (INIS)
Sanjines C, D.
1990-01-01
The analogy between the determination of the energy spectrum for periodic and localized 1 dimensional potentials and the stability for a particle under the influence of elastic forces is presented. For the particle to be confined in a bounded region of the phase space, it is necessary that the trace of the evolution matrix over a period of the periodic potential belongs to the interval (-2,2) [3]. Curiously, the same stability criterion might be applied to localized potentials and then determine the discrete spectrum for such potentials. We have found that for either periodic and localized potentials, the classical dynamical picture is more clarifying and the results are more compact. (Author)
Travelling Solitons in the Damped Driven Nonlinear Schroedinger Equation
Barashenkov, I V
2003-01-01
The well-known effect of the linear damping on the moving nonlinear Schrodinger soliton (even when there is energy supply via the spatially homogeneous driving) is to quench its momentum to zero. Surprisingly, the zero momentum does not necessarily mean zero velocity. We show that two or more parametrically driven damped solitons can form a complex travelling with zero momentum at a nonzero constant speed. All travelling complexes we have found so far, turned out to be unstable. Thus, the parametric driving is capable of sustaining the uniform motion of damped solitons, but some additional agent is required to make this motion stable.
Numerical methods for differential equations and applications
International Nuclear Information System (INIS)
Ixaru, L.G.
1984-01-01
This book is addressed to persons who, without being professionals in applied mathematics, are often faced with the problem of numerically solving differential equations. In each of the first three chapters a definite class of methods is discussed for the solution of the initial value problem for ordinary differential equations: multistep methods; one-step methods; and piecewise perturbation methods. The fourth chapter is mainly focussed on the boundary value problems for linear second-order equations, with a section devoted to the Schroedinger equation. In the fifth chapter the eigenvalue problem for the radial Schroedinger equation is solved in several ways, with computer programs included. (Auth.)
Interaction of Schroedinger electrons and photons
International Nuclear Information System (INIS)
Haller, K.; Sohn, R.B.
1979-01-01
The effect of transformations carried out on the Hamiltonian for the Schroedinger electron-photon system is studied. These transformations include gauge transformations and certain similarity and ''hybrid'' transformations. The last named involve unitary transformations of either operators or states, but not both. Unitary and hybrid transformation are discussed, which affect the transverse components of the electromagnetic vector potentials and therefore are distinct from gauge transformations. A hybrid transformation is identified which leads to a form of the Hamiltonian that contains no reference to the transverse vector potential and includes electric and magnetic fields as well as nonlocal interactions of charges and currents. The behavior of the scattering matrix under the influence of these hybrid transformations is discussed. Comments are made on two-photon absorption calculations
Formalism and physical interpretation in Schroedinger
International Nuclear Information System (INIS)
Paty, M.
1992-01-01
The question of the relation between a formalism and its physical interpretation arises not only when theoretical and conceptual systems are reorganized, but in the theoretical elaboration as well. The Schroedinger's work and thought are examined in this paper with this double concern. His work on the mathematical formalism is constantly sustained by a proper physical thought which takes the form of a wave intuition that guarantees him intelligibility. Concerning his interpretation of quantum mechanics, his thought remains characterized, through its evolution, by a w ave image of the world . The way he deals with space-time structure in General Relativity and favours the possibility of a direct interpretation of space-time geometrical quantities, is also studied. (author). 75 refs
Completely integrable operator evolution equations. II
International Nuclear Information System (INIS)
Chudnovsky, D.V.
1979-01-01
The author continues the investigation of operator classical completely integrable systems. The main attention is devoted to the stationary operator non-linear Schroedinger equation. It is shown that this equation can be used for separation of variables for a large class of completely integrable equations. (Auth.)
Toward an AdS/cold atoms correspondence: A geometric realization of the Schroedinger symmetry
International Nuclear Information System (INIS)
Son, D. T.
2008-01-01
We discuss a realization of the nonrelativistic conformal group (the Schroedinger group) as the symmetry of a spacetime. We write down a toy model in which this geometry is a solution to field equations. We discuss various issues related to nonrelativistic holography. In particular, we argue that free fermions and fermions at unitarity correspond to the same bulk theory with different choices for the near-boundary asymptotics corresponding to the source and the expectation value of one operator. We describe an extended version of nonrelativistic general coordinate invariance which is realized holographically.
Schroedinger operators - geometric estimates in terms of the occupation time
International Nuclear Information System (INIS)
Demuth, M.; Kirsch, W.; McGillivray, I.
1995-01-01
The difference of Schroedinger and Dirichlet semigroups is expressed in terms of the Laplace transform of the Brownian motion occupation time. This implies quantitative upper and lower bounds for the operator norms of the corresponding resolvent differences. One spectral theoretical consequence is an estimate for the eigenfunction for a Schroedinger operator in a ball where the potential is given as a cone indicator function. 12 refs
International Nuclear Information System (INIS)
Maksudov, F.G.; Gusejnov, G.Sh.
1986-01-01
Inverse scattering problem for the quadratic bundle of the Schroedinger one-dimensional operators in the whole axis is solved. The problem solution is given on the assumption of the discrete spectrum absence. In the discrete spectrum presence the inverse scattering problem solution is known for the Shroedinger differential equation considered
Modified Darboux transformations with foreign auxiliary equations
International Nuclear Information System (INIS)
Schulze-Halberg, Axel
2011-01-01
We construct a new type of first-order Darboux transformations for the stationary Schroedinger equation. In contrast to the conventional case, our Darboux transformations support arbitrary (foreign) auxiliary equations. We show that among other applications, our formalism can be used to systematically construct Darboux transformations for Schroedinger equations with energy-dependent potentials, including a recent result (Lin et al., 2007) as a special case. -- Highlights: → We generalize the Darboux transformation for the Schroedinger equation. → By admitting arbitrary auxiliary functions, we provide a new tool for generating solutions. → As a special case we recover a recent result on energy-dependent potentials. → We extend the latter result to very general energy-dependence.
Singular continuous spectrum for palindromic Schroedinger operators
International Nuclear Information System (INIS)
Hof, A.; Knill, O.; Simon, B.
1995-01-01
We give new examples of discrete Schroedinger operators with potentials taking finitely many values that have purely singular continuous spectrum. If the hull X of the potential is strictly ergodic, then the existence of just one potential x in X for which the operator has no eigenvalues implies that there is a generic set in X for which the operator has purely singular continuous spectrum. A sufficient condition for the existence of such an x is that there is a z element of X that contains arbitrarily long palindromes. Thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset in X. The class includes well-known substitutions like Fibonacci, Thue-Morse, Period Doubling, binary non-Pisot and ternary non-Pisot. We also show that the operator has no absolutely continuous spectrum for all x element of X if X derives from a primitive substitution. For potentials defined by circle maps, x n =l J (θ 0 +nα), we show that the operator has purely singular continuous spectrum for a generic subset in X for all irrational α and every half-open interval J. (orig.)
Independent particle Schroedinger Fluid: moments of inertia
International Nuclear Information System (INIS)
Kan, K.K.; Griffin, J.J.
1977-10-01
This philosophy of the Single Particle Schroedinger Fluid, especially as regards the velocity fields which find such a natural role therein, is applied to the study of the moments of inertia of independent Fermion system. It is shown that three simplified systems exhibit the rigid-body rotational velocity field in the limit of large A, and that the leading deviations, both on the average and fluctuating, from this large A limit can be described analytically, and verified numerically. For a single particle in a Hill-Wheeler box the moments are studied numerically, and their large fluctuations identified with the specific energy level degeneracies of its parallelepiped shape. The full assemblage of these new and old results is addressed to the question of the necessary and sufficient condition that the moment have the rigid value. Counterexamples are utilized to reject some conditions, and the conjecture is argued that Unconstrained Shape Equilibrium might be the necessary and sufficient condition. The spheroidal square well problem is identified as a promising test case
International Nuclear Information System (INIS)
Zhalij, Alexander
2002-01-01
We classify (1+3)-dimensional Pauli equations for a spin-(1/2) particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the 11 classes of vector-potentials of the electro-magnetic field A(t,x(vector sign))=(A 0 (t,x(vector sign)), A(vector sign)(t,x(vector sign))) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x(vector sign)) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field
Wave equation of hydrogen atom
International Nuclear Information System (INIS)
Suwito.
1977-01-01
The calculation of the energy levels of the hydrogen atom using Bohr, Schroedinger and Dirac theories is reviewed. The result is compared with that obtained from infinite component wave equations theory which developed recently. The conclusion can be stated that the latter theory is better to describe the composit system than the former. (author)
Deriving average soliton equations with a perturbative method
International Nuclear Information System (INIS)
Ballantyne, G.J.; Gough, P.T.; Taylor, D.P.
1995-01-01
The method of multiple scales is applied to periodically amplified, lossy media described by either the nonlinear Schroedinger (NLS) equation or the Korteweg--de Vries (KdV) equation. An existing result for the NLS equation, derived in the context of nonlinear optical communications, is confirmed. The method is then applied to the KdV equation and the result is confirmed numerically
International Nuclear Information System (INIS)
Wadia, S.R.
1979-01-01
A detailed formulation of the quantum theory of non-abelian gauge fields is presented in the Schroedinger picture. It is applied to the semiclassical quantization of the t'Hoft-Polyakov monopole, with special attention paid to the treatment of boundary conditions and local and global gauge symmetry. The perturbation expansion is then discussed with the aid of standard collective co-ordinates. In the Prasad-Sommerfield limit, all the eigenfunctions of the fluctuation equation are presented, the ground-state wave function is constructed in terms of gauge and translation invariant co-ordinates, and its total angular momentum is computed to be zero. Aspects of instanton phenomena are then examined in the Schroedinger picture; the role of euclidean time is elucidated. The precise relation between boundary conditions, choice of gauge, and the corresponding picture of the semiclassical vacuum is demonstrated
Spectral Target Detection using Schroedinger Eigenmaps
Dorado-Munoz, Leidy P.
Applications of optical remote sensing processes include environmental monitoring, military monitoring, meteorology, mapping, surveillance, etc. Many of these tasks include the detection of specific objects or materials, usually few or small, which are surrounded by other materials that clutter the scene and hide the relevant information. This target detection process has been boosted lately by the use of hyperspectral imagery (HSI) since its high spectral dimension provides more detailed spectral information that is desirable in data exploitation. Typical spectral target detectors rely on statistical or geometric models to characterize the spectral variability of the data. However, in many cases these parametric models do not fit well HSI data that impacts the detection performance. On the other hand, non-linear transformation methods, mainly based on manifold learning algorithms, have shown a potential use in HSI transformation, dimensionality reduction and classification. In target detection, non-linear transformation algorithms are used as preprocessing techniques that transform the data to a more suitable lower dimensional space, where the statistical or geometric detectors are applied. One of these non-linear manifold methods is the Schroedinger Eigenmaps (SE) algorithm that has been introduced as a technique for semi-supervised classification. The core tool of the SE algorithm is the Schroedinger operator that includes a potential term that encodes prior information about the materials present in a scene, and enables the embedding to be steered in some convenient directions in order to cluster similar pixels together. A completely novel target detection methodology based on SE algorithm is proposed for the first time in this thesis. The proposed methodology does not just include the transformation of the data to a lower dimensional space but also includes the definition of a detector that capitalizes on the theory behind SE. The fact that target pixels and
Schroedinger operators with singular perturbation potentials
International Nuclear Information System (INIS)
Harrell, E.M. II.
1976-01-01
This is a perturbative analysis of the eigenvalues and eigenfunctions of Schroedinger operators of the form -Δ + A + lambda V, defined on the Hilbert space L 2 (R/sup n/). A is a potential function (a smooth, real multiplication operator), and V is a ''spikelike'' perturbation, i.e., a perturbative potential function which diverges at some finite point. Lambda is a small real or complex parameter. The emphasis is on one-dimensional problems, and in particular the typical example is the ''spiked harmonic oscillator'' Hamiltonian, -d 2 /dx 2 + x 2 + lambda x/sup -α/, where α is a positive constant. An earlier study by L. Detwiler and J. R. Klauder [Phys. Rev. D 11 (1975) 1436] indicated that the lowest-order corrections to the ground-state eigenvalue of the spiked harmonic oscillator with lambda greater than 0 were proportional to lambda ln lambda when α = 3, and to lambda/sup 1/(α-2) when α is greater than 3. These and analogous results for a large class of operators and arbitrary eigenvalues are proved. Explicit constants in a modified perturbation series with a complicated dependence on lambda are determined and exhibited. Higher-order corrections for real lambda and lowest-order corrections for complex lambda are also discussed. While the substance of the dissertation is mathematical, its main applications are to quantum physics. The immediate cause of interest in such problems was the use of their peculiar convergence properties by J. R. Klauder as models for the behavior of nonrenormalizable quantum field theories. However, the results of this study are likely to be of greater importance in chemical or nuclear physics, as positive spikelike perturbations represent repulsive core interactions for quantum mechanical particles. The modified perturbation series are a new calculation technique for this situation
Nonlinear scalar field equations. Pt. 1
International Nuclear Information System (INIS)
Berestycki, H.; Lions, P.L.
1983-01-01
This paper as well as a subsequent one is concerned with the existence of nontrivial solutions for some semi-linear elliptic equations in Rsup(N). Such problems are motivated in particular by the search for certain kinds of solitary waves (stationary states) in nonlinear equations of the Klein-Gordon or Schroedinger type. (orig./HSI)
Some New Integrable Equations from the Self-Dual Yang-Mills Equations
International Nuclear Information System (INIS)
Ivanova, T.A.; Popov, A.D.
1994-01-01
Using the symmetry reductions of the self-dual Yang-Mills (SDYM) equations in (2+2) dimensions, we introduce new integrable equations which are 'deformations' of the chiral model in (2+1) dimensions, generalized nonlinear Schroedinger, Korteweg-de Vries, Toda lattice, Garnier, Euler-Arnold, generalized Calogero-Moser and Euler-Calogero-Moser equations. The Lax pairs for all of these equations are derived by the symmetry reductions of the Lax pair for the SDYM equations. 34 refs
On a complex differential Riccati equation
International Nuclear Information System (INIS)
Khmelnytskaya, Kira V; Kravchenko, Vladislav V
2008-01-01
We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schroedinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation such as the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical 'one-dimensional' results, we discuss new features of the considered equation including an analogue of the Cauchy integral theorem
International Nuclear Information System (INIS)
Chierchia, L.
1986-01-01
In the first chapter, the eigenvalue problem for a periodic Schroedinger operator, Lf = (-d 2 /dx 2 + v)f = Ef, is viewed as a two-dimensional Hamiltonian system which is integrable in the sense of Arnold and Liouville. With the aid of the Floquet-BLoch theory, it is shown that such a system is conjugate to two harmonic oscillators with frequencies α and omega, being the rotation number for L and 2π/omega the period of the potential v. This picture is generalized in the second chapter, to quasi periodic Schroedinger operators, L/sub epsilon/, with highly irrational frequencies (omega 1 , ..., omega/sub d/), which are a small perturbation of periodic operators. In the last chapter, the absolutely continuous spectrum σ/sub ac/ of a general quasi-periodic Schroedinger operators is considered. The Radon-Nikodym derivatives (with respect to Lebesgue measure) of the spectral measures are computed in terms of special independent eigensolutions existing for almost ever E in σ/sub ac/. Finally, it is shown that weak Bloch waves always exist for almost ever E in σ/sub ac/ and the question of the existence of genuine Bloch waves is turned into a regularity problem for a certain nonlinear partial differential equation on a d-dimensional torus
International Nuclear Information System (INIS)
Zhu, Zuo-nong; Tam, Hon-Wah; Ding, Qing
2003-01-01
In this Letter, by means of considering matrix form of a new Schroedinger discrete spectral operator equation, and constructing opportune time evolution equations, and using discrete zero curvature representation, two discrete integrable lattice hierarchies proposed by Boiti et al. [J. Phys. A: Math. Gen. 36 (2003) 139] are re-derived. From the matrix Lax representations, we demonstrate the existence of infinitely many conservation laws for the two lattice hierarchies and give the corresponding conserved densities and the associated fluxes by means of formulae. Thus their integrability is further confirmed. Specially we obtain the infinitely many conservation laws for a new discrete version of the KdV equation. A connection between the conservation laws of the discrete KdV equation and the ones of the KdV equation is discussed by two examples
Studying the gradient flow coupling in the Schroedinger functional
Energy Technology Data Exchange (ETDEWEB)
Fritzsch, P. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Ramos, A. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC
2013-08-15
We discuss the setup and features of a new definition of the running coupling in the Schroedinger functional scheme based on the gradient flow. Its suitability for a precise continuum limit in QCD is demonstrated on a set of N{sub f}=2 gauge field ensembles in a physical volume of L{proportional_to}0.4 fm.
Chronoprojective invariance of the five-dimensional Schroedinger formalism
International Nuclear Information System (INIS)
Perrin, M.; Burdet, G.; Duval, C.
1984-10-01
Invariance properties of the five-dimensional Schroedinger formalism describing a quantum test particle in the Newton-Cartan theory of gravitation are studied. The geometry which underlies these invariance properties is presented as a reduction of the 0(5,2) conformal geometry various applications are given
Remarks on the Schroedinger operator with singular complex potentials
International Nuclear Information System (INIS)
Brezis, Haim; Kato, Tosio
1979-01-01
To describe this method in a simple case Section 2 begin with real valued potentials. The main results in Section 2 are essentially known. In Section 3 the case of complex potentials is exposed. Schroedinger operators with complex potentials have been studied by Nelson. This results were extended. Here more general singularities are exposed
Erwin Schroedinger, Philosophy and the birth of quantum mechanics
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Bitbol, M.; Darrigol, O.
