The Bethe-Salpeter equation with fermions
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)
Glueball properties from the Bethe-Salpeter equation
Kellermann, Christian
2012-01-01
For over thirty years bound states of gluons are an outstanding problem of both theoretical and experimental physics. Being predicted by Quantum-Chromodynamics their experimental confirmation is one of the foremost goals of large experimental facilities currently under construction like FAIR in Darmstadt. This thesis presents a novel approach to the theoretical determination of physical properties of bound states of two gluons, called glueballs. It uses the consistent combination of Schwinger-Dyson equations for gluons and ghosts and appropriate Bethe-Salpeter equations describing their corresponding bound-states. A rigorous derivation of both sets of equations, starting from an 2PI effective action is given as well as a general determination of appropriate decompositions of Bethe-Salpeter amplitudes to a given set of quantum numbers of a glueball. As an application example bound state masses of glueballs in a simple truncation scheme are calculated. (orig.)
Excited charmonium states from Bethe-Salpeter Equation
Šauli, Vladimír; Bicudo, P.
2012-01-01
Roč. 7, 043 (2012), s. 1-10 ISSN 1824-8039. [International Workshop on QCD Green’s Functions. Tranto, 05.09.2011-09.09.2011] R&D Projects: GA MŠk(CZ) LG11005 Institutional research plan: CEZ:AV0Z10480505 Keywords : charmonium * Bethe-Salpeter Equation Subject RIV: BE - Theoretical Physics http:// pos .sissa.it/archive/conferences/136/043/QCD-TNT-II_043.pdf
Covariant solutions of the Bethe-Salpeter equation
Williams, A.G.; Kusaka, K.; Simpson, K.M.
1997-01-01
There is a need for covariant solutions of bound state equations in order to construct realistic QCD based models of mesons and baryons. Furthermore, we ideally need to know the structure of these bound states in all kinematical regimes, which makes a direct solution in Minkowski space (without any 3-dimensional reductions) desirable. The Bethe-Salpeter equation (BSE) for bound states in scalar theories is reformulated and solved for arbitrary scattering kernels in terms of a generalized spectral representation directly in Minkowski space. This differs from the conventional Euclidean approach, where the BSE can only be solved in ladder approximation after a Wick rotation. (author)
Bethe-salpeter equation from many-body perturbation theory
Sander, Tobias; Starke, Ronald; Kresse, Georg [Computational Materials Physics, University of Vienna, Sensengasse 8/12, 1090 Vienna (Austria)
2013-07-01
The Green function formalism is a powerful tool to calculate not only electronic structure within the quasi-particle (QP) picture, but it also gives access to optical absorption spectra. Starting from QP energies within the GW method, the polarizability, as central quantity, is calculated from the solution of a Bethe-Salpeter-like equation (BSE). It is usually solved within the Tamm-Dancoff Approximation (TDA) which neglects the coupling of resonant (positive frequency branch) and anti-resonant (negative frequency branch) excitations. In this work we solve the full BSE (beyond TDA) based on self-consistently calculated QP orbitals and energies for typical systems. The dielectric function is averaged over many low dimensional shifted k-meshes to obtain k-point converged results. We compare the results to recently introduced approximation to the BSE kernel. Additionally, the time-evolution ansatz is employed to calculate the polarizability, which avoids the direct solution of the BSE.
A systematic approach to sketch Bethe-Salpeter equation
Qin Si-xue
2016-01-01
Full Text Available To study meson properties, one needs to solve the gap equation for the quark propagator and the Bethe-Salpeter (BS equation for the meson wavefunction, self-consistently. The gluon propagator, the quark-gluon vertex, and the quark–anti-quark scattering kernel are key pieces to solve those equations. Predicted by lattice-QCD and Dyson-Schwinger analyses of QCD’s gauge sector, gluons are non-perturbatively massive. In the matter sector, the modeled gluon propagator which can produce a veracious description of meson properties needs to possess a mass scale, accordingly. Solving the well-known longitudinal Ward-Green-Takahashi identities (WGTIs and the less-known transverse counterparts together, one obtains a nontrivial solution which can shed light on the structure of the quark-gluon vertex. It is highlighted that the phenomenologically proposed anomalous chromomagnetic moment (ACM vertex originates from the QCD Lagrangian symmetries and its strength is proportional to the magnitude of dynamical chiral symmetry breaking (DCSB. The color-singlet vector and axial-vector WGTIs can relate the BS kernel and the dressed quark-gluon vertex to each other. Using the relation, one can truncate the gap equation and the BS equation, systematically, without violating crucial symmetries, e.g., gauge symmetry and chiral symmetry.
Solution of the Bethe-Salpeter equation in the field of a plane electromagnetic wave
Starostin, V.S.
1988-01-01
A solution is obtained of the Bethe--Salpeter equation for positronium in the field of linearly and circularly polarized plane electromagnetic waves at frequencies much higher than atomic. It is not assumed that the field is weak
Validity of various approximations for the Bethe-Salpeter equation and their WKB quantization
Silvestre-Brac, B.; Bilal, A.; Gignoux, C.; Schuck, P.
1984-01-01
The validity of the instantaneous approximation for the Bethe-Salpeter equation is questioned within the framework of the simple scalar-scalar model of Cutkosky. Detailed numerous results for various approximations are compared to the exact ones. WKB quantization is applied to these relativistic approximations. An unexpected question arises: is the currently used Bethe-Salpeter equation (i.e., the ladder approximation) well suited to describe two interacting relativistic particles
Stochastic integration of the Bethe-Salpeter equation for two bound fermions
Salomon, M.
1988-09-01
A non-perturbative method using a Monte Carlo algorithm is used to integrate the Bethe-Salpeter equation in momentum space. Solutions for two scalars and two fermions with an arbitrary coupling constant are calculated for bound states in the ladder approximation. The results are compared with other numerical methods. (Author) (13 refs., 2 figs.)
Dorkin, S M [Dal` nevostochnyj Gosudarstvennyj Univ., Vladivostok (Russian Federation); Kaptar` , L P; Semikh, S S [Joint Inst. for Nuclear Research, Dubna (Russian Federation). Lab. of Theoretical Physics
1997-12-31
The problem of calculating the energy spectrum of a two-fermion bound state within the Bethe-Salpeter formalism is discussed. An expansion of the kernel of the spinor-spinor Bethe-Salpeter equation in the ladder approximation is found in terms of a bi-orthogonal basis of the generalized Gilbert-Schmidt series for symmetric equations of the Fredholm type. According to this expansion, a new method of solving the Bethe-Salpeter equation and finding the mass spectrum is proposed. Methodological result of numerical solutions of equations with scalar interaction is presented. (author). 20 refs., 3 figs.
Perturbation theory for the Bethe-Salpeter equation in the field of a plane electromagnetic wave
Starostin, V.S.; Litskevich, I.K.