1992-01-01
The purpose of this collection of articles is to highlight the relation between Schroedinger's less well known research and his thoughts on quantum mechanics: philosophy, statistical mechanics, general relativity, cosmology, unified field theories, etc. Some articles are devoted to contemporary extensions of his work, and in particular on current echoes of his interpretation of quantum mechanics
Effective equations for the quantum pendulum from momentous quantum mechanics
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Hernandez, Hector H.; Chacon-Acosta, Guillermo [Universidad Autonoma de Chihuahua, Facultad de Ingenieria, Nuevo Campus Universitario, Chihuahua 31125 (Mexico); Departamento de Matematicas Aplicadas y Sistemas, Universidad Autonoma Metropolitana-Cuajimalpa, Artificios 40, Mexico D. F. 01120 (Mexico)
2012-08-24
In this work we study the quantum pendulum within the framework of momentous quantum mechanics. This description replaces the Schroedinger equation for the quantum evolution of the system with an infinite set of classical equations for expectation values of configuration variables, and quantum dispersions. We solve numerically the effective equations up to the second order, and describe its evolution.
Erwin Schroedinger: Collected papers V. 1. Contributions to statistical mechanics
International Nuclear Information System (INIS)
Schroedinger, E.
1984-01-01
38 publications reprinted in this volume show that the interest for statistical problems accompanied Schroedinger during his entire scientific career. Already in his second paper he worked on the magnetism of solid states. The classical considerations come close to the heart of diamagnetism and also to the origin of paramagnetism. In classical investigations of the specific heat Schroedinger helped through abstract theory but also by analysing a gigantic amount of experimental material. In 1926 he and F. Kohlrausch actually played the 'Urngame of Ehrenfest' as a model of the H-curve and published the results. Inclination towards experimenting, sequences of measurements and statistical evaluation of experimental data led to papers on the foundation of the theory of probability, where he tries to put the subjective probability concept on into a systematic framework. Two earlier papers on dynamics of the elastic chain remained particularly valuable. By solving the initial value problem with Bessel-functions this many-body-problem is led to an explicit discussion. These studies are likely to be the roots of another keynote in Schroedinger's thinking, namely, the irreversibility. 1945 a statistical theory of chain-reactions was published under the inconspicuous title of 'Probability Problems in Nuclear Chemistry'. In his last work Schroedinger turns off in a wrong direction: it is that energy should only be a statistical concept and should not be conserved in elementary processes, but somehow only in the mean. These short remarks only illuminate the diversity of the material in this volume, but testify Schroedinger's deep understanding in this field. (W.K.)
The Dirac equation for accountants
International Nuclear Information System (INIS)
Ord, G.N.
2006-01-01
In the context of relativistic quantum mechanics, derivations of the Dirac equation usually take the form of plausibility arguments based on experience with the Schroedinger equation. The primary reason for this is that we do not know what wavefunctions physically represent, so derivations have to rely on formal arguments. There is however a context in which the Dirac equation in one dimension is directly related to a classical generating function. In that context, the derivation of the Dirac equation is an exercise in counting. We provide this derivation here and discuss its relationship to quantum mechanics
On a new series of integrable nonlinear evolution equations
International Nuclear Information System (INIS)
Ichikawa, Y.H.; Wadati, Miki; Konno, Kimiaki; Shimizu, Tohru.
1980-10-01
Recent results of our research are surveyed in this report. The derivative nonlinear Schroedinger equation for the circular polarized Alfven wave admits the spiky soliton solutions for the plane wave boundary condition. The nonlinear equation for complex amplitude associated with the carrier wave is shown to be a generalized nonlinear Schroedinger equation, having the ordinary cubic nonlinear term and the derivative of cubic nonlinear term. A generalized scheme of the inverse scattering transformation has confirmed that superposition of the A-K-N-S scheme and the K-N scheme for the component equations valids for the generalized nonlinear Schroedinger equation. Then, two types of new integrable nonlinear evolution equation have been derived from our scheme of the inverse scattering transformation. One is the type of nonlinear Schroedinger equation, while the other is the type of Korteweg-de Vries equation. Brief discussions are presented for physical phenomena, which could be accounted by the second type of the new integrable nonlinear evolution equation. Lastly, the stationary solitary wave solutions have been constructed for the integrable nonlinear evolution equation of the second type. These solutions have peculiar structure that they are singular and discrete. It is a new challenge to construct singular potentials by the inverse scattering transformation. (author)
Schroedinger operators with point interactions and short range expansions
International Nuclear Information System (INIS)
Albeverio, S.; Hoeegh-Krohn, R.; Oslo Univ.
1984-01-01
We give a survey of recent results concerning Schroedinger operators with point interactions in R 3 . In the case where the point interactions are located at a discrete set of points we discuss results about the resolvent, the spectrum, the resonances and the scattering quantities. We also discuss the approximation of point interactions by short range local potentials (short range or low energy expansions) and the one electron model of a 3-dimensional crystal. Moreover we discuss Schroedinger operators with Coulomb plus point interactions, with applications to the determination of scattering lengths and of level shifts in mesic atoms. Further applications to the multiple well problem, to multiparticle systems, to crystals with random impurities, to polymers and quantum fields are also briefly discussed. (orig.)
The quantum nonlinear Schroedinger model with point-like defect
International Nuclear Information System (INIS)
Caudrelier, V; Mintchev, M; Ragoucy, E
2004-01-01
We establish a family of point-like impurities which preserve the quantum integrability of the nonlinear Schroedinger model in 1+1 spacetime dimensions. We briefly describe the construction of the exact second quantized solution of this model in terms of an appropriate reflection-transmission algebra. The basic physical properties of the solution, including the spacetime symmetry of the bulk scattering matrix, are also discussed. (letter to the editor)
Schroedinger propagation of initial discontinuities leads to divergence of moments
International Nuclear Information System (INIS)
Marchewka, A.; Schuss, Z.
2009-01-01
We show that the large phase expansion of the Schroedinger propagation of an initially discontinuous wave function leads to the divergence of average energy, momentum, and displacement, rendering them unphysical states. If initially discontinuous wave functions are considered to be approximations to continuous ones, the determinant of the spreading rate of these averages is the maximal gradient of the initial wave function. Therefore a dilemma arises between the inclusion of discontinuous wave functions in quantum mechanics and the requirement of finite moments.
Schroedinger propagation of initial discontinuities leads to divergence of moments
Energy Technology Data Exchange (ETDEWEB)
Marchewka, A., E-mail: avi.marchewka@gmail.co [Ruppin Academic Center, Emek-Hefer 40250 (Israel); Schuss, Z., E-mail: schuss@post.tau.ac.i [Department of Mathematics, Tel-Aviv University, Ramat-Aviv, 69978 Tel-Aviv (Israel)
2009-09-21
We show that the large phase expansion of the Schroedinger propagation of an initially discontinuous wave function leads to the divergence of average energy, momentum, and displacement, rendering them unphysical states. If initially discontinuous wave functions are considered to be approximations to continuous ones, the determinant of the spreading rate of these averages is the maximal gradient of the initial wave function. Therefore a dilemma arises between the inclusion of discontinuous wave functions in quantum mechanics and the requirement of finite moments.
The Schroedinger's paradox and the tranformation of quantum systems
International Nuclear Information System (INIS)
Bitsakis, E.I.
1980-01-01
The Schroedinger's paradox is analysed, as an illustration of certain weaknesses of the Copenhagen's interpretation of quantum mechanics and of the limits of the quantum-mechanical description of phenomena. A realistic approach of the paradox indicates the necessity of a theory that would permit not only the calculation of probabilities, but also the description of physical processes, as taking place in space and time
Improved Rosen's conditions on bound states of Schroedinger operators
International Nuclear Information System (INIS)
Exner, P.
1984-01-01
We derive a necessary condition on a Schroedinger operator H=-Δ+V on Lsup(2)(Rsup(d)), d>=3 to have a bound state below a given energy epsilon, and a lower bound to the ground-state energy of H. These conditions are expressed in terms of the potential V alone, and generalize the recent results of Rosen to the dimensions d>3 and to the potentials that are not necessarily rapidly decreasing. Some examples are given
Bound State Eigenvalues of the Schroedinger Eq. in two Spatial Variables.
Rawitscher, George H.; Koltracht, Israel
2002-08-01
An efficient spectral integral equation method (SIEM) has recently been developed for obtaining the scattering solution of a one-dimensional Schroedinger equation.(R.A. Gonzales, S.-Y. Kang, I. Koltracht and G. Rawitscher, J. of Comput. Phys. 153, 160 (1999).) The purpose of the present study is to extend this method to the case of bound-states in more than one dimension. Even though other methods have already been developed for this case, such as finite element methods, the application we have in mind is to solve the non-linear Bose-Einstein condensate case in the presence of an optical lattice. In the presence of a trapping potential alone, a B-E condensate solution has been obtained by a new iterative spectral method which solves the differential equation.(Y.-S. Choi, J. Javanainen, I. Koltracht, M. Koš)trun, P.J. McKenna and N. Savytska "A Fast Algorithm for the Solution of the Time-Independent Gross-Pitaevskii Equation," Submitted to Computational Physics. But this method becomes inadequate for the case that several potential barriers are also present. The reason that the SIEM is expected to be better suited is that it distributes the collocation points much more efficiently into partitions of variable size.
Quantum field theory in flat Robertson-Walker space-time functional Schroedinger picture
International Nuclear Information System (INIS)
Pi, S.Y.
1989-01-01
Quantum field theory in Robertson-Walker space-time is intrinsically time-dependent and the functional Schroedinger picture provides a useful description. We discuss free and self-interacting bosonic quantum field theories: Schroedinger picture quantization, time-dependent Gaussian approximations based on variational principles describing time evolution of pure and mixed states, and renormalizability of the Schroedinger picture. The techniques introduced can be used to study various dynamical questions in early universe processes. (author)
International Nuclear Information System (INIS)
Mehra, J.
1987-01-01
This paper, the first part of a three-part article, gives an account of Erwin Schroedinger's growing up and studies in Vienna, his scientific work--first in Vienna from 1911 to 1920, then in Zurich from 1920 to 1925--on the dielectric properties of matter, atmospheric electricity and radioactivity, general relativity, color theory and physiological optics, and on kinetic theory and statistical mechanics
Creation and annihilation of solitons in the string nonlinear equation
International Nuclear Information System (INIS)
Aguero G, M.A.; Espinosa G, A.A.; Martinez O, J.
1997-01-01
Starting from the cubic-quintic Schroedinger equation it is obtained the nonlinear string equation. This system supports regular and singular solitons. It is shown that two singular solitons could be generated after the interaction of two regular solitons and viceversa. (Author)
Prolongation Structure of Semi-discrete Nonlinear Evolution Equations
International Nuclear Information System (INIS)
Bai Yongqiang; Wu Ke; Zhao Weizhong; Guo Hanying
2007-01-01
Based on noncommutative differential calculus, we present a theory of prolongation structure for semi-discrete nonlinear evolution equations. As an illustrative example, a semi-discrete model of the nonlinear Schroedinger equation is discussed in terms of this theory and the corresponding Lax pairs are also given.
Energy Technology Data Exchange (ETDEWEB)
D`Agostino, S. [Rome Univ. (Italy)
1992-12-31
In the 50s, Schroedinger proposed a new conception of a continuous theory of Quantum Mechanics, which remarkably modified his 1926 ideas on ondulatory mechanics. The lack of individuality of the atomic particles presented in the new statistics, and in Heisenberg`s Indeterminacy Relations, was by him considered as an aspect of a more general crisis in the anthology itself of classical atomism. Unlike his 1926 ideas, he proposed now to represent the wave equation in an n-dimensional space and he considered second-quantization technique as the proper mathematical tool for his new physical conception. Although he accepted that space-time discontinuities and casual gaps may appear here and there on the observational level (e.g. in the Indeterminacy Relations), he was convinced that they could be made compatible with a continuous pure theory, provided one accepted a suitable conception of the theory`s epistemiological status. For him, only a continuous theory satisfied the conditions for a complete theory. On these matters, he thought he was somehow orthodox to the ideas of Hertz and Boltzmann, which were also reflected in the teaching of Exner. (author). 69 refs.
Solvable linear potentials in the Dirac equation
International Nuclear Information System (INIS)
Dominguez-Adame, F.; Gonzalez, M.A.
1990-01-01
The Dirac equation for some linear potentials leading to Schroedinger-like oscillator equations for the upper and lower components of the Dirac spinor have been solved. Energy levels for the bound states appear in pairs, so that both particles and antiparticles may be bound with the same energy. For weak coupling, the spacing between levels is proportional to the coupling constant while in the strong limit those levels are depressed compared to the nonrelativistic ones
Generalized quantal equation of motion
International Nuclear Information System (INIS)
Morsy, M.W.; Embaby, M.
1986-07-01
In the present paper, an attempt is made for establishing a generalized equation of motion for quantal objects, in which intrinsic self adjointness is naturally built in, independently of any prescribed representation. This is accomplished by adopting Hamilton's principle of least action, after incorporating, properly, the quantal features and employing the generalized calculus of variations, without being restricted to fixed end points representation. It turns out that our proposed equation of motion is an intrinsically self-adjoint Euler-Lagrange's differential equation that ensures extremization of the quantal action as required by Hamilton's principle. Time dependence is introduced and the corresponding equation of motion is derived, in which intrinsic self adjointness is also achieved. Reducibility of the proposed equation of motion to the conventional Schroedinger equation is examined. The corresponding continuity equation is established, and both of the probability density and the probability current density are identified. (author)
Schroedinger operators with Rudin-Shapiro potentials are not palindromic
International Nuclear Information System (INIS)
Allouche, J.
1997-01-01
We prove a conjecture of A. Hof, O. Knill and B. Simon [Commun. Math. Phys. 174, 149 endash 159 (1995)] by showing that the Rudin-Shapiro sequence is not palindromic, i.e., does not contain arbitrarily long palindromes. We prove actually this property for all paperfolding sequences and all Rudin-Shapiro sequences deduced from paperfolding sequences. As a consequence and as guessed by the above authors, their method cannot be used for establishing that discrete Schroedinger operators with Rudin-Shapiro potentials have a purely singular continuous spectrum. copyright 1997 American Institute of Physics
The gradient flow coupling in the Schroedinger functional
International Nuclear Information System (INIS)
Fritzsch, Patrick; Ramos, Alberto
2013-01-01
We study the perturbative behavior of the Yang-Mills gradient flow in the Schroedinger Functional, both in the continuum and on the lattice. The energy density of the flow field is used to define a running coupling at a scale given by the size of the finite volume box. From our perturbative computation we estimate the size of cutoff effects of this coupling to leading order in perturbation theory. On a set of N f =2 gauge field ensembles in a physical volume of L∝0.4 fm we finally demonstrate the suitability of the coupling for a precise continuum limit due to modest cutoff effects and high statistical precision.
Localization for random Schroedinger operators with correlated potentials
Energy Technology Data Exchange (ETDEWEB)
Von Dreifus, H [Princeton Univ., NJ (USA). Dept. of Physics; Klein, A [California Univ., Irvine (USA). Dept. of Mathematics
1991-08-01
We prove localization at high disorder or low energy for lattice Schroedinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance function C(x,y) decays as vertical strokex-yvertical stroke{sup -{theta}}, where {theta}>0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate. (orig.).
Schroedinger invariant solutions of type IIB with enhanced supersymmetry
Energy Technology Data Exchange (ETDEWEB)
Donos, Aristomenis [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Gauntlett, Jerome P. [Imperial College, London (United Kingdom). Theoretical Physics Group; Imperial College, London (United Kingdom). Inst. for Mathematical Sciences
2009-07-15
We construct the Killing spinors for a class of supersymmetric solutions of type IIB supergravity that are invariant under the non-relativistic Schroedinger algebra. The solutions depend on a five-dimensional Sasaki- Einstein space and it has been shown that they admit two Killing spinors. Here we will show that, for generic Sasaki-Einstein space, there are special subclasses of solutions which admit six Killing spinors and we determine the corresponding superisometry algebra. We also show that for the special case that the Sasaki-Einstein space is the round five-sphere, the number of Killing spinors can be increased to twelve. (orig.)
Numerical stochastic perturbation theory in the Schroedinger functional
International Nuclear Information System (INIS)
Brambilla, Michele; Di Renzo, Francesco; Hesse, Dirk; Dalla Brida, Mattia; Sint, Stefan; Deutsches Elektronen-Synchrotron
2013-11-01
The Schroedinger functional (SF) is a powerful and widely used tool for the treatment of a variety of problems in renormalization and related areas. Albeit offering many conceptual advantages, one major downside of the SF scheme is the fact that perturbative calculations quickly become cumbersome with the inclusion of higher orders in the gauge coupling and hence the use of an automated perturbation theory framework is desirable. We present the implementation of the SF in numerical stochastic perturbation theory (NSPT) and compare first results for the running coupling at two loops in pure SU(3) Yang-Mills theory with the literature.
Philosophical and methodological aspects of the Schroedinger paradox
International Nuclear Information System (INIS)
Juha, L.; Krajca, R.; Smatera, M.
1989-01-01
Methodological aspects of the foundations of quantum theory are dealt with in relation to the quantum description of macroscopic systems, biological in particular. Attention is paid to the philosophical content of the problems of 1) the logical status of the reduction postulate in quantum mechanics, and 2) the paradox of Schroedinger's cat, whose physical solution has not yet been attained. The problem of the quantum description of complex macroscopic systems is also treated, as is Herbert Froehlich's important concept of the excitation of dominant modes in biological systems. (author). 61 refs
Numerical stochastic perturbation theory in the Schroedinger functional
Energy Technology Data Exchange (ETDEWEB)
Brambilla, Michele; Di Renzo, Francesco; Hesse, Dirk [Parma Univ. (Italy); INFN, Parma (Italy); Dalla Brida, Mattia [Trinity College Dublin (Ireland). School of Mathematics; Sint, Stefan [Trinity College Dublin (Ireland). School of Mathematics; Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC
2013-11-15
The Schroedinger functional (SF) is a powerful and widely used tool for the treatment of a variety of problems in renormalization and related areas. Albeit offering many conceptual advantages, one major downside of the SF scheme is the fact that perturbative calculations quickly become cumbersome with the inclusion of higher orders in the gauge coupling and hence the use of an automated perturbation theory framework is desirable. We present the implementation of the SF in numerical stochastic perturbation theory (NSPT) and compare first results for the running coupling at two loops in pure SU(3) Yang-Mills theory with the literature.