1990-01-01
The completeness and orthogonality of the solutions of the Bethe-Salpeter equation is proven. A correct derivation of perturbation-theory equations is given. A generalization that includes the field of a plane electromagnetic wave is proposed. The rate of one-photon annihilation of positronium in this field is calculated. If the one-photon decay is allowed, the stationary states of the system are found (states of light-positronium)
The connection of two-particle relativistic quantum mechanics with the Bethe-Salpeter equation
Sazdjian, H.
1986-02-01
We show the formal equivalence between the wave equations of two-particle relativistic quantum mechanics, based on the manifestly covariant hamiltonian formalism with constraints, and the Bethe-Salpeter equation. This is achieved by algebraically transforming the latter so as to separate it into two independent equations which match the equations of hamiltonian relativistic quantum mechanics. The first equation determines the relative time evolution of the system, while the second one yields a three-dimensional eigenvalue equation. A connection is thus established between the Bethe-Salpeter wave function and its kernel on the one hand and the quantum mechanical wave function and interaction potential on the other. For the sector of solutions of the Bethe-Salpeter equation having non-relativistic limits, this relationship can be evaluated in perturbation theory. We also device a generalized form of the instantaneous approximation which simplifies the various expressions involved in the above relations. It also permits the evaluation of the normalization condition of the quantum mechanical wave function as a three-dimensional integral
Williams, A.G.
1998-01-01
There is a need for covariant solutions of bound state equations in order to construct realistic QCD based models of mesons and baryons. Furthermore, we ideally need to know the structure of these bound states in all kinematical regimes, which makes a direct solution in Minkowski space (without any 3-dimensional reductions) desirable. The Bethe-Salpeter equation (BSE) for bound states in scalar theories is reformulated and solved for arbitrary scattering kernels in terms of a generalized spectral representation directly in Minkowski space. This differs from the conventional Euclidean approach, where the BSE can only be solved in ladder approximation after a Wick rotation. An application of covariant Bethe-Salpeter solutions to a quark-diquark model of the nucleon is also briefly discussed. (orig.)
Numerical studies of the Bethe-Salpeter equation for a two-fermion bound state
de Paula, W.; Frederico, T.; Salmè, G.; Viviani, M.
2018-03-01
Some recent advances on the solution of the Bethe-Salpeter equation (BSE) for a two-fermion bound system directly in Minkowski space are presented. The calculations are based on the expression of the Bethe-Salpeter amplitude in terms of the so-called Nakanishi integral representation and on the light-front projection (i.e. the integration of the light-front variable k - = k 0 - k 3). The latter technique allows for the analytically exact treatment of the singularities plaguing the two-fermion BSE in Minkowski space. The good agreement observed between our results and those obtained using other existing numerical methods, based on both Minkowski and Euclidean space techniques, fully corroborate our analytical treatment.
Bound states of quarks calculated with stochastic integration of the Bethe-Salpeter equation
Salomon, M.
1992-07-01
We have computed the masses, wave functions and sea quark content of mesons in their ground state by integrating the Bethe-Salpeter equation with a stochastic algorithm. This method allows the inclusion of a large set of diagrams. Inspection of the kernel of the equation shows that q-q-bar pairs with similar constituent masses in a singlet spin state exhibit a high bound state which is not present in other pairs. The pion, kaon and eta belongs to this category. 19 refs., 2 figs., 2 tabs
A separable approximation of the NN-Paris-potential in the framework of the Bethe-Salpeter equation
Schwarz, K.; Haidenbauer, J.; Froehlich, J.
1985-09-01
The Bethe-Salpeter equation is solved with a separable kernel for the most important nucleon-nucleon partial wave states. We employ the Ernst Shakin-Thaler method in the framework of minimal relativity (Blankenbeckler-Sugar equation) to generate a separable representation of the meson-theoretical Paris potential. These separable interactions, which closely approximate the on-shell- and half-off-shell behaviour of the Paris potential, are then cast into a covariant form for application in the Bethe-Salpeter equation. The role of relativistic effects is discussed with respect to on-shell and off-shell properties of the NN-system. (Author)
Solution to Bethe-Salpeter equation via Mellin-Barnes transform
Allendes, Pedro; Kniehl, Bernd; Kondrashuk, Igor; Rojas Medar, Marko; Notte Cuello, Eduardo A.
2012-06-01
We consider Mellin-Barnes transform of triangle ladder-like scalar diagram in d=4 dimensions. It is shown how multi-fold MB transform of the momentum integral corresponding to any number of rungs is reduced to two-fold MB transform. For this purpose we use Belokurov-Usyukina reduction method for four-dimensional scalar integrals in the position space. The result is represented in terms of Euler ψ-function and its derivatives. We derive new formulas for MB two-fold integration in the complex planes of two complex variables. We demonstrate that these formulas solve Bethe-Salpeter equation. We comment on further applications of solution to Bethe-Salpeter equation for vertices in N=4 supersymmetric Yang-Mills theory. We show that the recursive property of MB transforms observed in the present work for that kind of diagrams has nothing to do with quantum field theory, theory of integral transforms, or with theory of polylogarithms in general, but has an origin in a simple recursive property for smooth functions which can be shown by using basic methods of mathematical analysis.
Solution to Bethe-Salpeter equation via Mellin-Barnes transform
Allendes, Pedro [Concepcion Univ. (Chile). Dept. de Fisica; Kniehl, Bernd [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Kondrashuk, Igor; Rojas Medar, Marko [Univ. del Bio-Bio, Chillan (Chile). Dept. de Ciencias Basicas; Notte Cuello, Eduardo A. [Univ. de La Serena (Chile). Facultad de Ciencias
2012-06-15
We consider Mellin-Barnes transform of triangle ladder-like scalar diagram in d=4 dimensions. It is shown how multi-fold MB transform of the momentum integral corresponding to any number of rungs is reduced to two-fold MB transform. For this purpose we use Belokurov-Usyukina reduction method for four-dimensional scalar integrals in the position space. The result is represented in terms of Euler {psi}-function and its derivatives. We derive new formulas for MB two-fold integration in the complex planes of two complex variables. We demonstrate that these formulas solve Bethe-Salpeter equation. We comment on further applications of solution to Bethe-Salpeter equation for vertices in N=4 supersymmetric Yang-Mills theory. We show that the recursive property of MB transforms observed in the present work for that kind of diagrams has nothing to do with quantum field theory, theory of integral transforms, or with theory of polylogarithms in general, but has an origin in a simple recursive property for smooth functions which can be shown by using basic methods of mathematical analysis.
Schwarz, K.; Froehlich, J.; Zingl, H.F.K.
1980-01-01
The Bethe-Salpeter equation is solved in closed form with the help of a four dimensional separable 'potential'. For possible applications to three-nucleon investigations the authors have fitted all nucleon-nucleon S-wave phase shifts in a sufficient way by this method; in addition they also present an example for a P-wave. (Auth.)
Low-lying qq(qq)-bar states in a relativistic model based on the Bethe-Salpeter equation
Ram, B.; Kriss, V.