Linear integral equations and soliton systems
International Nuclear Information System (INIS)
Quispel, G.R.W.
1983-01-01
A study is presented of classical integrable dynamical systems in one temporal and one spatial dimension. The direct linearizations are given of several nonlinear partial differential equations, for example the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the sine-Gordon equation, the nonlinear Schroedinger equation, and the equation of motion for the isotropic Heisenberg spin chain; the author also discusses several relations between these equations. The Baecklund transformations of these partial differential equations are treated on the basis of a singular transformation of the measure (or equivalently of the plane-wave factor) occurring in the corresponding linear integral equations, and the Baecklund transformations are used to derive the direct linearization of a chain of so-called modified partial differential equations. Finally it is shown that the singular linear integral equations lead in a natural way to the direct linearizations of various nonlinear difference-difference equations. (Auth.)
Matter-wave vortices in cigar-shaped and toroidal waveguides
International Nuclear Information System (INIS)
Salasnich, L.; Toigo, F.; Malomed, B. A.
2007-01-01
We study vortical states in a Bose-Einstein condensate (BEC) filling a cigar-shaped trap. An effective one-dimensional (1D) nonpolynomial Schroedinger equation (NPSE) is derived in this setting, for the models with both repulsive and attractive interatomic interactions. Analytical formulas for the density profiles are obtained from the NPSE in the case of self-repulsion within the Thomas-Fermi approximation, and in the case of the self-attraction as exact solutions (bright solitons). A crucially important ingredient of the analysis is the comparison of these predictions with direct numerical solutions for the vortex states in the underlying 3D Gross-Pitaevskii equation. The comparison demonstrates that the NPSE provides for a very accurate approximation, in all the cases, including the prediction of the stability of the bright solitons and collapse threshold for them. In addition to the straight cigar-shaped trap, we also consider a torus-shaped configuration. In that case, we find a threshold for the transition from the axially uniform state, with the transverse intrinsic vorticity, to a symmetry-breaking pattern, due to the instability in the self-attractive BEC filling the circular trap
Spectral transform and orthogonality relations for the Kadomtsev-Petviashvili I equation
Energy Technology Data Exchange (ETDEWEB)
Boiti, M; Leon, J J.P.; Pempinelli, F [Montpellier-2 Univ., 34 (France). Lab. de Physique Mathematique
1989-10-30
We define a new spectral transform r(k,l) of the potential u in the time dependent Schroedinger equation (associated to the KPI equation). Orthogonality relations for the sectionally holomorphic eigenfunctions of the Schroedinger equation are used to express the spectral transform f(k,l) previously introduced by Manakov and Fokas and Ablowitz in terms of r(k,l). The main advantage of the new spectral transform r(k,l) is that its definition does not require to introduce an additional nonanalytic eigenfunction N. Characterization equations for r(k,l) are also obtained. (orig.).
International Nuclear Information System (INIS)
Shore, B.W.; Sacks, R.; Karr, T.
1987-01-01
This memo discusses the equations of motion used to describe multilevel molecular excitation induced by Raman transitions. These equations are based upon the time-dependent Schroedinger equation expressed in a basis of molecular energy states. A partition of these states is made into two sets, those that are far from resonance (and hence unpopulated) and those that are close to resonance, either by one-photon transition or two-photon (Raman) processes. By adiabatic elimination an effective Schroedinger equation is obtained for the resonance states alone. The effective Hamiltonian is expressible in terms of a polarizibility operator
Spectral transform and orthogonality relations for the Kadomtsev-Petviashvili I equation
International Nuclear Information System (INIS)
Boiti, M.; Leon, J.J.P.; Pempinelli, F.
1989-01-01
We define a new spectral transform r(k,l) of the potential u in the time dependent Schroedinger equation (associated to the KPI equation). Orthogonality relations for the sectionally holomorphic eigenfunctions of the Schroedinger equation are used to express the spectral transform f(k,l) previously introduced by Manakov and Fokas and Ablowitz in terms of r(k,l). The main advantage of the new spectral transform r(k,l) is that its definition does not require to introduce an additional nonanalytic eigenfunction N. Characterization equations for r(k,l) are also obtained. (orig.)
Chern--Simons theory in the Schroedinger representation
International Nuclear Information System (INIS)
Dunne, G.V.; Jackiw, R.; Trugenberger, C.A.
1989-01-01
We quantize the (2+1)-dimensional Chern--Simons theory in the functional Schroedinger representation. The realization of gauge transformations on states involves a 1-cocycle. We determine this cocycle; we show how solving the Gauss law constraint in the non-Abelian theory requires quantizing the parameter that normalizes the action; we trivialize the 1-cocycle with a spatially non-local cochain related to a 2-dimensional fermion determinant and we find the physical states that satisfy the Gauss law constraint. The quantum holonomy of physical states involves a contribution that is missed when the constraint is solved before quantization. We compute this quantity for the Abelian theory in Minkowski space, where it exhibits an interesting group theoretic structure. (In a note added in proof the corresponding non-Abelian computation is presented.) Also we consider coupling to external sources and offer yet another derivation of the anomalous statistics and spin of the charge and flux carrying particles---a calculation which is especially simple in the functional Schroedinger representation. copyright 1989 Academic Press, Inc
The gradient flow coupling in the Schroedinger functional
Energy Technology Data Exchange (ETDEWEB)
Fritzsch, Patrick [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Ramos, Alberto [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC
2013-01-15
We study the perturbative behavior of the Yang-Mills gradient flow in the Schroedinger Functional, both in the continuum and on the lattice. The energy density of the flow field is used to define a running coupling at a scale given by the size of the finite volume box. From our perturbative computation we estimate the size of cutoff effects of this coupling to leading order in perturbation theory. On a set of N{sub f}=2 gauge field ensembles in a physical volume of L{proportional_to}0.4 fm we finally demonstrate the suitability of the coupling for a precise continuum limit due to modest cutoff effects and high statistical precision.
Fermionic covariant prolongation structure theory for supernonlinear evolution equation
International Nuclear Information System (INIS)
Cheng Jipeng; Wang Shikun; Wu Ke; Zhao Weizhong
2010-01-01
We investigate the superprincipal bundle and its associated superbundle. The super(nonlinear)connection on the superfiber bundle is constructed. Then by means of the connection theory, we establish the fermionic covariant prolongation structure theory of the supernonlinear evolution equation. In this geometry theory, the fermionic covariant fundamental equations determining the prolongation structure are presented. As an example, the supernonlinear Schroedinger equation is analyzed in the framework of this fermionic covariant prolongation structure theory. We obtain its Lax pairs and Baecklund transformation.
General method for reducing the two-body Dirac equation
International Nuclear Information System (INIS)
Galeao, A.P.; Ferreira, P.L.
1992-01-01
A semi relativistic two-body Dirac equation with an enlarged set of phenomenological potentials, including Breit-type terms, is investigated for the general case of unequal masses. Solutions corresponding to definite total angular momentum and parity are shown to fall into two classes, each one being obtained by solving a system of four coupled first-order radial differential equations. The reduction of each of these systems to a pair of coupled Schroedinger-type equations is also discussed. (author)
Global spacetime symmetries in the functional Schroedinger picture
International Nuclear Information System (INIS)
Halliwell, J.J.
1991-01-01
In the conventional functional Schroedinger quantization of field theory, the background spacetime manifold is foliated into a set of three-surfaces and the quantum state of the field is represented by a wave functional of the field configurations on each three-surface. Although this procedure may be covariantly described, the wave functionals generally fail to carry a representation of the complete spacetime symmetry group of the background, such as the Poincare group in Minkowski spacetime, because spacetime symmetries generally involve distortions or motions of the three-surfaces themselves within that spacetime. In this paper, we show that global spacetime symmetries in the functional Schroedinger picture may be represented by parametrizing the field theory---raising to the status of dynamical variables the embedding variables describing the spacetime location of each three-surface. In particular, we show that the embedding variables provide a connection between the purely geometrical operation of an isometry group on the spacetime and the operation of the usual global symmetry generators (constructed from the energy-momentum tensor) on the wave functionals of the theory. We study the path-integral representation of the wave functionals of the parametrized field theory. We show how to construct, from the path integral, wave functionals that are annihilated by the global symmetry generators, i.e., that are invariant under global spacetime symmetry groups. The invariance of the class of histories summed over in the path integral is identified as the source of the invariance of the wave functionals. We apply this understanding to a study of vacuum states in the de Sitter spacetime. We make mathematically precise a previously given heuristic argument for the de Sitter invariance of the matter wave functionals defined by the no-boundary proposal of Hartle and Hawking
The chirally rotated Schroedinger functional. Theoretical expectations and perturbative tests
International Nuclear Information System (INIS)
Dalla Brida, Mattia
2016-03-01
The chirally rotated Schroedinger functional (χSF) with massless Wilson-type fermions provides an alternative lattice regularization of the Schroedinger functional (SF), with different lattice symmetries and a common continuum limit expected from universality. The explicit breaking of flavour and parity symmetries needs to be repaired by tuning the bare fermion mass and the coefficient of a dimension 3 boundary counterterm. Once this is achieved one expects the mechanism of automatic O(a) improvement to be operational in the χSF, in contrast to the standard formulation of the SF. This is expected to significantly improve the attainable precision for step-scaling functions of some composite operators. Furthermore, the χSF offers new strategies to determine finite renormalization constants which are traditionally obtained from chiral Ward identities. In this paper we consider a complete set of fermion bilinear operators, define corresponding correlation functions and explain the relation to their standard SF counterparts. We discuss renormalization and O(a) improvement and then use this set-up to formulate the theoretical expectations which follow from universality. Expanding the correlation functions to one-loop order of perturbation theory we then perform a number of non-trivial checks. In the process we obtain the action counterterm coefficients to one-loop order and reproduce some known perturbative results for renormalization constants of fermion bilinears. By confirming the theoretical expectations, this perturbative study lends further support to the soundness of the χSF framework and prepares the ground for non-perturbative applications.
The Schroedinger functional for Gross-Neveu models
International Nuclear Information System (INIS)
Leder, B.
2007-01-01
Gross-Neveu type models with a finite number of fermion flavours are studied on a two-dimensional Euclidean space-time lattice. The models are asymptotically free and are invariant under a chiral symmetry. These similarities to QCD make them perfect benchmark systems for fermion actions used in large scale lattice QCD computations. The Schroedinger functional for the Gross-Neveu models is defined for both, Wilson and Ginsparg-Wilson fermions, and shown to be renormalisable in 1-loop lattice perturbation theory. In two dimensions four fermion interactions of the Gross-Neveu models have dimensionless coupling constants. The symmetry properties of the four fermion interaction terms and the relations among them are discussed. For Wilson fermions chiral symmetry is explicitly broken and additional terms must be included in the action. Chiral symmetry is restored up to cut-off effects by tuning the bare mass and one of the couplings. The critical mass and the symmetry restoring coupling are computed to second order in lattice perturbation theory. This result is used in the 1-loop computation of the renormalised couplings and the associated beta-functions. The renormalised couplings are defined in terms of suitable boundary-to-boundary correlation functions. In the computation the known first order coefficients of the beta-functions are reproduced. One of the couplings is found to have a vanishing betafunction. The calculation is repeated for the recently proposed Schroedinger functional with exact chiral symmetry, i.e. Ginsparg-Wilson fermions. The renormalisation pattern is found to be the same as in the Wilson case. Using the regularisation dependent finite part of the renormalised couplings, the ratio of the Lambda-parameters is computed. (orig.)
The relativistic electron wave equation
International Nuclear Information System (INIS)
Dirac, P.A.M.
1977-08-01
The paper was presented at the European Conference on Particle Physics held in Budapest between the 4th and 9th July of 1977. A short review is given on the birth of the relativistic electron wave equation. After Schroedinger has shown the equivalence of his wave mechanics and the matrix mechanics of Heisenberg, a general transformation theory was developed by the author. This theory required a relativistic wave equation linear in delta/delta t. As the Klein--Gordon equation available at this time did not satisfy this condition the development of a new equation became necessary. The equation which was found gave the value of the electron spin and magnetic moment automatically. (D.P.)
International Nuclear Information System (INIS)
Baik, M.; Pont, M.; Shakeshaft, R.
1995-01-01
We develop a method for calculating the (quasi)energy eigenvalue E(F) of a hydrogen atom in a nonperturbative ac field of strength F starting from a knowledge of the coefficients E (2m) of the Rayleigh-Schroedinger perturbation series E(F)=tsum m=0 M E (2m) F 2m . We first use the coefficients E (2m) (the unperturbed energy is E (0) ) to construct the inverse series F 2 (E)=tsum m=1 M F (m) (E-E (0) ) m . We resum the latter series using the Pade method, and solve the implicit equation F 2 (E)=bar F 2 for E(bar F). The reconstructed function E(F) has the singularity structure appropriate to the true E(F). We are able to obtain good results for the lifetime of a hydrogen atom in a high-frequency field up to very high intensities, well into the (highly nonperturbative) stabilization regime
Non-negative Feynman endash Kac kernels in Schroedinger close-quote s interpolation problem
International Nuclear Information System (INIS)
Blanchard, P.; Garbaczewski, P.; Olkiewicz, R.
1997-01-01
The local formulations of the Markovian interpolating dynamics, which is constrained by the prescribed input-output statistics data, usually utilize strictly positive Feynman endash Kac kernels. This implies that the related Markov diffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process. We discuss an extension of the framework to encompass singular potentials and associated non-negative Feynman endash Kac-type kernels. It allows us to deal with a class of continuous interpolations admitted by general non-negative solutions of the Schroedinger boundary data problem. The resulting nonstationary stochastic processes are capable of both developing and destroying nodes (zeros) of probability densities in the course of their evolution, also away from the spatial boundaries. This observation conforms with the general mathematical theory (due to M. Nagasawa and R. Aebi) that is based on the notion of multiplicative functionals, extending in turn the well known Doob close-quote s h-transformation technique. In view of emphasizing the role of the theory of non-negative solutions of parabolic partial differential equations and the link with open-quotes Wiener exclusionclose quotes techniques used to evaluate certain Wiener functionals, we give an alternative insight into the issue, that opens a transparent route towards applications.copyright 1997 American Institute of Physics
Electron confinement in quantum nanostructures: Self-consistent Poisson-Schroedinger theory
International Nuclear Information System (INIS)
Luscombe, J.H.; Bouchard, A.M.; Luban, M.
1992-01-01
We compute the self-consistent electron states and confining potential, V(r,T), for laterally confined cylindrical quantum wires at a temperature T from a numerical solution of the coupled Poisson and Schroedinger (PS) equations. Finite-temperature effects are included in the electron density function, n(r,T), via the single-particle density matrix in the grand-canonical ensemble using the self-consistent bound states. We compare our results for a GaAs quantum wire with those obtained previously [J. H. Luscombe and M. Luban, Appl. Phys. Lett. 57, 61 (1990)] from a finite-temperature Thomas-Fermi (TF) approximation. We find that the TF results agree well with those of the more realistic, but also more computationally intensive PS theory, except for low temperatures or for cases where the quantum wire is almost, but not totally, depleted due to a combination of either small geometry, surface boundary conditions, or low doping concentrations. In the latter situations, the number of subbands that are populated is relatively small, and both n(r,T) and V(r,T) exhibit Friedel-type oscillations. Otherwise the TF theory, which is based on free-particle states, is remarkably accurate. We also present results for the partial electron density functions associated with the angular momentum quantum numbers, and discuss their role in populating the quantum wire
Quantum mechanics of a free particle beyond differential equations ...
African Journals Online (AJOL)
With Feynman's path- integral method we can obtain the quantum mechanics of a quantum system like a free particle outside Schroedinger's method of differential equations and Heisenberg's method of algebra. The work involves obtaining the quantum propagator Kf, of the system which leads to summation over infinite ...
New trace formulae for a quadratic pencil of the Schroedinger operator
International Nuclear Information System (INIS)
Yang Chuanfu
2010-01-01
This work deals with the eigenvalue problem for a quadratic pencil of the Schroedinger operator on a finite closed interval with the two-point boundary conditions. We will obtain new regularized trace formulas for this class of differential pencil.
Finiteness of the discrete spectrum of the three-particle Schroedinger operator
International Nuclear Information System (INIS)
Abdullaev, Janikul I.; Khalkhujaev, Axmad, M.
2001-08-01
We analyse the spectrum of the three-particle Schroedinger operator with pair contact and three-particle interactions on the neighboring nodes on a three-dimensional lattice. We show that the essential spectrum of this operator is the union of two segments, one of which coincides with the spectrum of an unperturbed operator and the other called two-particle branch. We will prove finiteness of the discrete spectrum of the Schroedinger operator at all parameter values of the problem. (author)
International Nuclear Information System (INIS)
Griffin, J.J.; Lichtner, P.C.; Dworzecka, M.; Kan, K.K.
1979-01-01
It is suggested that the TDHF method be viewed, not as an approximation to but as a model of the exact Schroedinger system; that is, as a gedanken many-body experiment whose analysis with digital computers provides data worthy in itself of theoretical study. From such a viewpoint attention is focused on the structural analogies of the TDHF system with the exact theory rather than upon its quantitative equivalence, and the TDHF many-body system is studied as a challenge of its own which, although much simpler than the realistic problem, may still offer complexity enough to educate theorists in the present state of knowledge. In this spirit, the TDHF description of continuum reactions can be restructured from an initial-value problem into a form analogous to the S-matrix version of the Schroedinger theory. The resulting TD-S-HF theory involves only self-consistent single determinantal solutions of the TDHF equations and invokes time averaging to obtain a consistent interpretation of the TDHF analogs of quantities which are constant in the exact theory, such as the S-matrix and the asymptotic reaction channel characteristics. Periodic solutions then play the role of stationary eigenstates in the construction of suitable asymptotic reaction channels. If these periodic channel states occur only at discrete energies, then the resulting channels are mutually orthogonal (on the time average) and the theory exhibits a structure fully analogous to the exact theory. In certain special cases where the periodic solutions are known to occur as an energy continuum, the requirement that the periodicity of the channel solutions be gauge invariant provides a natural requantization condition which (suggestively) turns out to be identical with the Bohr-Sommerfeld quantization rule. 11 references
Complex nonlinear Lagrangian for the Hasegawa-Mima equation
International Nuclear Information System (INIS)
Dewar, R.L.; Abdullatif, R.F.; Sangeetha, G.G.