1985-01-01
Low-lying qq(qq)-bar states are analysed in a previously given relativistic model based on the Bethe-Salpeter equation. It is not got M-diquonia, P-mesonia, or meson molecules, but it is got T-diquonia
Projecting the Bethe-Salpeter Equation onto the Light-Front and Back: A Short Review
Frederico, T.; Salme, G.
2011-01-01
The technique of projecting the four-dimensional two-body Bethe-Salpeter equation onto the three-dimensional Light-Front hypersurface, combined with the quasi-potential approach, is briefly illustrated, by placing a particular emphasis on the relation between the projection method and the effective dynamics of the valence component of the Light-Front wave function. Some details on how to construct the Fock expansion of both (a) the Light-Front effective interaction and (b) the electromagnetic current operator, satisfying the proper Ward-Takahashi identity, will be presented, addressing the relevance of the Fock content in the operators living onto the Light-Front hypersurface. Finally, the generalization of the formalism to the three-particle case will be outlined. (author)
Bethe-Salpeter equation for non-self conjugate mesons in a power-law potential
Ikhdair, S.M.
1992-07-01
We develop an approach to the solution of the spinless Bethe-Salpeter equation for the different-mass case. Although the calculations are developed for spin-zero particles in any arbitrary spherically symmetric potential, the non-Coulombic effective power-law potential is used as a kernel to produce the spin-averaged bound states of the non-self-conjugate mesons. The analytical formulae are also applicable to the self-conjugate mesons in the equal-mass case. The flavor-independent case is investigated in this work. The calculations are carried out to the third-order correction of the energy series. Results are consistent with those obtained before. (author). 14 refs, 1 tab
Bethe-Salpeter equation for fermion-antifermion system in the ladder approximation
Fukui, Ichio; Seto, Noriaki; Yoshida, Toshihiro.
1977-01-01
The Bethe-Salpeter (B-S) equation is important for studying hadron physics. Especially intensive investigation on the fermion-antifermion B-S equation is indispensable for the phenomenological studies of hardrons. However, many components of the B-S amplitude and the Wick-rotated integral kernel of non-Fredholm type have prevented from knowing details the solutions even in the ladder approximation. Some particular solutions are known in case of the vanishing four-momenta of bound states. The B-S equation for the bound state of fermion-anti-fermion system interacting through vector (axial-vector) particle exchange was studied in the ladder approximation with Feynman gauge. The reduced equations were obtained for suitably decomposed amplitude, and it is shown that, in the S-wave case, the coupled equations separate into two parts. In the nonrelativistic limit, large components of the amplitude satisfy the Wick-Cutkosky equation, and small components are expressed in terms of the large ones. Equations are derived for the equal-time amplitudes. (Kobatake, H.)
Chen, Jiao-Kai
2018-04-01
We present one reduction of the Bethe-Salpeter equation for the bound states composed of two off-mass-shell constituents. Both the relativistic effects and the virtuality effects can be considered in the obtained spinless virtuality distribution equation. The eigenvalues of the spinless virtuality distribution equation are perturbatively calculated and the bound states e+e-, μ+μ-, τ+τ-, μ+e-, and τ+e- are discussed.
Efficient implementation of core-excitation Bethe-Salpeter equation calculations
Gilmore, K.; Vinson, John; Shirley, E. L.; Prendergast, D.; Pemmaraju, C. D.; Kas, J. J.; Vila, F. D.; Rehr, J. J.
2015-12-01
We present an efficient implementation of the Bethe-Salpeter equation (BSE) method for obtaining core-level spectra including X-ray absorption (XAS), X-ray emission (XES), and both resonant and non-resonant inelastic X-ray scattering spectra (N/RIXS). Calculations are based on density functional theory (DFT) electronic structures generated either by ABINIT or QuantumESPRESSO, both plane-wave basis, pseudopotential codes. This electronic structure is improved through the inclusion of a GW self energy. The projector augmented wave technique is used to evaluate transition matrix elements between core-level and band states. Final two-particle scattering states are obtained with the NIST core-level BSE solver (NBSE). We have previously reported this implementation, which we refer to as OCEAN (Obtaining Core Excitations from Ab initio electronic structure and NBSE) (Vinson et al., 2011). Here, we present additional efficiencies that enable us to evaluate spectra for systems ten times larger than previously possible; containing up to a few thousand electrons. These improvements include the implementation of optimal basis functions that reduce the cost of the initial DFT calculations, more complete parallelization of the screening calculation and of the action of the BSE Hamiltonian, and various memory reductions. Scaling is demonstrated on supercells of SrTiO3 and example spectra for the organic light emitting molecule Tris-(8-hydroxyquinoline)aluminum (Alq3) are presented. The ability to perform large-scale spectral calculations is particularly advantageous for investigating dilute or non-periodic systems such as doped materials, amorphous systems, or complex nano-structures.
Sagmeister, S.
2009-01-01
The aim of this work is to compare two state-of-the-art methods for the investigation of excitonic effects in solids, namely Time-Dependent Density Functional Theory (TDDFT) and Many-Body Perturbation Theory (MBPT), for selected simple gap systems as well as semiconducting polymers. Within TDDFT, the linear response framework is used and the Dyson equation for the density-density response function is solved, whereas within MBPT, the Bethe-Salpeter equation (BSE) for the electron-hole correlation function is solved. The dielectric function is obtained as a last step. Both techniques take into account the excitonic effects caused by the interaction of electron-hole pairs. In the former these effects are included in the exchange-correlation (xc) kernel, whereas in the latter they are located in the interaction kernel of the BSE. Kohn-Sham single-particle wave functions obtained from Density Functional Theory within the linearized augmented planewave (LAPW) method are used to calculate all relevant quantities of the formalism. For the simple systems GaAs, Si and LiF are chosen. The role of several approximations to the xc kernel is studied and it is found that for GaAs and Si simple semi-empirical models provide a dielectric function in accordance with the BSE. For the case of LiF, being a system with a weak screening and a strongly bound exciton, only an xc kernel derived from MBPT yields reasonable results but still a slight discrepancy to the BSE is observed. Finally, the semiconducting polymers poly-acetylene and poly(phenylene-vinylene) (PPV) are studied. For both materials the concept of semi-empirical approximations to the xc kernel turns out to be ambiguous due to their low-dimensional character. In the case of poly-acetylene, the xc kernel derived from MBPT yields a dielectric function which is in close but not exact agreement with the one obtained from the BSE. (author) [de
The average kinetic energy of the heavy quark in Λb in the Bethe-Salpeter equation approach
Guo, X.-H.; Wu, H.-K.