2005-01-01
The Hasegawa-Mima equation is the simplest nonlinear single-field model equation that captures the essence of drift wave dynamics. Like the Schroedinger equation it is first order in time. However its coefficients are real, so if the potential φ is initially real it remains real. However, by embedding φ in the space of complex functions a simple Lagrangian is found from which the Hasegawa-Mima equation may be derived from Hamilton's Principle. This Lagrangian is used to derive an action conservation equation which agrees with that of Biskamp and Horton. (author)
On the chirally rotated Schroedinger functional with Wilson fermions
International Nuclear Information System (INIS)
Gonzalez Lopez, Jenifer
2011-01-01
There are many phenomena in nature, which are closely linked to the low energy regime of QCD. From a theoretical point of view, these low energy phenomena can be dealt with only by means of non-perturbative methods. It is the central goal of this thesis to provide a framework for such a nonperturbative renormalization. For that purpose, we employ a 4-dimensional lattice as a regulator of QCD. As a renormalization scheme, we propose a finite volume Schroedinger functional scheme and here in particular, the chirally rotated Schroedinger functional (χSF). We first perform analytical studies of the χSF at tree-level of perturbation theory, in the continuum and on the lattice. We study the eigenvalue spectrum of the continuum Dirac operator, equipped with chirally rotated SF boundary conditions, and derive the corresponding quark propagator. We then determine the tree-level quark propagator on the lattice, employing massless Wilson fermions as a regulator of the theory. Beyond tree-level, all studies are performed in the quenched approximation of QCD, as a first, computationally much simpler step to understand the properties of the newly proposed χSF scheme. One of the main targets of the present work, has been to perform the non-perturbative tuning of the two required coefficients of the χSF scheme, such that a well defined continuum limit can be reached. We demonstrate, as the first main result of this thesis, that the tuning is feasible and that, moreover, physical quantities are insensitive to the particular tuning condition. As in any lattice regularization with SF-like boundary conditions, there are also in the χSF a couple of counterterms at the boundaries, whose coefficients need to be tuned in order to remove the O(a) discretization effects originated at the boundaries. However, besides these boundary O(a) effects, the χSF is expected to be compatible with bulk automatic O(a)-improvement. We show here that, indeed, the scaling behavior of physical
On the chirally rotated Schroedinger functional with Wilson fermions
Energy Technology Data Exchange (ETDEWEB)
Gonzalez Lopez, Jenifer
2011-05-25
There are many phenomena in nature, which are closely linked to the low energy regime of QCD. From a theoretical point of view, these low energy phenomena can be dealt with only by means of non-perturbative methods. It is the central goal of this thesis to provide a framework for such a nonperturbative renormalization. For that purpose, we employ a 4-dimensional lattice as a regulator of QCD. As a renormalization scheme, we propose a finite volume Schroedinger functional scheme and here in particular, the chirally rotated Schroedinger functional ({chi}SF). We first perform analytical studies of the {chi}SF at tree-level of perturbation theory, in the continuum and on the lattice. We study the eigenvalue spectrum of the continuum Dirac operator, equipped with chirally rotated SF boundary conditions, and derive the corresponding quark propagator. We then determine the tree-level quark propagator on the lattice, employing massless Wilson fermions as a regulator of the theory. Beyond tree-level, all studies are performed in the quenched approximation of QCD, as a first, computationally much simpler step to understand the properties of the newly proposed {chi}SF scheme. One of the main targets of the present work, has been to perform the non-perturbative tuning of the two required coefficients of the {chi}SF scheme, such that a well defined continuum limit can be reached. We demonstrate, as the first main result of this thesis, that the tuning is feasible and that, moreover, physical quantities are insensitive to the particular tuning condition. As in any lattice regularization with SF-like boundary conditions, there are also in the {chi}SF a couple of counterterms at the boundaries, whose coefficients need to be tuned in order to remove the O(a) discretization effects originated at the boundaries. However, besides these boundary O(a) effects, the {chi}SF is expected to be compatible with bulk automatic O(a)-improvement. We show here that, indeed, the scaling behavior
Vortex Nucleation in a Dissipative Variant of the Nonlinear Schroedinger Equation Under Rotation
2014-12-01
iΩrotuθ, (2.1) where (·)t = d(·)/dt and (·)θ = d(·)/dθ and (r, θ) are the polar coordinates. Here the potential is assumed as representing a parabolic ... inclusion of dissipation in a Hamiltonian model leads modes of different energy (Krein signature) to move differently, due to their distinct topological...captures the initial stages of the dynamical evolution and the eventual asymptotic behavior may well be different. This can be due to symmetry
International Nuclear Information System (INIS)
Zhang Mincang; Sun Guohua; Dong Shihai
2010-01-01
A spherically harmonic oscillatory ring-shaped potential is proposed and its exactly complete solutions are presented by the Nikiforov-Uvarov method. The effect of the angle-dependent part on the radial solutions is discussed.
Quantitive theory of the Fermi-Pasta-Ulam recurrence in the nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Infeld, E.
1981-01-01
By limiting attention to the lowest-order Fourier modes we obtain a theory of the Fermi-Pasta-Ulam recurrence that gives excellent agreement with recent numerical results. Both the predicted period of the recurrence and the temporal development of the n = 0 mode are very good fits. The maximum of the n = 1 mode, however, is off by about 30%
Blow-up in nonlinear Schroedinger equations. II. Similarity structure of the blow-up singularity
DEFF Research Database (Denmark)
Rypdal, K.; Juul Rasmussen, Jens
1986-01-01
invariance and generalizations of the latter. This generalized "quasi-invariance" reveals the nature of the blow-up singularity and resolves an old controversy. Most of the previous work has been done on the cubic nonlinearity. We generalize the results to an arbitrary power nonlinearity....
Numerical solver of the time-dependent Schroedinger equation with Coulomb singularities
International Nuclear Information System (INIS)
Gordon, Ariel; Jirauschek, Christian; Kaertner, Franz X.
2006-01-01
This paper addresses a very fundamental and important problem in the numerical analysis of atomic and molecular systems: How to discretize Hamiltonians with divergent potential terms, such as Coulomb singularities. At the point of a Coulomb singularity, the wave function cannot be described by a Taylor series expansion, which results in problems when standard discretization schemes are used. We propose using the known asymptotic form of the wave function near the singularity instead of the (nonexistent) Taylor series. This principle, namely discretization by asymptotic behavior correspondence (ABC), is employed in this paper for obtaining grid-discretizations for the Coulomb potential in Cartesian, cylindrical and spherical coordinate systems. We show that computations with the ABC discretization are faster and more precise than with a naive discretization by orders of magnitude. The ABC discretization is well suited for the standard numerical time propagators, such as the Crank-Nicholson, Peaceman-Rachford, and leapfrog schemes. We use the latter, since it is faster and has the same order of accuracy. The leapfrog scheme is generalized to allow absorbing potentials at the grid boundaries
International Nuclear Information System (INIS)
Hoffmann-Ostenhof, M.; Hoffmann-Ostenhof, T.; Swetina, J.
1986-01-01
Let (- Δ + V 1 - E) psi = 0 in Ωsub(R) = (x is an element of Rsup(n)| |x| > R), psi is an element of L 2 (Ωsub(R)), where E 1 (|x|) + V 2 (|x|) with V 1 , V 2 tending to zero for |x| → infinity and satisfying suitable regularity assumption. Further let (- Δ + V 2 (|x|) - E) v(|x|) = 0 for |x| > R where v > 0 and v → 0 for |x| → infinity. Previous results on the asymptotics on psi/v for n = 2 are here extended to the n-dimensional case: It is shown that psi/v (|x| x/|x|) satisfies certain regularity properties uniformly for |x| → infinity as a map from Ssup(n-1) to R. Furthermore using a certain scaling it is shown that the asymptotic behaviour of psi/v can be characterized by eigenfunctions of the isotropic (n-1)-dimensional harmonic oscillator. (Author)
International Nuclear Information System (INIS)
Arians, S.
1997-01-01
We consider the Hamiltonian H=(p-A(x)) 2 /(2m)+V(x) of a quantum particle in a magnetic field B=rotA and a potential V in space dimensions ν≥2. If V is of short range, then the high-velocity limit of the scattering operator uniquely determines the magnetic field B and the potential V. If, in addition, long-range potentials V l are present, some knowledge of (the far out tail of) V l is needed to define a modified Dollard wave operator and a scattering operator S D . Again its high- velocity limit uniquely determines B and V=V s +V l . Moreover, we give explicit error bounds which are inverse proportional to the velocity. copyright 1997 American Institute of Physics
The resonance expansion for the Green's function of the Schroedinger and wave equations
International Nuclear Information System (INIS)
Albeverio, S.; Aix-Marseille-2 Univ., 13 - Marseille; Hoeegh-Krohn, R.; Oslo Univ.
1984-01-01
We give a survey of some recent mathematical work on resonances, in particular on perturbation series, low energy expansions and on resonances for point interactions. Expansions of the kernels of esup(-it)√sup(H+) and esup(-itH) in terms of resonances are also given (where Hsub(+) is the positive part of the Hamiltonian). (orig.)
Darboux transformation for the NLS equation
International Nuclear Information System (INIS)
Aktosun, Tuncay; Mee, Cornelis van der
2010-01-01
We analyze a certain class of integral equations associated with Marchenko equations and Gel'fand-Levitan equations. Such integral equations arise through a Fourier transformation on various ordinary differential equations involving a spectral parameter. When the integral operator is perturbed by a finite-rank perturbation, we explicitly evaluate the change in the solution in terms of the unperturbed quantities and the finite-rank perturbation. We show that this result provides a fundamental approach to derive Darboux transformations for various systems of ordinary differential operators. We illustrate our theory by providing the explicit Darboux transformation for the Zakharov-Shabat system and show how the potential and wave function change when a simple discrete eigenvalue is added to the spectrum, and thus we also provide a one-parameter family of Darboux transformations for the nonlinear Schroedinger equation.
Estimate of the difference between the Kac operator and the Schroedinger semigroup
International Nuclear Information System (INIS)
Ichinose, T.; Satoshi, S.
1997-01-01
An operator norm estimate of the difference between the Kac operator and the Schroedinger semigroup is proved and used to give a variant of the Trotter product formula for Schroedinger operators in the L p operator norm. This extends Helffer's result in the L 2 operator norm to the case in the L p operator norm for more general scalar potentials and with vector potentials. The method of the proof is probabilistic based on the Feynman-Kac a nd Feynman-Kac-Ito formula. (orig.)
Exact solutions of the one-dimensional generalized modified complex Ginzburg-Landau equation
International Nuclear Information System (INIS)
Yomba, Emmanuel; Kofane, Timoleon Crepin
2003-01-01
The one-dimensional (1D) generalized modified complex Ginzburg-Landau (MCGL) equation for the traveling wave systems is analytically studied. Exact solutions of this equation are obtained using a method which combines the Painleve test for integrability in the formalism of Weiss-Tabor-Carnevale and Hirota technique of bilinearization. We show that pulses, fronts, periodic unbounded waves, sources, sinks and solution as collision between two fronts are the important coherent structures that organize much of the dynamical properties of these traveling wave systems. The degeneracies of the 1D generalized MCGL equation are examined as well as several of their solutions. These degeneracies include two important equations: the 1D generalized modified Schroedinger equation and the 1D generalized real modified Ginzburg-Landau equation. We obtain that the one parameter family of traveling localized source solutions called 'Nozaki-Bekki holes' become a subfamily of the dark soliton solutions in the 1D generalized modified Schroedinger limit
Geometry, heat equation and path integrals on the Poincare upper half-plane
International Nuclear Information System (INIS)
Kubo, Reijiro.
1987-08-01
Geometry, heat equation and Feynman's path integrals are studied on the Poincare upper half-plane. The fundamental solution to the heat equation δf/δt = Δ H f is expressed in terms of a path integral defined on the upper half-plane. It is shown that Kac's proof that Feynman's path integral satisfies the Schroedinger equation is also valid for our case. (author)
Local and non-local Schroedinger cat states in cavity QED
International Nuclear Information System (INIS)
Haroche, S.
2005-01-01
Full text: I will review recent experiments performed on mesoscopic state superpositions of field states in cavity QED. Proposals to extend these studies to Schroedinger cat states delocalized in two cavities will be discussed. New versions of Bell's inequality tests will probe the non-local behavior of these cats and study their sensitivity to decoherence. (author)
Random Schroedinger operators and the theory of disordered systems: some rigorous results
International Nuclear Information System (INIS)
Kunz, H.; Souillard, B.
1981-01-01
The authors report results on a class of finite difference Schroedinger operators with stochastic potentials. The Hamiltonian is then H(V)=-Δ+V; where Δ is the discretized Laplacian and the potential V acts as a multiplication operator. The potential V is random. (Auth.)
Bounds on resolvents of dilated Schroedinger operators with non trapping potentials
International Nuclear Information System (INIS)
Duclos, P.; Klein, M.
1985-06-01
We provide bounds on resolvents of dilated Schroedinger operators via an exterior scaling. It is done under a non trapping condition on the potential which has a clear interpretation in classical mechanics. These bounds are a powerful tool to prove absence of resonances due to the tail of the potential in the shape resonance problem
International Nuclear Information System (INIS)
D'Agostino, S.
1992-01-01
In the 50s, Schroedinger proposed a new conception of a continuous theory of Quantum Mechanics, which remarkably modified his 1926 ideas on ondulatory mechanics. The lack of individuality of the atomic particles presented in the new statistics, and in Heisenberg's Indeterminacy Relations, was by him considered as an aspect of a more general crisis in the anthology itself of classical atomism. Unlike his 1926 ideas, he proposed now to represent the wave equation in an n-dimensional space and he considered second-quantization technique as the proper mathematical tool for his new physical conception. Although he accepted that space-time discontinuities and casual gaps may appear here and there on the observational level (e.g. in the Indeterminacy Relations), he was convinced that they could be made compatible with a continuous pure theory, provided one accepted a suitable conception of the theory's epistemiological status. For him, only a continuous theory satisfied the conditions for a complete theory. On these matters, he thought he was somehow orthodox to the ideas of Hertz and Boltzmann, which were also reflected in the teaching of Exner. (author). 69 refs
A variational Integro-Differential Equation for three identical particles in an S-state
International Nuclear Information System (INIS)
Fabre de la Ripelle, M.; Braun, M.; Sofianos, S.A.
1997-01-01
Starting from the Schroedinger equation, a new Variational Integro-Differential Equation (VIDE) for three bosons in S-state is derived. The wave function has the simple structure of a sum of two-body amplitudes. It is shown that the new equation gives results which are three orders of magnitude better than the corresponding results obtained from a single Faddeev equation, where the pairs are in an S-state. The latter equation generates an exact solution only for S-state projected potentials. Moreover, the ghost contributions occurring in the Faddeev amplitudes for three bosons in an S-state do not exist in the new equation. (author)
Simple functional-differential equations for the bound-state wave-function components
International Nuclear Information System (INIS)
Kamuntavicius, G.P.
1986-01-01
The author presents a new method of a direct derivation of differential equations for the wave-function components of identical-particles systems. The method generates in a simple manner all the possible variants of these equations. In some cases they are the differential equations of Faddeev or Yakubovskii. It is shown that the case of the bound states allows to formulate very simple equations for the components which are equivalent to the Schroedinger equation for the complete wave function. The components with a minimal antisymmetry are defined and the corresponding equations are derived. (Auth.)
Convergence of method of lines approximations to partial differential equations
International Nuclear Information System (INIS)
Verwer, J.G.; Sanz-Serna, J.M.
1984-01-01
Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. The main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C-stability. A nonlinear parabolic equation and the cubic Schroedinger equation are used for illustrating the ideas. (Auth.)
On the evolution equations, solvable through the inverse scattering method
International Nuclear Information System (INIS)
Gerdjikov, V.S.; Khristov, E.Kh.
1979-01-01
The nonlinear evolution equations (NLEE), related to the one-parameter family of Dirac operators are considered in a uniform manner. The class of NLEE solvable through the inverse scatterina method and their conservation laws are described. The description of the hierarchy of Hamiltonian structures and the proof of complete integrability of the NLEE is presented. The class of Baecklund transformations for these NLEE is derived. The general formulae are illustrated by two important examples: the nonlinear Schroedinger equation and the sine-Gordon equation
Erwin Schroedinger: Collected papers V. 4. General scientific and popular papers
International Nuclear Information System (INIS)
Schroedinger, E.
1984-01-01
The present volume contains all of Schroedinger's papers, which did not fit naturally into his earlier volumes. It certainly does not contain only popular writings and is perhaps more so than the other volumes a testimony of the spiritual breadth of its author. Schroedinger occupied himself very extensively with the physiological optics. His papers reach from survey articles, where an enormous experimental material is being sifted, to theoretical explorations. A leitmotiv in Schroedinger's thinking, which never left him in peace, was the duality of particle and wave. He looked upon the wave-picture as the one more to the point and in this connection expressed various fruitful thoughts as e.g. that the state of macroscopic bodies would not be eigenstates of the number of particles. He objected against provisional formulations and thus controverted against quantum jumps, which suggest a discontinuous time-development. Schroedinger rejected the radical positivism, which tolerates only directly observable quantities in a theory and he did not look upon the Copenhagen interpretation as the complete solution to the problem. The collection of 59 papers is a goldmine for all pedagogues. Numerous popularizations of Schroedinger's spheres of interest, which go widely beyond his field of exploration prove how simple and clear a topic can be presented. Let it be astronomy or biology, theory of knowledge or Greek mythology, he always discovered the heart of the questions and illustrated it with simple means. He never pretends learning and deepness of thoughts by cryptical remarks and by vague or contradictory formulations. All writings show his way of thinking, which cannot be restricted by authority and objects to any claim of eternal validity. For instance, he analyses various epoche-making arguements of Galileo and finds that his plausible explanation of the tides is wrong. His conceptions on consciousness, free will and human soul are documented in some essays too
Energy Technology Data Exchange (ETDEWEB)
Gaffney, J M
1975-01-01
A reappraisal of electromagnetic field theories is made and an account is given of the radiation gauge, Gupta-Bleuler and Fermi methods of quantitising the electromagnetic fields. The Weyl algebra of the vector potential is constructed and the Fermi method is then related to a certain representation of the algebra. The representation is specified by a generating functional for a state on the algebra. The Weyl algebra of the physical field is then constructed as a factor algebra. The Schroedinger representation of the algebra is then studied and it was found that the Fermi method is just a generalization of this representation to an infinite number of degrees of freedom. The Schroedinger representation of Fermi method is constructed.