2007-01-01
In the previous paper, based on the SU(2) f xSU(2) s heavy quark symmetries of the QCD Lagrangian in the heavy quark limit, the Bethe-Salpeter equation for the heavy baryon Λ b was established with the picture that Λ b is composed of a heavy quark and a scalar light diquark. In the present work, we apply this model to calculate μ π 2 for Λ b , the average kinetic energy of the heavy quark inside Λ b . This quantity is particularly interesting since it can be measured in experiments and since it contributes to the inclusive semileptonic decays of Λ b when contributions from higher order terms in 1/M b expansions are taken into account and consequently influences the determination of the Cabibbo-Kobayashi-Maskawa matrix elements V ub and V cb . We find that μ π 2 for Λ b is 0.25GeV 2 ∼0.95GeV 2 , depending on the parameters in the model including the light diquark mass and the interaction strength between the heavy quark and the light diquark in the kernel of the BS equation. We also find that this result is consistent with the value of μ π 2 for Λ b which is derived from the experimental value of μ π 2 for the B meson with the aid of the heavy quark effective theory
Hilger, T.; Krassnigg, A. [University of Graz, NAWI Graz, Institute of Physics, Graz (Austria); Gomez-Rocha, M. [ECT*, Villazzano, Trento (Italy)
2017-09-15
We investigate the light-quarkonium spectrum using a covariant Dyson-Schwinger-Bethe-Salpeter-equation approach to QCD. We discuss splittings among as well as orbital angular momentum properties of various states in detail and analyze common features of mass splittings with regard to properties of the effective interaction. In particular, we predict the mass of anti ss exotic 1{sup -+} states, and identify orbital angular momentum content in the excitations of the ρ meson. Comparing our covariant model results, the ρ and its second excitation being predominantly S-wave, the first excitation being predominantly D-wave, to corresponding conflicting lattice-QCD studies, we investigate the pion-mass dependence of the orbital-angular-momentum assignment and find a crossing at a scale of m{sub π} ∝ 1.4 GeV. If this crossing turns out to be a feature of the spectrum generated by lattice-QCD studies as well, it may reconcile the different results, since they have been obtained at different values of m{sub π}. (orig.)
Yan, Jun; Jacobsen, Karsten W.; Thygesen, Kristian S.
2012-01-01
-dimensional systems of graphene and hexagonal boron-nitride (h-BN) we find good agreement with previous many-body calculations. For the graphene/h-BN interface we find that the fundamental and optical gaps of the h-BN layer are reduced by 2.0 and 0.7 eV, respectively, compared to freestanding h-BN. This reduction......We present an efficient implementation of the Bethe-Salpeter equation (BSE) for optical properties of materials in the projector augmented wave method Grid-based projector-augmented wave method (GPAW). Single-particle energies and wave functions are obtained from the Gritsenko, Leeuwen, Lenthe...
Bethe-Salpeter amplitudes and static properties of the deuteron
Kaptari, L.P.; Bondarenko, S.G.; Khanna, F.C.; Kaempfer, B.; Technische Univ. Dresden
1996-04-01
Extended calculations of the deuteron's static properties, based on the numerical solution of the Bethe-Salpeter equation, are presented. A formalism is developed, which provides a comparative analysis of the covariant amplitudes in various representations and nonrelativistic wave functions. The magnetic and quadrupole moments of the deuteron are calculated in the Bethe-Salpeter formalism and the role of relativistic corrections is discussed. (orig.)
Janyszek, H [Uniwersytet Mikolaja Kopernika, Torun (Poland). Instytut Fizyki
1974-01-01
A new modified quasirelativistic equation (different from that of Breit) for N charged Dirac particles in the external stationary electromagnetic field is proposed. This equation is an amplified quantum-mechanical Bethe-Salpeter equation obtained by adding (in a semi-phenomenological manner) terms which take into account radiative corrections. The application of this approximate equations is limited to third order terms in the fine structure constant ..cap alpha...
Bhatnagar, Shashank; Alemu, Lmenew
2018-02-01
In this work we calculate the mass spectra of charmonium for 1 P ,…,4 P states of 0++ and 1++, for 1 S ,…,5 S states of 0-+, and for 1 S ,…,4 D states of 1- along with the two-photon decay widths of the ground and first excited states of 0++ quarkonia for the process O++→γ γ in the framework of a QCD-motivated Bethe-Salpeter equation (BSE). In this 4 ×4 BSE framework, the coupled Salpeter equations are first shown to decouple for the confining part of the interaction (under the heavy-quark approximation) and are analytically solved, and later the one-gluon-exchange interaction is perturbatively incorporated, leading to mass spectral equations for various quarkonia. The analytic forms of wave functions obtained are used for the calculation of the two-photon decay widths of χc 0. Our results are in reasonable agreement with data (where available) and other models.
Hilger, Thomas Uwe
2012-04-11
The interplay of hadron properties and their modification in an ambient nuclear medium on the one hand and spontaneous chiral symmetry breaking and its restoration on the other hand is investigated. QCD sum rules for D and B mesons embedded in cold nuclear matter are evaluated. We quantify the mass splitting of D- anti D and B- anti B mesons as a function of the nuclear matter density and investigate the impact of various condensates in linear density approximation. The analysis also includes D{sub s} and D{sup *}{sub 0} mesons. QCD sum rules for chiral partners in the open-charm meson sector are presented at nonzero baryon net density or temperature. We focus on the differences between pseudo-scalar and scalar as well as vector and axial-vector D mesons and derive the corresponding Weinberg type sum rules. Based on QCD sum rules we explore the consequences of a scenario for the ρ meson, where the chiral symmetry breaking condensates are set to zero whereas the chirally symmetric condensates remain at their vacuum values. The complementarity of mass shift and broadening is discussed. An alternative approach which utilizes coupled Dyson-Schwinger and Bethe-Salpeter equations for quark-antiquark bound states is investigated. For this purpose we analyze the analytic structure of the quark propagators in the complex plane numerically and test the possibility to widen the applicability of the method to the sector of heavy-light mesons in the scalar and pseudo-scalar channels, such as the D mesons, by varying the momentum partitioning parameter. The solutions of the Dyson-Schwinger equation in the Wigner-Weyl phase of chiral symmetry at nonzero bare quark masses are used to investigate a scenario with explicit but without dynamical chiral symmetry breaking.
Malik, G P
2016-01-01
Given the Debye temperature of an elemental superconductor (SC) and its Tc, BCS theory enables one to predict the value of its gap 0 at T = 0, or vice versa. This monograph shows that non-elemental SCs can be similarly dealt with via the generalized BCS equations (GBCSEs) which, given any two parameters of the set {Tc, 10, 20 > 10}, enable one to predict the third. Also given herein are new equations for the critical magnetic field and critical current density of an elemental and a non-elemental SC — equations that are derived directly from those that govern pairing in them. The monograph includes topics that are usually not covered in any one text on superconductivity, e.g., BCS-BEC crossover physics, the long-standing puzzle posed by SrTiO3, and heavy-fermion superconductors — all of which are still imperfectly understood and therefore continue to avidly engage theoreticians. It suggests that addressing the Tcs, s and other properties (e.g., number densities of charge carriers) of high-Tc SCs via GBCSE...
Anomalous magnetic nucleon moments in a Bethe-Salpeter model
Chak Wing Chan.
1978-01-01
We investigate the anomalous magnetic moment of the nucleon in a field theoretic many-channel model for the electromagnetic form factors of the N anti N, the ππ, the K anti K, the πω and the πrho systems. Propagator self-energy corrections from the Ward idendity and phenomenological strong vertex corrections are both included. The photon is coupled minimally to pions, kaons and nucleons with power multiplicative renormalization. With solutions in the framework of the Bethe-Salpeter equation we obtain a value 1.84 for the isovector moment and a value -0.02 for the isoscalar moment. (orig.)
Quarkonia in the Bethe--Salpeter formalism with background fields
Mathur, Y.K.; Mitra, A.N.
1989-01-01
A QCD-oriented Bethe--Salpeter (BS) equation for a q bar q system is formulated in which the quark 4-momenta p μ are modified as p μ →p μ -gA μ (x) in the inverse propagators therein, and a Fock--Schwinger (FS) gauge expansion is employed for the gluon fields A μ (x). The first term (∼x μ ) of the FS representation yields a harmonic kernel when the BS equation is reduced to a 3-dimensional level via the null-plane ansatz (NPA). It also generates a spin-dependent interaction proportional to (j 1 +s 1 )·(j 2 +s 2 ), in close parallel to a J·S term generated by a vector-like (γ (1) gamma(2)) harmonic model for the q bar q interaction proposed earlier by the Delhi Group. A possible mechanism for confinement in an asymptotically linear scene is proposed within the BS framework, taking cue partly from the suggestions of multiple correlation effects (Shifman), and partly from the postulation of stochastic fields (Simonov)
Single-time reduction of bethe-salpeter formalism for two-fermion system
Arkhipov, A.A.
1988-01-01
The single-time reduction method proposed in other refs. for the system of two scalar particles is generalized for the case of two-fermion system. A self-consistent procedure of single-time reduction has been constructed both in terms of the Bethe-Salpeter wave function and in terms of the Green's function of two-fermion system. Three-dimensional dynamic equations have been obtained for single-time wave functions and two-time Green's functions of a two-fermion system and the Schroedinger structure of the equations obtained is shown to be a consequence of the causality structure of the local QFT. 32 refs
Euclidean to Minkowski Bethe-Salpeter amplitude and observables
Carbonell, J.; Frederico, T.; Karmanov, V.A.
2017-01-01
We propose a method to reconstruct the Bethe-Salpeter amplitude in Minkowski space given the Euclidean Bethe-Salpeter amplitude - or alternatively the light-front wave function - as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function. (orig.)
Euclidean to Minkowski Bethe-Salpeter amplitude and observables
Carbonell, J. [Universite Paris-Sud, IN2P3-CNRS, Institut de Physique Nucleaire, Orsay Cedex (France); Frederico, T. [Instituto Tecnologico de Aeronautica, DCTA, Sao Jose dos Campos (Brazil); Karmanov, V.A. [Lebedev Physical Institute, Moscow (Russian Federation)
2017-01-15
We propose a method to reconstruct the Bethe-Salpeter amplitude in Minkowski space given the Euclidean Bethe-Salpeter amplitude - or alternatively the light-front wave function - as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function. (orig.)
Delta and Omega electromagnetic form factors in a Dyson-Schwinger/Bethe-Salpeter approach
Diana Nicmorus, Gernot Eichmann, Reinhard Alkofer
2010-12-01
We investigate the electromagnetic form factors of the Delta and the Omega baryons within the Poincare-covariant framework of Dyson-Schwinger and Bethe-Salpeter equations. The three-quark core contributions of the form factors are evaluated by employing a quark-diquark approximation. We use a consistent setup for the quark-gluon dressing, the quark-quark bound-state kernel and the quark-photon interaction. Our predictions for the multipole form factors are compatible with available experimental data and quark-model estimates. The current-quark mass evolution of the static electromagnetic properties agrees with results provided by lattice calculations.
Bethe-Salpeter analysis of the radiative pion disintegration
Abad, J.; Pacheco, A.F. (Zaragoza Univ. (Spain). Dept. de Fisica Teorica); Rodriguez-Trias, R.; Esteve, J.G. (Paris-11 Univ., 91 - Orsay (France). Lab. de Physique Theorique et Hautes Energies)
1990-04-01
The structure-dependent amplitude of the decay {pi}{yields}e{nu}{gamma} is evaluated in the framework of a Bethe-Salpeter description for the pion. We assume a general B-S wave function in the S-wave. Within this hypothesis, we show that the gauge invariance constrains the different contributions of the wave functions to the amplitude, resulting in the vanishing of the axial form factor. (orig.).
Construction of Bethe Salpeter wave functions and applications in QCD
Gromes, D.
1993-01-01
We suggest an ansatz for the Bethe Salpeter wave function which is strictly covariant, obeys the spectrum conditions, and has the correct non relativistic limit. As a first simple application we present a wave function for the pion. It contains two parameters, one of them being the quark mass. The decay constant and the form factor derived from this are in excellent agreement with the data. (orig.)
Covariant Bethe-Salpeter wave functions for heavy hadrons
Hussain, F.
1992-09-01
In recent years the dynamics of heavy mesons and baryons has considerably simplified by the development of the so-called heavy quark effective theory (HQET). A covariant formulation of heavy meson and heavy baryon decays in the leading order of the HQET is presented. The method is based on a Bethe-Salpeter formulation in the limit of the heavy quark mass going to infinity. 15 refs, 4 figs
GW and Bethe-Salpeter study of small water clusters
Blase, Xavier, E-mail: xavier.blase@neel.cnrs.fr; Boulanger, Paul [CNRS, Institut NEEL, F-38042 Grenoble (France); Bruneval, Fabien [CEA, DEN, Service de Recherches de Métallurgie Physique, F-91191 Gif-sur-Yvette (France); Fernandez-Serra, Marivi [Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800 (United States); Institute for Advanced Computational Sciences, Stony Brook University, Stony Brook, New York 11794-3800 (United States); Duchemin, Ivan [INAC, SP2M/L-Sim, CEA/UJF Cedex 09, 38054 Grenoble (France)
2016-01-21
We study within the GW and Bethe-Salpeter many-body perturbation theories the electronic and optical properties of small (H{sub 2}O){sub n} water clusters (n = 1-6). Comparison with high-level CCSD(T) Coupled-Cluster at the Single Double (Triple) levels and ADC(3) Green’s function third order algebraic diagrammatic construction calculations indicates that the standard non-self-consistent G{sub 0}W{sub 0}@PBE or G{sub 0}W{sub 0}@PBE0 approaches significantly underestimate the ionization energy by about 1.1 eV and 0.5 eV, respectively. Consequently, the related Bethe-Salpeter lowest optical excitations are found to be located much too low in energy when building transitions from a non-self-consistent G{sub 0}W{sub 0} description of the quasiparticle spectrum. Simple self-consistent schemes, with update of the eigenvalues only, are shown to provide a weak dependence on the Kohn-Sham starting point and a much better agreement with reference calculations. The present findings rationalize the theory to experiment possible discrepancies observed in previous G{sub 0}W{sub 0} and Bethe-Salpeter studies of bulk water. The increase of the optical gap with increasing cluster size is consistent with the evolution from gas to dense ice or water phases and results from an enhanced screening of the electron-hole interaction.