The general Klein-Gordon-Schroedinger system: modulational instability and exact solutions
International Nuclear Information System (INIS)
Tang Xiaoyan; Ding Wei
2008-01-01
The general Klein-Gordon-Schroedinger (gKGS) system is studied where the cubic auto-interactions are introduced in both the nonlinear Schroedinger and the nonlinear Klein-Gordon fields. We first investigate the modulational instability (MI) of the system, and thus derive the general dispersion relation between the frequency and wavenumber of the modulating perturbations, which demonstrates many possibilities for the MI regions. Using the travelling wave reduction, the gKGS system is greatly simplified. Via a simple function expansion method, we obtain some exact travelling wave solutions. Under some special parameter values, some representative wave structures are graphically displayed including the kink, anti-kink, bright, dark, grey and periodic solitons
Optical soliton solutions for two coupled nonlinear Schroedinger systems via Darboux transformation
International Nuclear Information System (INIS)
Zhang Haiqiang; Li Juan; Xu Tao; Zhang Yaxing; Hu Wei; Tian Bo
2007-01-01
In nonlinear optical fibers, the vector solitons can be governed by the systems of coupled nonlinear Schroedinger from polarized optical waves in an isotropic medium. Based on the Ablowitz-Kaup-Newell-Segur technology, the Darboux transformation method is successfully applied to two coupled nonlinear Schroedinger systems. With the help of symbolic computation, the bright vector one- and two-soliton solutions including one-peak and two-peak solitons are further constructed via the iterative algorithm of Darboux transformation. Through the figures for several sample solutions, the stable propagation and elastic collisions for these kinds of bright vector solitons are discussed and the possible applications are pointed out in optical communications and relevant optical experiments.In addition, the conserved quantities of such two systems, i.e., the energy, momentum and Hamiltonian, are also presented
Bell's theorem and quantum realism. Reassessment in light of the Schroedinger paradox
International Nuclear Information System (INIS)
Shakur, Asif M.; Hemmick, Douglas L.
2012-01-01
Quantum theory presents a strange picture of the world, offering no real account of physical properties apart from observation. Neils Bohr felt that this reflected a core truth of nature: ''There is no quantum world. There is only an abstract mathematical description.'' Among the most significant developments since Bohr's day has been the theorem of John S. Bell. It is important to consider whether Bell's analysis supports such a denial of microrealism. In this book, we evaluate the situation in terms of an early work of Erwin Schroedinger. Doing so, we see how Bell's theorem is conceptually related to the Conway and Kochen Free Will theorem and also to all the major anti-realism efforts. It is easy to show that none of these analyses imply the impossibility of objective realism. We find that Schroedinger's work leads to the derivation of a new series of theoretical proofs and potential experiments, each involving ''entanglement,'' the link between particles in some quantum systems. (orig.)
One-dimensional Schroedinger operators with interactions singular on a discrete set
International Nuclear Information System (INIS)
Gesztesy, F.; Kirsch, W.
We study the self-adjointness of Schroedinger operators -d 2 /dx 2 +V(x) on an arbitrary interval, (a,b) with V(x) locally integrable on (a,b)inverse slantX where X is a discrete set. The treatment of quantum mechanical systems describing point interactions or periodic (possibly strongly singular) potentials is thereby included and explicit examples are presented. (orig.)
On a minimization of the eigenvalues of Schroedinger operator relatively domains
International Nuclear Information System (INIS)
Gasymov, Yu.S.; Niftiev, A.A.
2001-01-01
Minimization of the eigenvalues plays an important role in the operators spectral theory. The problem on the minimization of the eigenvalues of the Schroedinger operator by areas is considered in this work. The algorithm, analogous to the conditional gradient method, is proposed for the numerical solution of this problem in the common case. The result is generalized for the case of the positively determined completely continuous operator [ru
Berry phases for 3D Hartree-type equations with a quadratic potential and a uniform magnetic field
International Nuclear Information System (INIS)
Litvinets, F N; Shapovalov, A V; Trifonov, A Yu
2007-01-01
A countable set of asymptotic space-localized solutions is constructed for a 3D Hartree-type equation with a quadratic potential by the complex germ method in the adiabatic approximation. The asymptotic parameter is 1/T, where T >> 1 is the adiabatic evolution time. A generalization of the Berry phase of the linear Schroedinger equation is formulated for the Hartree-type equation. For the solutions constructed, the Berry phases are found in an explicit form
International Nuclear Information System (INIS)
Ziqi Sun
1993-01-01
During the past few years a considerable interest has been focused on the inverse boundary value problem for the Schroedinger operator with a scalar (electric) potential. The popularity gained by this subject seems to be due to its connection with the inverse scattering problem at fixed energy, the inverse conductivity problem and other important inverse problems. This paper deals with an inverse boundary value problem for the Schroedinger operator with vector (electric and magnetic) potentials. As in the case of the scalar potential, results of this study would have immediate consequences in the inverse scattering problem for magnetic field at fixed energy. On the other hand, inverse boundary value problems for elliptic operators are of independent interest. The study is partly devoted to the understanding of the inverse boundary value problem for a class of general elliptic operator of second order. Note that a self-adjoint elliptic operator of second order with Δ as its principal symbol can always be written as a Schroedinger operator with vector potentials
Phase-space formalism: Operational calculus and solution of evolution equations in phase-space
International Nuclear Information System (INIS)
Dattoli, G.; Torre, A.
1995-05-01
Phase-space formulation of physical problems offers conceptual and practical advantages. A class of evolution type equations, describing the time behaviour of a physical system, using an operational formalism useful to handle time ordering problems has been described. The methods proposed generalize the algebraic ordering techniques developed to deal with the ordinary Schroedinger equation, and how they are taylored suited to treat evolution problems both in classical and quantum dynamics has been studied
Darboux transformations and linear parabolic partial differential equations
International Nuclear Information System (INIS)
Arrigo, Daniel J.; Hickling, Fred
2002-01-01
Solutions for a class of linear parabolic partial differential equation are provided. These solutions are obtained by first solving a system of (n+1) nonlinear partial differential equations. This system arises as the coefficients of a Darboux transformation and is equivalent to a matrix Burgers' equation. This matrix equation is solved using a generalized Hopf-Cole transformation. The solutions for the original equation are given in terms of solutions of the heat equation. These results are applied to the (1+1)-dimensional Schroedinger equation where all bound state solutions are obtained for a 2n-parameter family of potentials. As a special case, the solutions for integral members of the regular and modified Poeschl-Teller potentials are recovered. (author). Letter-to-the-editor
Two-body Dirac equations for nucleon-nucleon scattering
International Nuclear Information System (INIS)
Liu Bin; Crater, Horace
2003-01-01
We investigate the nucleon-nucleon interaction by using the meson exchange model and the two-body Dirac equations of constraint dynamics. This approach to the two-body problem has been successfully tested for QED and QCD relativistic bound states. An important question we wish to address is whether or not the two-body nucleon-nucleon scattering problem can be reasonably described in this approach as well. This test involves a number of related problems. First we must reduce our two-body Dirac equations exactly to a Schroedinger-like equation in such a way that allows us to use techniques to solve them already developed for Schroedinger-like systems in nonrelativistic quantum mechanics. Related to this, we present a new derivation of Calogero's variable phase shift differential equation for coupled Schroedinger-like equations. Then we determine if the use of nine meson exchanges in our equations gives a reasonable fit to the experimental scattering phase shifts for n-p scattering. The data involve seven angular momentum states including the singlet states 1 S 0 , 1 P 1 , 1 D 2 and the triplet states 3 P 0 , 3 P 1 , 3 S 1 , 3 D 1 . Two models that we have tested give us a fairly good fit. The parameters obtained by fitting the n-p experimental scattering phase shift give a fairly good prediction for most of the p-p experimental scattering phase shifts examined (for the singlet states 1 S 0 , 1 D 2 and triplet states 3 P 0 , 3 P 1 ). Thus the two-body Dirac equations of constraint dynamics present us with a fit that encourages the exploration of a more realistic model. We outline generalizations of the meson exchange model for invariant potentials that may possibly improve the fit
Complete integrability of the difference evolution equations
International Nuclear Information System (INIS)
Gerdjikov, V.S.; Ivanov, M.I.; Kulish, P.P.
1980-01-01
The class of exactly solvable nonlinear difference evolution equations (DEE) related to the discrete analog of the one-dimensional Dirac problem L is studied. For this starting from L we construct a special linear non-local operator Λ and obtain the expansions of w and σ 3 deltaw over its eigenfunctions, w being the potential in L. This allows us to obtain compact expressions for the integrals of motion and to prove that these DEE are completely integrable Hamiltonian systems. Moreover, it is shown that there exists a hierarchy of Hamiltonian structures, generated by Λ, and the action-angle variables are explicity calculated. As particular cases the difference analog of the non-linear Schroedinger equation and the modified Korteweg-de-Vries equation are considered. The quantization of these Hamiltonian system through the use of the quantum inverse scattering method is briefly discussed [ru
A Solution of Time Dependent Schrodinger Equation by Quantum Walk
International Nuclear Information System (INIS)
Sekino, Hideo; Kawahata, Masayuki; Hamada, Shinji
2012-01-01
Time Dependent Schroedinger Equation (TDSE) with an initial Gaussian distribution, is solved by a discrete time/space Quantum Walk (QW) representing consecutive operations corresponding to a dot product of Pauli matrix and momentum operators. We call it as Schroedinger Walk (SW). Though an Hadamard Walk (HW) provides same dynamics of the probability distribution for delta-function-like initial distributions as that of the SW with a delta-function-like initial distribution, the former with a Gaussian initial distribution leads to a solution for advection of the probability distribution; the initial distribution splits into two distinctive distributions moving in opposite directions. Both mechanisms are analysed by investigating the evolution of the both amplitude components. Decoherence of the oscillating amplitudes in central region is found to be responsible for the splitting of the probability distribution in the HW.
Eigenvalues of the simplified ideal MHD ballooning equation
International Nuclear Information System (INIS)
Paris, R.B.; Auby, N.; Dagazian, R.Y.
1986-01-01
The investigation of the spectrum of the simplified differential equation describing the variation of the amplitude of the ideal MHD ballooning instability along magnetic field lines constitutes a multiparameter Schroedinger eigenvalue problem. An exact eigenvalue relation for the discrete part of the spectrum is obtained in terms of the oblate spheroidal functions. The dependence of the eigenvalues lambda on the two free parameters γ 2 and μ 2 of the equation is discussed, together with certain analytical approximations in the limits of small and large γ 2 . A brief review of the principal properties of the spheroidal functions is given in an appendix
Samples of noncommutative products in certain differential equations
International Nuclear Information System (INIS)
Legare, M
2010-01-01
A set of associative noncommutative products is considered in different differential equations of the ordinary and partial types. A method of separation of variables is considered for a large set of those systems. The products involved include for example some * products and some products based on Nijenhuis tensors, which are embedded in the differential equations of the Laplace/Poisson, Lax and Schroedinger styles. A comment on the *-products of Reshetikhin-Jambor-Sykora type is also given in relation to *-products of Vey type.
On norm resolvent convergence of Schroedinger operators with δ'-like potentials
International Nuclear Information System (INIS)
Golovaty, Yu D; Hryniv, R O
2010-01-01
For a function V:R→R that is integrable and compactly supported, we prove the norm resolvent convergence, as ε → 0, of a family S ε of one-dimensional Schroedinger operators on the line of the form S ε :=-d 2 /dx 2 + 1/ε 2 V(x/ε). If the potential V satisfies the conditions ∫ R V(ξ)dξ=0, ∫ R ξV(ξ)dξ=-1, then the functions ε -2 V(x/ε) converge in the sense of distributions as ε → 0 to δ'(x), and the limit S 0 of S ε might be considered as a 'physically motivated' interpretation of the one-dimensional Schroedinger operator with a potential δ'. In 1985, Seba claimed that the limit operator S 0 is the direct sum of the free Schroedinger operators on positive and negative semi-axes subject to the Dirichlet condition at x = 0, which suggested that in dimension 1 there is no non-trivial Hamiltonian with the potential δ'. In this paper, we show that in fact S 0 essentially depends on V: although the above results are true generically, in the exceptional (or 'resonant') case, the limit S 0 is non-trivial and is determined by the properties of an auxiliary Sturm-Liouville spectral problem associated with V. We then set V(ξ) = αΨ(ξ) with a fixed Ψ and show that there exists a countable set of resonances {α k } ∞ k=-∞ for which a partial transmission of the wave package occurs for S 0 .
Integrability and structural stability of solutions to the Ginzburg-Landau equation
Keefe, Laurence R.
1986-01-01
The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).
Solution of the Korteweg--de Vries equation in a half-space bounded by a wall
International Nuclear Information System (INIS)
Moses, H.E.
1976-01-01
A solution of the Korteweg--de Vries equation in the half-space 0 less than r less than infinity with the boundary condition V(0) = 0 is given. The boundary condition may be interpreted as the requirement that the plane which bounds the half-space be a rigid wall. Aside from possible physical interest, this solution, which is obtained from one of the potentials for the radial Schroedinger equation which do not scatter, appears to indicate that the radial Schroedinger equation and the corresponding Gel'fand--Levitan equation play a role in the case of the half-space bounded by a wall similar to that of the one-dimensional Schroedinger equation (-- infinity less than x less than infinity) and its corresponding Gel'fand--Levitan equation in the more usual full space treatment of the KdV equation. A possible interpretation of the solution presented is that it corresponds to the reflection of a wave by a wall, in which the incident wave is singular and the reflected wave is nonsingular but highly dispersive
Transfer matrix in 1D Schroedinger problems with constant and position-dependent mass
International Nuclear Information System (INIS)
Perez-Alvarez, R.; Rodriguez-Coppola, H.
1987-10-01
We consider the transfer matrix method for obtaining properties of standard wells and barriers in one-dimensional Schroedinger problems with constant and position-dependent mass. We report the formulae for the energy levels of a well and the transmission coefficient of a barrier. We demonstrate the continuity between virtual bound states and bound states in a well of position-dependent mass and the relation between the zero energy gap states of a periodic potential problem with the corresponding energies of the non-periodic ones with transmission coefficient equal to one. The calculations were carried out for a wide class of potential profiles. (author). 30 refs, 2 figs
Energy Technology Data Exchange (ETDEWEB)
Lopez, J. Gonzalez [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Jansen, K. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Renner, D.B. [Jefferson Lab, Newport News, VA (United States); Shindler, A. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik
2012-08-23
The use of chirally rotated boundary conditions provides a formulation of the Schroedinger functional that is compatible with automatic O(a) improvement of Wilson fermions up to O(a) boundary contributions. The elimination of bulk O(a) effects requires the non-perturbative tuning of the critical mass and one additional boundary counterterm. We present the results of such a tuning in a quenched setup for several values of the renormalized gauge coupling, from perturbative to nonperturbative regimes, and for a range of lattice spacings. We also check that the correct boundary conditions and symmetries are restored in the continuum limit. (orig.)
International Nuclear Information System (INIS)
Lopez, J. Gonzalez; Jansen, K.; Renner, D.B.; Shindler, A.
2012-01-01
The use of chirally rotated boundary conditions provides a formulation of the Schroedinger functional that is compatible with automatic O(a) improvement of Wilson fermions up to O(a) boundary contributions. The elimination of bulk O(a) effects requires the non-perturbative tuning of the critical mass and one additional boundary counterterm. We present the results of such a tuning in a quenched setup for several values of the renormalized gauge coupling, from perturbative to nonperturbative regimes, and for a range of lattice spacings. We also check that the correct boundary conditions and symmetries are restored in the continuum limit. (orig.)
On form factors of the conjugated field in the non-linear Schroedinger model
Energy Technology Data Exchange (ETDEWEB)
Kozlowski, K.K.
2011-05-15
Izergin-Korepin's lattice discretization of the non-linear Schroedinger model along with Oota's inverse problem provides one with determinant representations for the form factors of the lattice discretized conjugated field operator. We prove that these form factors converge, in the zero lattice spacing limit, to those of the conjugated field operator in the continuous model. We also compute the large-volume asymptotic behavior of such form factors in the continuous model. These are in particular characterized by Fredholm determinants of operators acting on closed contours. We provide a way of defining these Fredholm determinants in the case of generic paramaters. (orig.)
Basic properties of the current-current correlation measure for random Schroedinger operators
International Nuclear Information System (INIS)
Hislop, Peter D.; Lenoble, Olivier
2006-01-01
The current-current correlation measure plays a crucial role in the theory of conductivity for disordered systems. We prove a Pastur-Shubin-type formula for the current-current correlation measure expressing it as a thermodynamic limit for random Schroedinger operators on the lattice and the continuum. We prove that the limit is independent of the self-adjoint boundary conditions and independent of a large family of expanding regions. We relate this finite-volume definition to the definition obtained by using the infinite-volume operators and the trace-per-unit volume
Energy Technology Data Exchange (ETDEWEB)
Lopez, J. Gonzalez [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Jansen, K. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Renner, D.B. [Jefferson Lab, Newport News, VA (United States); Shindler, A. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik
2012-08-24
In a previous paper (J. G. Lopez et al.,2012) we have discussed the non-perturbative tuning of the chirally rotated Schroedinger functional ({chi}SF). This tuning is required to eliminate bulk O(a) cutoff effects in physical correlation functions. Using our tuning results obtained in this paper we perform scaling and universality tests analyzing the residual O(a) cutoff effects of several step-scaling functions and we compute renormalization factors at the matching scale. As an example of possible application of the {chi}SF we compute the renormalized strange quark mass using large volume data obtained from Wilson twisted mass fermions at maximal twist. (orig.)
International Nuclear Information System (INIS)
Lopez, J. Gonzalez; Jansen, K.; Renner, D.B.; Shindler, A.
2012-01-01
In a previous paper (J. G. Lopez et al.,2012) we have discussed the non-perturbative tuning of the chirally rotated Schroedinger functional (χSF). This tuning is required to eliminate bulk O(a) cutoff effects in physical correlation functions. Using our tuning results obtained in this paper we perform scaling and universality tests analyzing the residual O(a) cutoff effects of several step-scaling functions and we compute renormalization factors at the matching scale. As an example of possible application of the χSF we compute the renormalized strange quark mass using large volume data obtained from Wilson twisted mass fermions at maximal twist. (orig.)
Differential equations and integrable models: the SU(3) case
International Nuclear Information System (INIS)
Dorey, Patrick; Tateo, Roberto
2000-01-01
We exhibit a relationship between the massless a 2 (2) integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schroedinger equation. This forms part of a more general correspondence involving A 2 -related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the non-linear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators phi 12 , phi 21 and phi 15 . This is checked against previous results obtained using the thermodynamic Bethe ansatz
International Nuclear Information System (INIS)
Hesse, Dirk
2012-01-01
The author developed the pastor software package for automated lattice perturbation theory calculations in the Schroedinger functional scheme. The pastor code consists of two building blocks, dealing with the generation of Feynman rules and Feynman diagrams respectively. Accepting a rather generic class of lattice gauge and fermion actions, passed to the code in a symbolic form as input, a low level part of pastor will generate Feynman rules to an arbitrary order in the bare coupling with a trivial or an Abelian background field. The second, high level part of pastor is a code generator whose output relies on the vertex generator. It writes programs that evaluate Feynman diagrams for a class of Schroedinger functional observables up to one loop order automatically, the relevant O(a) improvement terms are taken into account. We will describe the algorithms used for implementation of both parts of the code in detail, and provide cross checks with perturbative and non-perturbative data to demonstrate the correctness of our code. We demonstrate the usefulness of the pastor package through various applications taken from the matching process of heavy quark effective theory with quantum chromodynamics. We have e.g. completed a one loop analysis for new candidates for matching observables timely and with rather small effort, highlighting two advantages of an automated software setup. The results that were obtained so far will be useful as a guideline for further non-perturbative studies.
Finite-time quantum-to-classical transition for a Schroedinger-cat state
International Nuclear Information System (INIS)
Paavola, Janika; Hall, Michael J. W.; Paris, Matteo G. A.; Maniscalco, Sabrina
2011-01-01
The transition from quantum to classical, in the case of a quantum harmonic oscillator, is typically identified with the transition from a quantum superposition of macroscopically distinguishable states, such as the Schroedinger-cat state, into the corresponding statistical mixture. This transition is commonly characterized by the asymptotic loss of the interference term in the Wigner representation of the cat state. In this paper we show that the quantum-to-classical transition has different dynamical features depending on the measure for nonclassicality used. Measures based on an operatorial definition have well-defined physical meaning and allow a deeper understanding of the quantum-to-classical transition. Our analysis shows that, for most nonclassicality measures, the Schroedinger-cat state becomes classical after a finite time. Moreover, our results challenge the prevailing idea that more macroscopic states are more susceptible to decoherence in the sense that the transition from quantum to classical occurs faster. Since nonclassicality is a prerequisite for entanglement generation our results also bridge the gap between decoherence, which is lost only asymptotically, and entanglement, which may show a ''sudden death''. In fact, whereas the loss of coherences still remains asymptotic, we emphasize that the transition from quantum to classical can indeed occur at a finite time.
Energy Technology Data Exchange (ETDEWEB)
Hesse, Dirk
2012-07-13
The author developed the pastor software package for automated lattice perturbation theory calculations in the Schroedinger functional scheme. The pastor code consists of two building blocks, dealing with the generation of Feynman rules and Feynman diagrams respectively. Accepting a rather generic class of lattice gauge and fermion actions, passed to the code in a symbolic form as input, a low level part of pastor will generate Feynman rules to an arbitrary order in the bare coupling with a trivial or an Abelian background field. The second, high level part of pastor is a code generator whose output relies on the vertex generator. It writes programs that evaluate Feynman diagrams for a class of Schroedinger functional observables up to one loop order automatically, the relevant O(a) improvement terms are taken into account. We will describe the algorithms used for implementation of both parts of the code in detail, and provide cross checks with perturbative and non-perturbative data to demonstrate the correctness of our code. We demonstrate the usefulness of the pastor package through various applications taken from the matching process of heavy quark effective theory with quantum chromodynamics. We have e.g. completed a one loop analysis for new candidates for matching observables timely and with rather small effort, highlighting two advantages of an automated software setup. The results that were obtained so far will be useful as a guideline for further non-perturbative studies.
Darboux transformations for the time-dependent nonhomogeneous Burgers equation in (1+1) dimensions
International Nuclear Information System (INIS)
Schulze-Halberg, Axel; Manuel Carballo Jimenez, Juan
2009-01-01
We extend the formalism of nth order Darboux transformations to the time-dependent nonhomogeneous Burgers equation (NBE) in (1+1) dimensions. Similar to the Schroedinger case, our Darboux transformation retains the form of the NBE, while changing the nonhomogeneous term. The transformed solution of the NBE and the corresponding transformed nonhomogeneity are given in closed form. Furthermore, properties of the transformation are discussed and an application is given.
International Nuclear Information System (INIS)
Bellissard, J.
1981-07-01
We exhibit an example of a one-dimensional discrete Schroedinger operator with an almost periodic potential for which the steps of the density of states do not belong to the frequency module. This example is suggested by the K-theory
International Nuclear Information System (INIS)
Inahama, Yuzuru; Shirai, Shin-ichi
2003-01-01
We study the essential spectrum of the magnetic Schroedinger operators on the Poincare upper-half plane and establish a hyperbolic analog of Iwatsuka's result [J. Math. Kyoto Univ. 23(3), 475-480 (1983)] on the stability of the essential spectrum under perturbations from constant magnetic fields
International Nuclear Information System (INIS)
Olive, D.
1987-01-01
The centenary of the birth of polymath Erwin Schrodinger was marked by a suitably multidisciplinary conference in April at London's Imperial College, reflecting the impact of the man's work on physics, chemistry, molecular biology and the history and philosophy of science
Energy Technology Data Exchange (ETDEWEB)
Olive, D.
1987-07-15
The centenary of the birth of polymath Erwin Schrodinger was marked by a suitably multidisciplinary conference in April at London's Imperial College, reflecting the impact of the man's work on physics, chemistry, molecular biology and the history and philosophy of science.
Spurious solutions in few-body equations
International Nuclear Information System (INIS)
Adhikari, S.K.; Gloeckle, W.
1979-01-01
After Faddeev and Yakubovskii showed how to write connected few-body equations which are free from discrete spurious solutions various authors have proposed different connected few-body scattering equations. Federbush first pointed out that Weinberg's formulation admits the existence of discrete spurious solutions. In this paper we investigate the possibility and consequence of the existence of spurious solutions in some of the few-body formulations. Contrary to a proof by Hahn, Kouri, and Levin and by Bencze and Tandy the channel coupling array scheme of Kouri, Levin, and Tobocman which is also the starting point of a formulation by Hahn is shown to admit spurious solutions. We can show that the set of six coupled four-body equations proposed independently by Mitra, Gillespie, Sugar, and Panchapakesan, by Rosenberg, by Alessandrini, and by Takahashi and Mishima and the seven coupled four-body equations proposed by Sloan and related by matrix multipliers to basic sets which correspond uniquely to the Schroedinger equation. These multipliers are likely to give spurious solutions to these equations. In all these cases spuriosities are shown to have no hazardous consequence if one is interested in studying the scattering problem
Imaginary Time Step Method to Solve the Dirac Equation with Nonlocal Potential
International Nuclear Information System (INIS)
Zhang Ying; Liang Haozhao; Meng Jie
2009-01-01
The imaginary time step (ITS) method is applied to solve the Dirac equation with nonlocal potentials in coordinate space. Taking the nucleus 12 C as an example, even with nonlocal potentials, the direct ITS evolution for the Dirac equation still meets the disaster of the Dirac sea. However, following the recipe in our former investigation, the disaster can be avoided by the ITS evolution for the corresponding Schroedinger-like equation without localization, which gives the convergent results exactly the same with those obtained iteratively by the shooting method with localized effective potentials.
Role of statistical linearization in the solution of nonlinear stochastic equations
International Nuclear Information System (INIS)
Budgor, A.B.
1977-01-01
The solution of a generalized Langevin equation is referred to as a stochastic process. If the external forcing function is Gaussian white noise, the forward Kolmogarov equation yields the transition probability density function. Nonlinear problems must be handled by approximation procedures e.g., perturbation theories, eigenfunction expansions, and nonlinear optimization procedures. After some comments on the first two of these, attention is directed to the third, and the method of statistical linearization is used to demonstrate a relation to the former two. Nonlinear stochastic systems exhibiting sustained or forced oscillations and the centered nonlinear Schroedinger equation in the presence of Gaussian white noise excitation are considered as examples. 5 figures, 2 tables
Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation
Energy Technology Data Exchange (ETDEWEB)
Kurzke, Matthias; Melcher, Christof; Moser, Roger; Spirn, Daniel
2009-06-15
A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of Ginzburg-Landau vortices for sphere-valued maps. In particular we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization is ruled by the Landau-Lifshitz-Gilbert equation, which combines characteristic properties of a nonlinear Schroedinger equation and a gradient flow. This paper studies the motion of the vortex centers under this evolution equation. (orig.)
Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation
International Nuclear Information System (INIS)
Kurzke, Matthias; Melcher, Christof; Moser, Roger; Spirn, Daniel
2009-01-01
A simplified model for the energy of the magnetization of a thin ferromagnetic film gives rise to a version of the theory of Ginzburg-Landau vortices for sphere-valued maps. In particular we have the development of vortices as a certain parameter tends to 0. The dynamics of the magnetization is ruled by the Landau-Lifshitz-Gilbert equation, which combines characteristic properties of a nonlinear Schroedinger equation and a gradient flow. This paper studies the motion of the vortex centers under this evolution equation. (orig.)
International Nuclear Information System (INIS)
LaChapelle, J.
2004-01-01
A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schroedinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette
Superfield generating equation of field-antifield formalism as a hyper-gauge theory
Energy Technology Data Exchange (ETDEWEB)
Batalin, Igor A. [P.N. Lebedev Physics Institute, Moscow (Russian Federation); Tomsk State Pedagogical University, Tomsk (Russian Federation); Lavrov, Peter M. [Tomsk State Pedagogical University, Tomsk (Russian Federation); National Research Tomsk State University, Tomsk (Russian Federation)
2017-02-15
Within a superfield approach, we formulate a simple quantum generating equation of the field-antifield formalism. Then we derive the Schroedinger equation with the Hamiltonian whose Δ-exact part serves as a generator to the quantum master transformations. We show that these generators do satisfy a nice composition law in terms of the quantum antibrackets. We also present an Sp(2) symmetric extension to the main construction, with specific features caused by the principal fact that all basic equations become Sp(2) vector-valued ones. (orig.)
International Nuclear Information System (INIS)
Fedele, Renato; Jovanovic, Dusan; De Nicola, Sergio; Eliasson, Bengt; Shukla, Padma K.
2010-01-01
The possibility of the decomposition of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) into a pair of coupled Schroedinger-type equations, is investigated. It is shown that, under suitable mathematical conditions, it is possible to construct the exact controlled solutions of the 3D GPE from the solutions of a linear 2D Schroedinger equation coupled with a 1D nonlinear Schroedinger equation (the transverse and longitudinal components of the GPE, respectively). The coupling between these two equations is the functional of the transverse and the longitudinal profiles. The applied method of nonlinear decomposition, called the controlling potential method (CPM), yields the full 3D solution in the form of the product of the solutions of the transverse and longitudinal components of the GPE. It is shown that the CPM constitutes a variational principle and sets up a condition on the controlling potential well. Its physical interpretation is given in terms of the minimization of the (energy) effects introduced by the control. The method is applied to the case of a parabolic external potential to construct analytically an exact BEC state in the form of a bright soliton, for which the quantitative comparison between the external and controlling potentials is presented.
International Nuclear Information System (INIS)
Zakhariev, B.N.; Chabanov, V.M.
2008-01-01
It was an important examination to give a review talk at the previous Conference on Inverse Quantum Scattering (1996, Lake Balaton) about computer visualization of this science in front of its fathers - creators, B.M. Levitan and V.A. Marchenko. We have achieved a new understanding that the discovered main rules of transformations of a single wave function bump, e.g., for the ground bound states of one dimensional quantum systems are applicable to any state of any potential with arbitrary number of bumps from finite to unlimited ones as scattering states and bound states embedded into continuum. It appeared that we need only to repeat the rule mentally the necessary number of times. That uttermost simplification and unification of physical notion of spectral, scattering and decay control for any potential have got an obligatory praise from B.M. Levitan at the conference and was a mighty stimulus for our further research. After that we have written both Russian (2002) and improved English editions of 'Submissive Quantum Mechanics. New Status of the Theory in Inverse Problem Approach' (appeared at the very end of 2007). This book was written for correction of the present defect in quantum education throughout the world. Recently the quantum IP intuition helped us to discover a new concept of permanent wave resonance with potential spatial oscillations. This means the constant wave swinging frequency on the whole energy intervals of spectral forbidden zones destroying physical solutions and deepening the theory of waves in periodic potentials. It also shows the other side of strengthening the fundamentally important magic structures. A 'new language' of wave bending will be presented to enrich our quantum intuition, e.g., the paradoxical effective attraction of barriers and repulsion of wells in multichannel systems, etc. (author)
Symmetries of the Schrodinger Equation and Algebra/Superalgebra Duality
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Toppan, Francesco
2014-12-01
Some key features of the symmetries of the Schroedinger equation that are common to a much broader class of dynamical systems (some under construction) are illustrated. I discuss the algebra/superalgebra duality involving rst and second-order differential operators. It provides different viewpoints for the spectrum-generating subalgebras. The representation dependent notion of on-shell symmetry is introduced. The difference in associating the time derivative symmetry operator with either a root or a Cartan generator of the sl(2) subalgebra is discussed. In application to one-dimensional Lagrangian superconformal sigma-models it implies superconformal actions which are either supersymmetric or non-supersymmetric. (author)
Hamiltonian models for the Madelung fluid and generalized Langevin equations
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Nonnenmacher, T.F.
1985-01-01
We present a Hamiltonian formulation of some type of an 'electromagnetic' Madelung fluid leading to a fluid mechanics interpretation of the Aharonov-Bohm effect and to a subsidary condition to be required in order to make the correspondence between Schroedinger's quantum mechanics and Madelung's fluid mechanics unique. Then we discuss some problems related with the Brownian oscillator. Our aim is to start out with a Hamiltonian for the composite system with surrounding heat bath) and to finally arrive at a stochastic differential equation with completely determined statistical properties. (orig./HSI)
Multibreather solitons in the diffraction managed NLS equation
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Panayotaros, Panayotis
2006-01-01
We study analytically and numerically localized breather solutions in the averaged discrete nonlinear Schroedinger equation (NLS) with diffraction management, a system that models coupled waveguide arrays with periodic diffraction management geometries. Localized breathers can be characterized as constrained critical points of the Hamiltonian of the averaged diffraction managed NLS. In addition to local extrema, we find numerically more general solutions that are saddle points of the constrained Hamiltonian. An interesting class of saddle points are 'multi-bump' solutions that are close to superpositions of translates of simpler breathers. In the case of zero residual diffraction and small diffraction management, the existence of multibumps can be shown rigorously by a continuation argument
The number of bound states for a discrete Schroedinger operator on ZN, N≥1, lattices
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Karachalios, N I
2008-01-01
We consider the discrete Schroedinger operator -Δ d +U in Z N , N≥1 in the case of a potential with negative part in an appropriate l σ -space (decays with an appropriate rate). We present a discrete analog of the method of Li and Yau (1983 Commun. Math. Phys. 88 309-18), proving an explicit upper estimate on the number of bound states N d (0)={j:μ j ≤0}, which is independent of the dimension of the lattice. This is a major difference with the continuous counterpart estimate, which is not valid when N = 1, 2. As a consequence, a dimension-independent smallness criterion for the existence of bound states is derived in contrast to the continuous case as well as to the discrete case of vanishing potential. A short comment is made on possible applications of the results to the study of the dynamics of some particular spatially discrete nonlinear systems
Antibound states for a class of one-dimensional Schroedinger Operators
Energy Technology Data Exchange (ETDEWEB)
Angeletti, A [Camerino Univ. (Italy). Ist. di Matematica
1980-11-01
Let delta(x) be the Dirac's delta, q(x) element of L/sup 1/(R) L/sup 2/(R) be a real valued function, and lambda, ..mu.. element of R; we will consider the following class of one-dimensional formal Schroedinger operators on L/sup 2/(R) H(lambda,..mu..) = - (d/sup 2//dx/sup 2/) + lambdadelta(x) + ..mu..q(x). It is known that to the formal operator H(lambda, ..mu..) may be associated a selfadjoint operator H(lambda, ..mu..) on L/sup 2/(R). If q is of finite range, for lambda < 0 and /..mu../ is small enough, we prove that H(lambda,..mu..) has an antibound state; that is the resolvent of H(lambda,..mu..) has a pole on the negative real axis on the second Riemann sheet.