Null-plane formulation of Bethe-Salpeter qqq dynamics: Baryon mass spectra
Kulshreshtha, D.S.; Mitra, A.N.
1988-01-01
The Bethe-Salpeter (BS) equation for a qqq system is formulated in the null-plane approximation (NPA) for the BS wave function, as a direct generalization of a corresponding QCD-motivated formalism developed earlier for qq-bar systems. The confinement kernel is assumed vector type (γ/sub μ//sup (1)/γ/sub μ//sup (2)/) for both qq-bar and qq pairs, with identical harmonic structures, and with the spring constant proportional, among other things, to the running coupling constant α/sub s/ (for an explicit QCD motivation). The harmonic kernel is given a suitable Lorentz-invariant definition [not D'Alembertian 2 δ 4 (q)], which is amenable to NPA reduction in a covariant form. The reduced qqq equation in NPA is solved algebraically in a six-dimensional harmonic-oscillator (HO) basis, using the techniques of SO(2,1) algebra interlinked with S 3 symmetry. The results on the nonstrange baryon mass spectra agree well with the data all the way up to N = 6, thus confirming the asymptotic prediction M∼N/sup 2/3/ characteristic of vector confinement in HO form. There are no extra parameters beyond the three basic constants (ω 0 ,C 0 ,m/sub u//sub d/) which were earlier found to provide excellent fits to meson spectra (qq-bar)
Bayesian extraction of the parton distribution amplitude from the Bethe-Salpeter wave function
Gao, Fei; Chang, Lei; Liu, Yu-xin
2017-07-01
We propose a new numerical method to compute the parton distribution amplitude (PDA) from the Euclidean Bethe-Salpeter wave function. The essential step is to extract the weight function in the Nakanishi representation of the Bethe-Salpeter wave function in Euclidean space, which is an ill-posed inversion problem, via the maximum entropy method (MEM). The Nakanishi weight function as well as the corresponding light-front parton distribution amplitude (PDA) can be well determined. We confirm prior work on PDA computations, which was based on different methods.
Comment on the analysis of Bethe-Salpeter scattering states by Hormozdiari and Huang
Tryon, E.P.
1978-01-01
The analysis of Bethe-Salpeter scattering states by Hormozdiari and Huang appears to contain invalid mathematical arguments. When these arguments are rectified, one arrives at substantially different conclusions. In particular, the prescription of Hormozdiari and Huang for constructing such states does not seem applicable to any process occurring in nature
Deep inelastic scattering on the deuteron in the Bethe-Salpeter formalism
Kaptari, L.P.; Kazakov, K.Yu.; Umnikov, A.Yu.; Khanna, F.C.
1996-01-01
The nuclear effects in the spin structure functions of the deuteron g 1 and b 2 are estimated in a fully covariant approach of the Bethe-Salpeter formalism. The construction of the relativistic wave function of the deuteron is discussed in detail. Numerical results for g 1 and b 2 are compared with nonrelativistic results and relativistic corrections are discussed [ru
Heavy quark effective theory, interpolating fields and Bethe-Salpeter amplitudes
Hussain, F.; Thomspon, G.
1994-07-01
We use the LSZ reduction theorem and interpolating fields, along with the heavy quark effective theory, to investigate the structure of the Bethe-Salpeter amplitude for heavy hadrons. We show how a simple form of this amplitude, used extensively in heavy hadron decay calculations, follows naturally up to O(1/M) from these field theoretic considerations. (author). 13 refs, 1 tab
Bhatnagar, Shashank [Department of Physics, Addis Ababa University, PO Box 101739, Addis Ababa (Ethiopia); Li Shiyuan [Department of Physics, Shandong University, Jinan, 250100 (China)
2006-07-15
We employ the framework of the Bethe-Salpeter equation under a covariant instantaneous ansatz to study the leptonic decays of vector mesons. The structure of the hadron-quark vertex function {gamma} is generalized to include various Dirac covariants (other than i{gamma} . {epsilon}) from their complete set. They are incorporated in accordance with a naive power counting rule order-by-order in powers of the inverse of the meson mass. The decay constants for {rho}, {omega} and {phi} mesons are calculated with the incorporation of leading-order covariants.
Bhatnagar, Shashank; Li Shiyuan
2006-01-01
We employ the framework of the Bethe-Salpeter equation under a covariant instantaneous ansatz to study the leptonic decays of vector mesons. The structure of the hadron-quark vertex function Γ is generalized to include various Dirac covariants (other than iγ . ε) from their complete set. They are incorporated in accordance with a naive power counting rule order-by-order in powers of the inverse of the meson mass. The decay constants for ρ, ω and φ mesons are calculated with the incorporation of leading-order covariants
Bethe-Salpeter kernels and particle structure in the Yukawa2 quantum field theory
Cooper, A.S.
1981-01-01
The author discusses the extension to the (weakly coupled) Yukawa quantum field theory in two space-time dimensions (Y 2 ), with equal bare masses, of some techniques used in the analysis of particle structure for weakly coupled even P(PHI) 2 . In particular he considers existence, regularity, and decay properties for the inverse two point functions and various Bethe-Salpeter kernels of the theory. These properties suffice to ensure that in the +-2 fermion sectors the mass spectrum is discrete below 2m 0 and the S-matrix is unitary up to 2m 0 + epsilon. (Auth.)
Gupta, K.K.; Mitra, A.N.; Singh, N.N.
1990-01-01
A new relativistic definition of the reduced mass (μ 12 ) of a q bar q pair, so as to be in conformity with the standard Wightman-Garding definition of its relative four-momenta q μ , is introduced into the kernel of an ongoing Bethe-Salpeter (BS) program on a two-tier basis. The new definition of μ 12 (involving the hadron mass M) is found to produce a natural Regge asymptotic behavior (M 2 ∼N) in the hadron mass spectra, while retaining the property of an asymptotically linear (∼r) confinement in the three-dimensional structure of the BS kernel. The relativistic structure of μ 12 is responsible for a significant improvement in the fits to the ground-state masses of q bar q and Q bar q mesons as compared to its nonrelativistic definition m 1 m 2 /(m 1 +m 2 ). The leptonic decay constants f p and the charge radii thus calculated are also in excellent agreement with data (π,k) where available, while f p predictions for Q bar q mesons have good overlap with recent lattice predictions. Further, the scaling property (∼k μ -2 ) of the hadron's electromagnetic form factor at large k 2 is a consequence of the ''on-shell'' form of its null-plane wave function. All these results (which are indicated in the barest outline) are preceded by a perspective summary of the theoretical premises and practical working of the BS equation with a four-fermion interaction kernel as a necessary background on a two-tier basis
QCD-oriented Bethe-Salpeter dynamics for all flavours, light and heavy
Mitra, A.N.