Algebraic Bethe ansatz for a quantum integrable derivative nonlinear Schroedinger model
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Basu-Mallick, B.; Bhattacharyya, Tanaya
2002-01-01
We find that the quantum monodromy matrix associated with a derivative nonlinear Schroedinger (DNLS) model exhibits U(2) or U(1,1) symmetry depending on the sign of the related coupling constant. By using a variant of quantum inverse scattering method which is directly applicable to field theoretical models, we derive all possible commutation relations among the operator valued elements of such monodromy matrix. Thus, we obtain the commutation relation between creation and annihilation operators of quasi-particles associated with DNLS model and find out the S-matrix for two-body scattering. We also observe that, for some special values of the coupling constant, there exists an upper bound on the number of quasi-particles which can form a soliton state for the quantum DNLS model
Some simple conditions of bound states of Schroedinger operators in dimension d >= 3
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Exner, P.
1984-01-01
A necessary condition for existence of bound states below a given energy of a Schroedinger operator H=-Δ+V on L 2 (Rsup(d)), d>=3, together with a lower bound to the ground-state energy of H are derived using the Sobolev inequalities. It generalizes some recent results to the dimensions d>3 and to the potentials that are not necessarily rapidly decreasing. Comparison to other known necessary conditions is given. The examples of the d-dimensional hydrogen-like atom and the d-dimensional harmonic oscillator are discussed. In both of them the bound to the ground-state energy becomes remarkably tight for large values of d
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Silverman, J.N.
1983-01-01
A generalized Euler transformation (GET) is introduced which provides a powerful alternative method of accurately summing strongly divergent Rayleigh-Schroedinger (RS) perturbation series when other summability methods fail or are difficult to apply. The GET is simple to implement and, unlike a number of other summation procedures, requires no a priori knowledge of the analytic properties of the function underlying the RS series. Application of the GET to the difficult problem of the RS weak-field ground-state eigenvalue series of the hydrogen atom in a magnetic field (quadratic Zeeman effect) yields sums of good accuracy over a very wide range of field strengths up to the most intense fields of 10 14 G. The GET results are compared with those obtained by other summing methods
Gauge invariance of the Rayleigh--Schroedinger time-independent perturbation theory
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Yang, K.H.
1977-08-01
It is shown that the Rayleigh-Schroedinger time-independent perturbation theory is gauge invariant when the operator concerned is the particle's instantaneous energy operator H/sub B/ = (1/2m)[vector p - (e/c) vector A] 2 + eV 0 . More explicitly, it is shown that the energy perturbation corrections of each individual order of every state is gauge invariant. When the vector potential is curlless, the energy corrections of all orders are shown to vanish identically regardless of the explicit form of the vector potential. The relation between causality and gauge invariance is investigated. It is shown that gauge invariance guarantees conformity with causality and violation of gauge invariance implies violation of causality
Antibound states for a class of one-dimensional Schroedinger Operators
International Nuclear Information System (INIS)
Angeletti, A.
1980-01-01
Let delta(x) be the Dirac's delta, q(x) element of L 1 (R) L 2 (R) be a real valued function, and lambda, μ element of R; we will consider the following class of one-dimensional formal Schroedinger operators on L 2 (R) H(lambda,μ) = - (d 2 /dx 2 ) + lambdadelta(x) + μq(x). It is known that to the formal operator H(lambda, μ) may be associated a selfadjoint operator H(lambda, μ) on L 2 (R). If q is of finite range, for lambda < 0 and /μ/ is small enough, we prove that H(lambda,μ) has an antibound state; that is the resolvent of H(lambda,μ) has a pole on the negative real axis on the second Riemann sheet. (orig.)
Degenerate RS perturbation theory. [Rayleigh-Schroedinger energies and wave functions
Hirschfelder, J. O.; Certain, P. R.
1974-01-01
A concise, systematic procedure is given for determining the Rayleigh-Schroedinger energies and wave functions of degenerate states to arbitrarily high orders even when the degeneracies of the various states are resolved in arbitrary orders. The procedure is expressed in terms of an iterative cycle in which the energy through the (2n + 1)-th order is expressed in terms of the partially determined wave function through the n-th order. Both a direct and an operator derivation are given. The two approaches are equivalent and can be transcribed into each other. The direct approach deals with the wave functions (without the use of formal operators) and has the advantage that it resembles the usual treatment of nondegenerate perturbations and maintains close contact with the basic physics. In the operator approach, the wave functions are expressed in terms of infinite-order operators which are determined by the successive resolution of the space of the zeroth-order functions.
Computation of a long-time evolution in a Schroedinger system
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Girard, R.; Kroeger, H.; Labelle, P.; Bajzer, Z.
1988-01-01
We compare different techniques for the computation of a long-time evolution and the S matrix in a Schroedinger system. As an application we consider a two-nucleon system interacting via the Yamaguchi potential. We suggest computation of the time evolution for a very short time using Pade approximants, the long-time evolution being obtained by iterative squaring. Within the technique of strong approximation of Moller wave operators (SAM) we compare our calculation with computation of the time evolution in the eigenrepresentation of the Hamiltonian and with the standard Lippmann-Schwinger solution for the S matrix. We find numerical agreement between these alternative methods for time-evolution computation up to half the number of digits of internal machine precision, and fairly rapid convergence of both techniques towards the Lippmann-Schwinger solution
Low-mode truncation methods in the sine-Gordon equation
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Xiong Chuyu.
1991-01-01
In this dissertation, the author studies the chaotic and coherent motions (i.e., low-dimensional chaotic attractor) in some near integrable partial differential equations, particularly the sine-Gordon equation and the nonlinear Schroedinger equation. In order to study the motions, he uses low mode truncation methods to reduce these partial differential equations to some truncated models (low-dimensional ordinary differential equations). By applying many methods available to low-dimensional ordinary differential equations, he can understand the low-dimensional chaotic attractor of PDE's much better. However, there are two important questions one needs to answer: (1) How many modes is good enough for the low mode truncated models to capture the dynamics uniformly? (2) Is the chaotic attractor in a low mode truncated model close to the chaotic attractor in the original PDE? And how close is? He has developed two groups of powerful methods to help to answer these two questions. They are the computation methods of continuation and local bifurcation, and local Lyapunov exponents and Lyapunov exponents. Using these methods, he concludes that the 2N-nls ODE is a good model for the sine-Gordon equation and the nonlinear Schroedinger equation provided one chooses a 'good' basis and uses 'enough' modes (where 'enough' depends on the parameters of the system but is small for the parameter studied here). Therefore, one can use 2N-nls ODE to study the chaos of PDE's in more depth
A novel method to solve functional differential equations
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Tapia, V.
1990-01-01
A method to solve differential equations containing the variational operator as the derivation operation is presented. They are called variational differential equations (VDE). The solution to a VDE should be a function containing the derivatives, with respect to the base space coordinates, of the fields up to a generic order s: a s-th-order function. The variational operator doubles the order of the function on which it acts. Therefore, in order to make compatible the orders of the different terms appearing in a VDE, the solution should be a function containing the derivatives of the fields at all orders. But this takes us again back to the functional methods. In order to avoid this, one must restrict the considerations, in the case of second-order VDEs, to the space of s-th-order functions on which the variational operator acts transitively. These functions have been characterized for a one-dimensional base space for the first- and second-order cases. These functions turn out to be polynomial in the highest-order derivatives of the fields with functions of the lower-order derivatives as coefficients. Then VDEs reduce to a system of coupled partial differential equations for the coefficients above mentioned. The importance of the method lies on the fact that the solutions to VDEs are in a one-to-one correspondence with the solutions of functional differential equations. The previous method finds direct applications in quantum field theory, where the Schroedinger equation plays a central role. Since the Schroedinger equation is reduced to a system of coupled partial differential equations, this provides a nonperturbative scheme for quantum field theory. As an example, the massless scalar field is considered
Solutions for confluent and double-confluent Heun equations and some applications
Energy Technology Data Exchange (ETDEWEB)
El-Jaick, Lea Jaccoud; Figueiredo, Bartolomeu D.B. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil)
2008-07-01
This paper examines some solutions for confluent and double-confluent Heun equations and their applications to the Schroedinger equation with quasi-exactly solvable potentials. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)] and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. In the third place, solutions of the Heun equations are used to solve the one-dimensional Schroedinger equation for quasi-exactly solvable potentials. We consider a symmetric and an asymmetric double-Morse potentials which appear in the theory of quantum spin systems [O. B. Zaslavskii and V. V. Ulyanov, Sov. Phys. JETP 60, 991 (1984)], a bottomless volcano-type potential which gives degenerate eigenstates [S. Kar and R. R. Parwani, Europhys. Lett., 80, 30004 (2007)], and a potential which leads to quasi normal modes, that is, to solutions presenting complex energies [H. T. Cho and C. L. Ho, J. Phys. A: Math. Theor. 40, 1325 (2007)]. (author)
Solutions for confluent and double-confluent Heun equations and some applications
International Nuclear Information System (INIS)
El-Jaick, Lea Jaccoud; Figueiredo, Bartolomeu D.B.
2008-01-01
This paper examines some solutions for confluent and double-confluent Heun equations and their applications to the Schroedinger equation with quasi-exactly solvable potentials. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation [E. W. Leaver, J. Math. Phys. 27, 1238 (1986)] and introduce an additional expansion in series of irregular confluent hypergeometric functions. Then, we find the conditions under which one of these solutions can be written as a linear combination of the others. In the second place, by means of limiting procedures we generate solutions for the double-confluent equation as well as for special limits of both the confluent and double-confluent equations. In the third place, solutions of the Heun equations are used to solve the one-dimensional Schroedinger equation for quasi-exactly solvable potentials. We consider a symmetric and an asymmetric double-Morse potentials which appear in the theory of quantum spin systems [O. B. Zaslavskii and V. V. Ulyanov, Sov. Phys. JETP 60, 991 (1984)], a bottomless volcano-type potential which gives degenerate eigenstates [S. Kar and R. R. Parwani, Europhys. Lett., 80, 30004 (2007)], and a potential which leads to quasi normal modes, that is, to solutions presenting complex energies [H. T. Cho and C. L. Ho, J. Phys. A: Math. Theor. 40, 1325 (2007)]. (author)
Single-site Green function of the Dirac equation for full-potential electron scattering
Energy Technology Data Exchange (ETDEWEB)
Kordt, Pascal
2012-05-30
I present an elaborated analytical examination of the Green function of an electron scattered at a single-site potential, for both the Schroedinger and the Dirac equation, followed by an efficient numerical solution, in both cases for potentials of arbitrary shape without an atomic sphere approximation. A numerically stable way to calculate the corresponding regular and irregular wave functions and the Green function is via the angular Lippmann-Schwinger integral equations. These are solved based on an expansion in Chebyshev polynomials and their recursion relations, allowing to rewrite the Lippmann-Schwinger equations into a system of algebraic linear equations. Gonzales et al. developed this method for the Schroedinger equation, where it gives a much higher accuracy compared to previous perturbation methods, with only modest increase in computational effort. In order to apply it to the Dirac equation, I developed relativistic Lippmann-Schwinger equations, based on a decomposition of the potential matrix into spin spherical harmonics, exploiting certain properties of this matrix. The resulting method was embedded into a Korringa-Kohn-Rostoker code for density functional calculations. As an example, the method is applied by calculating phase shifts and the Mott scattering of a tungsten impurity. (orig.)
Single-site Green function of the Dirac equation for full-potential electron scattering
International Nuclear Information System (INIS)
Kordt, Pascal
2012-01-01
I present an elaborated analytical examination of the Green function of an electron scattered at a single-site potential, for both the Schroedinger and the Dirac equation, followed by an efficient numerical solution, in both cases for potentials of arbitrary shape without an atomic sphere approximation. A numerically stable way to calculate the corresponding regular and irregular wave functions and the Green function is via the angular Lippmann-Schwinger integral equations. These are solved based on an expansion in Chebyshev polynomials and their recursion relations, allowing to rewrite the Lippmann-Schwinger equations into a system of algebraic linear equations. Gonzales et al. developed this method for the Schroedinger equation, where it gives a much higher accuracy compared to previous perturbation methods, with only modest increase in computational effort. In order to apply it to the Dirac equation, I developed relativistic Lippmann-Schwinger equations, based on a decomposition of the potential matrix into spin spherical harmonics, exploiting certain properties of this matrix. The resulting method was embedded into a Korringa-Kohn-Rostoker code for density functional calculations. As an example, the method is applied by calculating phase shifts and the Mott scattering of a tungsten impurity. (orig.)
International Nuclear Information System (INIS)
Amour, L.; Raoux, T.; Faupin, J.
2009-01-01
We pursue the analysis of the Schroedinger operator on the unit interval in inverse spectral theory initiated in the work of Amour and Raoux [''Inverse spectral results for Schroedinger operators on the unit interval with potentials in Lp spaces,'' Inverse Probl. 23, 2367 (2007)]. While the potentials in the work of Amour and Raoux belong to L 1 with their difference in L p (1≤p k,1 spaces having their difference in W k,p , where 1≤p≤+∞, k(set-membership sign)(0,1,2). It is proved that two potentials in W k,1 ([0,1]) being equal on [a,1] are also equal on [0,1] if their difference belongs to W k,p ([0,a]) and if the number of their common eigenvalues is sufficiently high. Naturally, this number decreases as the parameter a decreases and as the parameters k and p increase
Energy Technology Data Exchange (ETDEWEB)
Tang, Jau
1996-02-01
As an alternative to better physical explanations of the mechanisms of quantum interference and the origins of uncertainty broadening, a linear hopping model is proposed with ``color-varying`` dynamics to reflect fast exchange between time-reversed states. Intricate relations between this model, particle-wave dualism, and relativity are discussed. The wave function is shown to possess dual characteristics of a stable, localized ``soliton-like`` de Broglie wavelet and a delocalized, interfering Schroedinger carrier wave function.
Integrability of the Gross-Pitaevskii equation with Feshbach resonance management
International Nuclear Information System (INIS)
Zhao Dun; Luo Honggang; Chai Huayue
2008-01-01
In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schroedinger equation. By this transformation, each exact solution of the standard nonlinear Schroedinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique
Nonlinear q-Generalizations of Quantum Equations: Homogeneous and Nonhomogeneous Cases—An Overview
Directory of Open Access Journals (Sweden)
Fernando D. Nobre
2017-01-01
Full Text Available Recent developments on the generalizations of two important equations of quantum physics, namely the Schroedinger and Klein–Gordon equations, are reviewed. These generalizations present nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard linear equations are recovered in the limit q → 1 . Interestingly, these equations present a common, soliton-like, traveling solution, which is written in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics. In both cases, the corresponding well-known Einstein energy-momentum relations, as well as the Planck and the de Broglie ones, are preserved for arbitrary values of q. In order to deal appropriately with the continuity equation, a classical field theory has been developed, where besides the usual Ψ ( x → , t , a new field Φ ( x → , t must be introduced; this latter field becomes Ψ * ( x → , t only when q → 1 . A class of linear nonhomogeneous Schroedinger equations, characterized by position-dependent masses, for which the extra field Φ ( x → , t becomes necessary, is also investigated. In this case, an appropriate transformation connecting Ψ ( x → , t and Φ ( x → , t is proposed, opening the possibility for finding a connection between these fields in the nonlinear cases. The solutions presented herein are potential candidates for applications to nonlinear excitations in plasma physics, nonlinear optics, in structures, such as those of graphene, as well as in shallow and deep water waves.
A new approach to the I/N-expansion for the Dirac equation
International Nuclear Information System (INIS)
Stepanov, S.S.; Tutik, R.S.
1991-01-01
The difficulties associated with application of the I/N-expansion to the Dirac equation have been resolved by applying the method of (h/2π)-expansion. This technique does not involve converting the initial equation into the Schroedinger-like or Klein-Gordon-like form. Obtained recurrence formulae have a simple form and allow one to find the I/N-corrections of an arbitrary order in any of the I/N-expansion scheme. The method restores the exact results for the Coulomb potential. 17 refs. (author)
The (ℎ/2π)-expansion for Regge-trajectories. 2. Relativistic equations
International Nuclear Information System (INIS)
Stepanov, S.S.; Tutik, R.S.
1992-01-01
The (h/2π)-expansion method, proposed earlier for deriving Regge trajectories for bound states of central potentials in the Schroedinger equation framework, is extended to the Klein-Gordon and Dirac equations with potentials having vector and scalar components. The simple recursion formulae, with the same form both for the parent and daughter Regge trajectories, are obtained. They provide, in principle, the calculation of the (h/2π)-expansion terms up to an arbitrary order. As an illustration, a superposition of the vector and scalar Coulomb potentials, and the funnel-shaped potential are treated with the technique developed. 20 refs.; 3 figs.; 1 table. (author)
Moiseiwitsch, B L
2005-01-01
Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco
Ince's limits for confluent and double-confluent Heun equations
International Nuclear Information System (INIS)
Figueiredo, B.D. Bonorino
2005-01-01
We find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation (this is also known as confluent Heun equation). One solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z, while the other is given by a series of modified Bessel functions and converges for vertical bar z vertical bar > vertical bar z 0 vertical bar, where z 0 denotes a regular singularity. For short, the preceding limit is called Ince's limit after Ince who have used the same procedure to get the Mathieu equations from the Whittaker-Hill ones. We find as well that, when z 0 tends to zero, the Ince limit of the generalized spheroidal wave equation turns out to be the Ince limit of a double-confluent Heun equation, for which solutions are provided. Finally, we show that the Schroedinger equation for inverse fourth- and sixth-power potentials reduces to peculiar cases of the double-confluent Heun equation and its Ince's limit, respectively
The Bethe-Salpeter equation with fermions
International Nuclear Information System (INIS)
Efimov, G.V.