1987-01-01
A QCD oriented Bethe-Salpeter approach with a vector-like confinement which looks harmonic in the instantaneous approximation (IA), was found to describe several features of qanti q and qqq systems (mass spectra and structural properties) adequately, with little extra assumptions. The model had the following drawbacks: (a) The IA, being non-covariant, had restricted the application of the model to slow hadrons; (b) The zero-point energies were poorly described without additional assumptions; (c) The predicted Qanti Q spacings were too large for the data. These limitations have recently been overcome and the model now has a wide sweep of successful predictions, not only on the mass spectra for both light and heavy sectors, but also for transition amplitudes involving fast moving hadrons. A sketchy but consolidated account is given of the main features of the model (as modified through the new assumptions) and the nature of its successes on the experimental front (mostly mass spectra) are indicated. 20 refs
Short-distance behavior of the Bethe--Salpeter wave function in the ladder approximation
Guth, A.H.; Soper, D.E.
1975-01-01
We investigate the short-distance behavior of the (Wick-rotated) Bethe--Salpeter wave function for the two spin-1/2 quarks bound by the exchange of a massive vector meson. We use the ladder-model kernel, which has the same p -4 scaling behavior as the true kernel in a theory with a fixed point of the renormalization group at g not equal to 0. For a bound state with the quantum numbers of the pion, the leading asymptotic behavior is chi (q/sup μ/) approx. cq/sup -4 + epsilon(g)/γ 5 , where epsilon (g) =1- (1-g 2 /π 2 ) 1 / 2 . Our method also provides the full asymptotic series, although it should be noted that the nonleading terms will depend on the nonleading behavior of the ladder-model kernel. A general term has the form cq - /sup a/(lnq)/sup n/phi (q/sup μ/), where c is an unknown constant, a may be integral or nonintegral, n is an integer, and phi (q/sup μ/) is a representation function of the rotation group in four dimensions
The decay constants of heavy-light mesons in a two-tier Bethe-Salpeter model
Pagnamenta, A.; Illinois Univ., Chicago, IL; Gupta, K.K.; Mitra, A.N.; Singh, N.N.; Ramanathan, R.
1990-01-01
Leptonic decay widths for pseudoscalar mesons are calculated in a QCD-motivated Bethe-Salpeter formalism on a two-tier basis. This model, which is characterised by a Lorentz-invariant confining kernel that depends on the relativistic reduced mass defined in accordance with the Wightman-Garding definition of the internal 4-momenta q μ , has already shown precision fits to the spectra of qq-bar, qQ-bar and qqq hadrons apart from exhibiting an eplicit Regge-asymptotic behaviour (M 2 ∼N). The leptonic decay constants thus obtained with no free parameters, are (in MeV units), f P =134(π), 166(K); 158(D); 94.9(B), 114(B S ). (author)
Intriguing solutions of the Bethe-Salpeter equation for radially excited pseudoscalar charmonia
Šauli, Vladimír
2014-01-01
Roč. 90, č. 1 (2014), 016005 ISSN 1550-7998 Institutional support: RVO:61389005 Keywords : quantum chromodynamics * confinement * quarks * gluons Subject RIV: BE - Theoretical Physics Impact factor: 4.643, year: 2014
Bondarenko, S.G.; Burov, V.V.; Hamamoto, N.; Manabe, Y.; Hosaka, A.; Toki, H.
2005-01-01
Recent results obtained by the application of the Bethe-Salpeter approach to the analysis of elastic electron-deuteron scattering with the separable NN kernel are presented. We analyze the impact of the P waves (negative-energy components) on the electromagnetic properties of the deuteron and compare it with experimental data. It was shown that the contribution of the P waves must be taken into account to explain tensor polarization and charge form factor of the deuteron
Rebolini, Elisa, E-mail: elisa.rebolini@kjemi.uio.no; Toulouse, Julien, E-mail: julien.toulouse@upmc.fr [Laboratoire de Chimie Théorique, Sorbonne Universités, UPMC Univ Paris 06, CNRS, 4 place Jussieu, F-75005 Paris (France)
2016-03-07
We present a range-separated linear-response time-dependent density-functional theory (TDDFT) which combines a density-functional approximation for the short-range response kernel and a frequency-dependent second-order Bethe-Salpeter approximation for the long-range response kernel. This approach goes beyond the adiabatic approximation usually used in linear-response TDDFT and aims at improving the accuracy of calculations of electronic excitation energies of molecular systems. A detailed derivation of the frequency-dependent second-order Bethe-Salpeter correlation kernel is given using many-body Green-function theory. Preliminary tests of this range-separated TDDFT method are presented for the calculation of excitation energies of the He and Be atoms and small molecules (H{sub 2}, N{sub 2}, CO{sub 2}, H{sub 2}CO, and C{sub 2}H{sub 4}). The results suggest that the addition of the long-range second-order Bethe-Salpeter correlation kernel overall slightly improves the excitation energies.
Bethe-Salpeter dynamics and the constituent mass concept for heavy quark mesons
Souchlas, N.; Stratakis, D.
2010-01-01
The definition of a quark as heavy requires a comparison of its mass with the nonperturbative chiral symmetry breaking scale which is about 1 GeV (Λ χ ∼1 GeV) or with the scale Λ QCD ∼0.2 GeV that characterizes the distinction between perturbative and nonperturbative QCD. For quark masses significantly larger than these scales, nonperturbative dressing effects, or equivalently nonperturbative self-energy contributions, and relativistic effects are believed to be less important for physical observables. We explore the concept of a constituent mass for heavy quarks in the Dyson-Schwinger equations formalism, for light-heavy and heavy-heavy quark mesons by studying their masses and electroweak decay constants.
Supersymmetric two-particle equations
Sissakyan, A.N.; Skachkov, N.B.; Shevchenko, O.Yu.
1986-01-01
In the framework of the scalar superfield model, a particular case of which is the well-known Wess-Zumino model, the supersymmetric Schwinger equations are found. On their basis with the use of the second Legendre transformation the two-particle supersymmetric Edwards and Bethe-Salpeter equations are derived. A connection of the kernels and inhomogeneous terms of these equations with generating functional of the second Legendre transformation is found
Relativistic three-particle dynamical equations: I. Theoretical development
Adhikari, S.K.; Tomio, L.; Frederico, T.