2007-01-01
The Bethe-Salpeter (BS) equation in the ladder approximation is studied within a fermion theory: two fermion fields (constituents) with mass m interacting via an exchange of a scalar field with mass μ. The BS equation can be written in the form of an integral equation in the configuration Euclidean x-space with the symmetric kernel K for which Tr K 2 = ∞ due to the singular character of the fermion propagator. This kernel is represented in the form K = K 0 + K I . The operator K 0 with Tr K 0 2 ∞ is of the 'fall at the center' potential type and describes a continuous spectrum only. Besides the presence of this operator leads to a restriction on the value of the coupling constant. The kernel K I with Tr K I 2 2 c 2 and the variational procedure of calculations of eigenvalues and eigenfunctions can be applied. The quantum pseudoscalar and scalar mesodynamics is considered. The binding energy of the state 1 + (deuteron) as a function of the coupling constant is calculated in the framework of the procedure formulated above. It is shown that this bound state is absent in the pseudoscalar mesodynamics and does exist in the scalar mesodynamics. A comparison with the non-relativistic Schroedinger picture is made. (author)
Tricomi, FG
2013-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and diff
Barbu, Viorel
2016-01-01
This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.
Single particle Schroedinger fluid and moments of inertia of deformed nuclei
International Nuclear Information System (INIS)
Doma, S.B.
2002-01-01
The authors have applied the theory of the single-particle Schroedinger fluid to the nuclear collective motion of axially deformed nuclei. A counter example of an arbitrary number of independent nucleons in the anisotropic harmonic oscillator potential at the equilibrium deformation has been also given. Moreover, the ground states of the doubly even nuclei in the s-d shell 20 Ne, 24 Mg, 28 Si, 32 S and 36 Ar are constructed by filling the single-particle states corresponding to the possible values of the number of quanta of excitations n x , n y and n z . Accordingly, the cranking-model, the rigid-body model and the equilibrium-model moments of inertia of these nuclei are calculated as functions of the oscillator parameters ℎω x , ℎω y and ℎω z which are given in terms of the non deformed value ℎω 0 0 , depending on the mass number A, the number of neutrons N, the number of protons Z, and the deformation parameter β. The calculated values of the cranking-model moments of inertia of these nuclei are in good agreement with the corresponding experiential values and show that the considered axially deformed nuclei may have oblate as well as prolate shapes and that the nucleus 24 Mg is the only one which is highly deformed. The rigid-body model and the equilibrium-model moments of inertia of the two nuclei 20 Ne and 24 Mg are also in good agreement with the corresponding experimental values
International Nuclear Information System (INIS)
Hagedorn, G.A.
1979-01-01
We investigate elastic and inelastic (2 cluster)→(2 cluster)scattering for classes of two, three, and four body Schroedinger operators H=H 0 +ΣVij. Formulas are derived for those generalized eigenfunctions of H which correspond asymptotically in the past to two freely moving clusters. With these eigenfunctions, we establish a formula for the (2 cluster)→(2 cluster) T-matrix and prove the convergence of a Born series for the T-matrix at high energy. (orig.) [de
Energy Technology Data Exchange (ETDEWEB)
Dubrovsky, V. G.; Topovsky, A. V. [Novosibirsk State Technical University, Karl Marx prosp. 20, Novosibirsk 630092 (Russian Federation)
2013-03-15
New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u{sup (n)}, n= 1, Horizontal-Ellipsis , N are constructed via Zakharov and Manakov {partial_derivative}-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u{sup (n)} and calculated by {partial_derivative}-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schroedinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums of special solutions u{sup (n)}. It is shown that the sums u=u{sup (k{sub 1})}+...+u{sup (k{sub m})}, 1 Less-Than-Or-Slanted-Equal-To k{sub 1} < k{sub 2} < Horizontal-Ellipsis < k{sub m} Less-Than-Or-Slanted-Equal-To N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schroedinger equation and can serve as model potentials for electrons in planar structures of modern electronics.
Energy Technology Data Exchange (ETDEWEB)
Garcia-Ravelo, J.; Trujillo, A. L. [Escuela Superior de Fisica y Matematicas, Instituto Politecnico Nacional, Unidad Profesional Adolfo Lopez Mateos, Zacatenco, 07738 Mexico D.F. (Mexico); Schulze-Halberg, A. [Department of Mathematics and Actuarial Science, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States)
2012-10-15
We obtain explicit formulas for perturbative corrections of the infinite quantum well model. The formulas we obtain are based on a class of matrix elements that we construct by means of two-parameter ladder operators associated with the infinite quantum well system. Our approach can be used to construct solutions to Schroedinger-type equations that involve generalized harmonic perturbations of their potentials, such as cosine powers, Fourier series, and more general functions. As a particular case, we obtain characteristic values for odd periodic solutions of the Mathieu equation.
Analytic structure of solutions to multiconfiguration equations
Energy Technology Data Exchange (ETDEWEB)
Fournais, Soeren [Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Arhus C (Denmark); Hoffmann-Ostenhof, Maria [Fakultaet fuer Mathematik, Universitaet Wien, Nordbergstrasse 15, A-1090 Vienna (Austria); Hoffmann-Ostenhof, Thomas [Institut fuer Theoretische Chemie, Waehringerstrasse 17, Universitaet Wien, A-1090 Vienna (Austria); Soerensen, Thomas Oestergaard [Department of Mathematics, Imperial College London, Huxley Building, 180 Queen' s Gate, London SW7 2AZ (United Kingdom)], E-mail: fournais@imf.au.dk, E-mail: Maria.Hoffmann-Ostenhof@univie.ac.at, E-mail: thoffman@esi.ac.at, E-mail: t.sorensen@imperial.ac.uk
2009-08-07
We study the regularity at the positions of the (fixed) nuclei of solutions to (non-relativistic) multiconfiguration equations (including Hartree-Fock) of Coulomb systems. We prove the following: let {l_brace}{psi}{sub 1}, ..., {psi}{sub M}{r_brace} be any solution to the rank-M multiconfiguration equations for a molecule with L fixed nuclei at R{sub 1},...,R{sub L} element of R{sup 3}. Then, for any j in {l_brace}1, ..., M{r_brace}, k in {l_brace}1, ..., L{r_brace}, there exists a neighborhood U{sub j,k} subset or equal R{sup 3} of R{sub k}, and functions {psi}{sup (1)}{sub j,k}, {psi}{sup (2)}{sub j,k}, real analytic in U{sub j,k}, such that {phi}{sub j}(x)={phi}{sub j,k}{sup (1)}(x)+|x-R{sub k}|{phi}{sub j,k}{sup (2)}(x), x element of U{sub j,k}. A similar result holds for the corresponding electron density. The proof uses the Kustaanheimo-Stiefel transformation, as applied in [9] to the study of the eigenfunctions of the Schroedinger operator of atoms and molecules near two-particle coalescence points.
Indian Academy of Sciences (India)
regarding nature of forces hold equally for liquids, even though the ... particle. Figure A. A fluid particle is a very small imaginary blob of fluid, here shown sche- matically in .... picture gives important information about the flow field. ... Bernoulli's equation is derived assuming ideal flow, .... weight acting in the flow direction S is.
International Nuclear Information System (INIS)
Gross, F.
1986-01-01
Relativistic equations for two and three body scattering are discussed. Particular attention is paid to relativistic three body kinetics because of recent form factor measurements of the Helium 3 - Hydrogen 3 system recently completed at Saclay and Bates and the accompanying speculation that relativistic effects are important for understanding the three nucleon system. 16 refs., 4 figs
Relativistic n-body wave equations in scalar quantum field theory
International Nuclear Information System (INIS)
Emami-Razavi, Mohsen
2006-01-01
The variational method in a reformulated Hamiltonian formalism of Quantum Field Theory (QFT) is used to derive relativistic n-body wave equations for scalar particles (bosons) interacting via a massive or massless mediating scalar field (the scalar Yukawa model). Simple Fock-space variational trial states are used to derive relativistic n-body wave equations. The equations are shown to have the Schroedinger non-relativistic limits, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Some examples of approximate ground state solutions of the n-body relativistic equations are obtained for various strengths of coupling, for both massive and massless mediating fields
Born approximation to a perturbative numerical method for the solution of the Schrodinger equation
International Nuclear Information System (INIS)
Adam, Gh.
1978-05-01
A perturbative numerical (PN) method is given for the solution of a regular one-dimensional Cauchy problem arising from the Schroedinger equation. The present method uses a step function approximation for the potential. Global, free of scaling difficulty, forward and backward PN algorithms are derived within first order perturbation theory (Born approximation). A rigorous analysis of the local truncation errors is performed. This shows that the order of accuracy of the method is equal to four. In between the mesh points, the global formula for the wavefunction is accurate within O(h 4 ), while that for the first order derivative is accurate within O(h 3 ). (author)
International Nuclear Information System (INIS)
Adhikari, Sadhan K.
2005-01-01
We demonstrate the formation of bright solitons in coupled self-defocusing nonlinear Schroedinger (NLS) equation supported by attractive coupling. As an application we use a time-dependent dynamical mean-field model to study the formation of stable bright solitons in two-component repulsive Bose-Einstein condensates (BECs) supported by interspecies attraction in a quasi one-dimensional geometry. When all interactions are repulsive, there cannot be bright solitons. However, bright solitons can be formed in two-component repulsive BECs for a sufficiently attractive interspecies interaction, which induces an attractive effective interaction among bosons of same type
International Nuclear Information System (INIS)
Halpern, L.
1987-01-01
The author, who was Schroedinger's assistant during his last years in Vienna, gives an account of Schroedinger's views and activities during that time which lead him to a different approach to research on the relations between gravitation and quantum phenomena. Various features of past research are outlined in nontechnical terms. A heuristic argument is presented for the role of the zero-point energy of massive particles in counteracting gravitational collapse and the formation of horizons. Arguments are presented for the view that progress in describing extreme gravitational phenomena can be achieved by the new outlook obtained from the introduction of the analog of Maxwell's vacuum displacement term with a quasiconstant parameter, rather than from renormalization of special processes, even if this is successful. The results can be expected to be in accord with Schroedinger's conjectures. A physical interpretation for the change of sign of the differential invariant of Karlhede, Lindstroem, and Aman at the horizon is suggested. Some important historical details about Schroedinger are touched upon
Force-balance and differential equation for the ground-state electron density in atoms and molecules
International Nuclear Information System (INIS)
Amovilli, C.; March, N.H.; Gal, T.; Nagy, A.
2000-01-01
Holas and March (1995) established a force-balance equation from the many-electron Schroedinger equation. Here, the authors propose this as a basis for the construction of a (usually approximate) differential equation for the ground-state electron density. By way of example they present the simple case of two-electron systems with different external potentials but with weak electron-electron Coulomb repulsion λe 2 /r 12 . In this case first-order Rayleigh-Schroedinger (RS) perturbation theory of the ground-state wave function is known to lead to a compact expression for the first-order density matrix γ(r,rprime) in terms of its diagonal density ρ(r) and the density corresponding to λ = 0. This result allows the force-balance equation to be written as a third-order linear, differential homogeneous equation for the ground-state electron density ρ(r). The example of the two-electron Hookean atom is treated: For this case one can also transcend the first-order RS perturbation theory and get exact results for discrete choices of force constants (external potential)
Travelling wave solutions to the Kuramoto-Sivashinsky equation
International Nuclear Information System (INIS)
Nickel, J.
2007-01-01
Combining the approaches given by Baldwin [Baldwin D et al. Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. J Symbol Comput 2004;37:669-705], Peng [Peng YZ. A polynomial expansion method and new general solitary wave solutions to KS equation. Comm Theor Phys 2003;39:641-2] and by Schuermann [Schuermann HW, Serov VS. Weierstrass' solutions to certain nonlinear wave and evolution equations. Proc progress electromagnetics research symposium, 28-31 March 2004, Pisa. p. 651-4; Schuermann HW. Traveling-wave solutions to the cubic-quintic nonlinear Schroedinger equation. Phys Rev E 1996;54:4312-20] leads to a method for finding exact travelling wave solutions of nonlinear wave and evolution equations (NLWEE). The first idea is to generalize ansaetze given by Baldwin and Peng to find elliptic solutions of NLWEEs. Secondly, conditions used by Schuermann to find physical (real and bounded) solutions and to discriminate between periodic and solitary wave solutions are used. The method is shown in detail by evaluating new solutions of the Kuramoto-Sivashinsky equation
Spinless Salpeter equation: Laguerre bounds on energy levels
International Nuclear Information System (INIS)
Lucha, W.; Schoeberl, F.F.
1996-08-01
The spinless Salpeter equation may be considered either as a standard approximation to the Bethe-Salpeter formalism, designed for the description of bound states within a relativistic quantum field theory, or as the most simple, to a certain extent relativistic generalization of the customary non relativistic Schroedinger formalism. Because of the presence of the rather difficult-to-handle square-root operator of the relativistic kinetic energy in the corresponding Hamiltonian, very frequently the corresponding (discrete) spectrum of energy eigenvalues cannot be determined analytically. Therefore, we show how to calculate, by some clever choice of basis vectors in the Hilbert space of solutions, for the rather large class of power-law potentials, at least (sometimes excellent) upper bounds on these energy eigenvalues, for the lowest-lying levels this even analytically. (author)
International Nuclear Information System (INIS)
Schubert, R.
1995-05-01
We investigate the behaviour of the remainder term R(E) in the Weyl formula {nvertical stroke E n ≤E}=Vol(M).E d/2 /[(4π) d/2 Γ(d/2+1)]+R(E) for the eigenvalues E n of a Schroedinger operator on a d-dimensional compact Riemannian manifold all of whose geodesics are closed. We show that R(E) is of the form E (d-1)/2 Θ(√E), where Θ(x) is an almost periodic function of Besicovitch class B 2 which has a limit distribution whose density is a box-shaped function. Furthermore we derive a trace formula and study higher order terms in the asymptotics of the coefficients related to the periodic orbits. The periodicity of the geodesic flow leads to a very simple structure of the trace formula which is the reason why the limit distribution can be computed explicitly. (orig.)
Differential Equations Compatible with KZ Equations
International Nuclear Information System (INIS)
Felder, G.; Markov, Y.; Tarasov, V.; Varchenko, A.
2000-01-01
We define a system of 'dynamical' differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra g. These are equations on a function of n complex variables z i taking values in the tensor product of n finite dimensional g-modules. The KZ equations depend on the 'dual' variable in the Cartan subalgebra of g. The dynamical differential equations are differential equations with respect to the dual variable. We prove that the standard hypergeometric solutions of the KZ equations also satisfy the dynamical equations. As an application we give a new determinant formula for the coordinates of a basis of hypergeometric solutions
Formation of double layers: shocklike solutions of an mKdV-equation
International Nuclear Information System (INIS)
Raadu, M.A.; Chanteur, G.
1985-10-01
Small amplitude double layers (DLs) in a plasma with a suitable electron distribution may be identified with shocklike solutions of a modified Korteweg-deVries (mKdV) equation. A thought experiment for the formation of such DLs is specified to clarify the physical constraints and to demonstrate the emergence of a DL from an initial disturbance. A scattering formulation of the mKdV initial value problem may be diagonalised to give a pair of Schroedinger equations with a scattering potential satisfying the ordinary KdV equation. The initial value problem can then be treated using Khruslov's generalisation of the inverse scattering method which allows a difference in the asymptotic values of the potential. A necessary and sufficient condition for the emergence of a shocklike soliton (wave) train and of a finite number of isolated solitons may also be determined from the scattering properties of the initial potential. With 26 refs and 5 figures. (Author)
Operations involving momentum variables in non-Hamiltonian evolution equations
International Nuclear Information System (INIS)
Benatti, F.; Ghirardi, G.C.; Rimini, A.; Weber, T.
1988-02-01
Non-Hamiltonian evolution equations have been recently considered for the description of various physical processes. Among this type of equations the class which has been more extensively studied is the one usually referred to as Quantum Dynamical Semigroup equations (QDS). In particular an equation of the QDS type has been considered as the basis for a model, called Quantum Mechanics with Spontaneous Localization (QMSL), which has been shown to exhibit some very interesting features allowing to overcome most of the conceptual difficulties of standard quantum theory, QMSL assumes a modification of the pure Schroedinger evolution by assuming the occurrence, at random times, of stochastic processes for the wave function corresponding formally to approximate position measurements. In this paper, we investigate the consequences of modifying and/or enlarging the class of the considered stochastic processes, by considering the spontaeous occurrence of approximate momentum and of simultaneous position and momentum measurements. It is shown that the considered changes in the elementary processes have unacceptable consequences. In particular they either lead to drastic modifications in the dynamics of microsystems or are completely useless from the point of view of the conceptual advantages that one was trying to get from QMSL. The present work supports therefore the idea that QMSL, as originally formulated, can be taken as the basic scheme for the generalizations which are still necessary in order to make it appropriate for the description of systems of identical particles and to meet relativistic requirements. (author). 14 refs
Operations involving momentum variables in non-Hamiltonian evolution equation
International Nuclear Information System (INIS)
Benatti, F.; Ghirardi, G.C.; Weber, T.; Rimini, A.
1988-01-01
Non-Hamiltonian evolution equations have been recently considered for the description of various physical processes. Among these types of equations the class which has been more extensively studied is the one usually referred to as quantum-dynamical semi-group equations (QDS). In particular an equation of the QDS type has been considered as the basis for a model, called quantum mechanics with spontaneous localization (QMSL), which has been shown to exhibit some very interesting features allowing us to overcome most of the conceptual difficulties of standard quantum theory. QMSL assumes a modification of the pure Schroedinger evolution by assuming the occurrence, at random times, of stochastic processes for the wave function corresponding formally to approximate position measurements. In this paper the consequences of modifying and/or enlarging the class of the considered stochastic processes, by considering the spontaneous occurrence of approximate momentum and of simultaneous position and momentum measurements, are investigated. It is shown that the considered changes in the elementary processes have unacceptable consequences. In particular they either lead to drastic modification in the dynamics of microsystems or are completely useless from the point of view of the conceptual advantages that one was trying to get from QMSL. The present work supports therefore the idea that QMSL, as originally formulated, can be taken as the basic scheme for the generalizations which are still necessary in order to make it appropriate for the description of systems of identical particles and to meet relativistic requirements