1993-11-01
Starting from the two-particle Bethe-Salpeter equation in the ladder approximation and integrating over the time component of momentum, three dimensional scattering integral equations satisfying constrains of relativistic unitarity and covariance are rederived. These equations were first derived by Weinberg and by Blankenbecler and Sugar. These two-particle equations are shown to be related by a transformation of variables. Hence it is shown to perform and relate dynamical calculation using these two equations. Similarly, starting from the Bethe-Salpeter-Faddeev equation for the three-particle system and integrating over the time component of momentum, several three dimensional three-particle scattering equations satisfying constraints of relativistic unitary and covariance are derived. Two of these three-particle equations are related by a transformation of variables as in the two-particle case. The three-particle equations obtained are very practical and suitable for performing relativistic scattering calculations. (author)
Nochi, Kazuki; Kawanai, Taichi; Sasaki, Shoichi
2018-03-01
The quark potential models with an energy-independent central potential have been successful for understanding the conventional charmonium states especially below the open charm threshold. As one might consider, however, the interquark potential is in general energy-dependent, and its tendency gets stronger in higher lying states. Confirmation of whether the interquark potential is energy-independent is also important to verify the validity of the quark potential models. In this talk, we examine the energy dependence of the charmonium potential, which can be determined from the Bethe-Salpeter (BS) amplitudes of cc̅ mesons in lattice QCD.We first calculate the BS amplitudes of radially excited charmonium states, the ηc(2S) and ψ(2S) states, using the variational method and then determine both the quark kinetic mass and the charmonium potential within the HAL QCD method. Through a direct comparison of charmonium potentials determined from both the 1S and 2S states, we confirm that neither the central nor spin-spin potential shows visible energy dependence at least up to 2S state.
LSZ asymptotic condition and dynamic equations in quantum field theory
Arkhipov, A.A.; Savrin, V.I.
1983-01-01
Some techniques that may be appropriate for the derivation of dynamic equations in quantum field theory are considered. A new method of deriving equations based on the use of LSZ asymptotic condition is described. It is proved that with the help of this method it becomes possible to obtain equations for wave functions both of scattering and bound states. Work is described in several papers under the dame title. The first paper is devoted to the Bethe-Salpeter equation
Dynamic equations for gauge-invariant wave functions
Kapshaj, V.N.; Skachkov, N.B.; Solovtsov, I.L.
1984-01-01
The Bethe-Salpeter and quasipotential dynamic equations for wave functions of relative quark motion, have been derived. Wave functions are determined by the gauge invariant method. The V.A. Fock gauge condition is used in the construction. Despite the transl tional noninvariance of the gauge condition the standard separation of variables has been obtained and wave function doesn't contain gauge exponents
Supersymmetric quasipotential equations
Zaikov, R.P.
1981-01-01
A supersymmetric extension of the Logunov-Tavkhelidze quasipotential approach is suggested. The supersymmetric Bethe- Salpeter equation is an initial equation. The transition from the four-time to the two-time Green function is made in the super- center-of-mass system. The two-time Green function has no inverse function in the whole spinor space. The resolvent operator if found using the Majorana character of the spinor wave function. The supersymmetric quasipotential equation is written. The consideration is carried out in the framework of the theory of chiral scalar superfields [ru
SU(N)-QCD2 meson equation in next-to-leading order
Durgut, M.; Pak, N.K.
1982-08-01
We compute the 1/N corrections to the meson equation in the regular cut-off scheme. We illustrate that although the quark and gluon self energy and vertex corrections do not vanish explicitly as in the singular cut-off scheme, their contributions to the meson Bethe-Salpeter equation get cancelled within the whole set of contributing diagrams. We also argue that 0(1/N) corrections to the meson equation remove the massless boson from the spectrum in accordance with the Coleman theorem. (author)
Hadronic bound states in SU(2) from Dyson-Schwinger equations
Vujinovic, Milan [Karl-Franzens-Universitaet Graz, Institut fuer Physik, Graz (Austria); Williams, Richard [Justus-Liebig-Universitaet Giessen, Institut fuer Theoretische Physik, Giessen (Germany)
2015-03-01
By using the Dyson-Schwinger/Bethe-Salpeter formalism in Euclidean spacetime, we calculate the ground state spectrum of J ≤ 1 hadrons in an SU(2) gauge theory with two fundamental fermions. We show that the rainbow-ladder truncation, commonly employed in QCD studies, is unsuitable for a description of an SU(2) theory. This we remedy by truncating at the level of the quark-gluon vertex Dyson-Schwinger equation in a diagrammatic expansion. Results obtained within this novel approach show good agreement with lattice studies. These findings emphasize the need to use techniques more sophisticated than rainbow-ladder when investigating generic strongly interacting gauge theories. (orig.)
A discussion of the relativistic equal-time equation
Chengrui, Q.; Danhua, Q.
1981-03-01
Ruan Tu-nan et al have proposed an equal-time equation for composite particles which is derived from Bethe-Salpeter (B-S) equation. Its advantage is that the kernel of this equation is a completely definite single rearrangement of the B-S irreducible kernel without any artificial assumptions. In this paper we shall give a further discussion of the properties of this equation. We discuss the behaviour of this equation as the mass of one of the two particles approaches the limit M 2 → infinite in the ladder approximation of single photon exchange. We show that up to order O(α 4 ) this equation is consistent with the Dirac equation. If the crossed two photon exchange diagrams are taken into account the difference between them is of order O(α 6 ). (author)
Self-consistence equations for extended Feynman rules in quantum chromodynamics
Wielenberg, A.
2005-01-01
In this thesis improved solutions for Green's functions are obtained. First the for this thesis essential techniques and concepts of QCD as euclidean field theory are presented. After a discussion of the foundations of the extended approach for the Feynman rules of QCD with a systematic approach for the 4-gluon vertex a modified renormalization scheme for the extended approach is developed. Thereafter the resummation of the Dyson-Schwinger equations (DSE) by the appropriately modified Bethe-Salpeter equation is discussed. Then the leading divergences for the 1-loop graphs of the resummed DSE are determined. Thereafter the equation-of-motion condensate is defined as result of an operator-product expansion. Then the self-consistency equations for the extended approaches are defined and numerically solved. (HSI)
Spinless Salpeter equation: Laguerre bounds on energy levels
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)
A comprehensive treatment of electromagnetic interactions and the three-body spectator equations
Jiri Adam; Jay Van Orden
2004-10-01
We present a general derivation the three-body spectator (Gross) equations and the corresponding electromagnetic currents. As in previous paper on two-body systems, the wave equations and currents are derived from those for Bethe-Salpeter equation with the help of algebraic method using a concise matrix notation. The three-body interactions and currents introduced by the transition to the spectator approach are isolated and the matrix elements of the e.m. current are presented in detail for system of three indistinguishable particles, namely for elastic scattering and for two and three body break-up. The general expressions are reduced to the one-boson-exchange approximation to make contact with previous work. The method is general in that it does not rely on introduction of the electromagnetic interaction with the help of the minimal replacement. It would therefore work also for other external fields.
Covariant equations for the three-body bound state
Stadler, A.; Gross, F.; Frank, M.
1997-01-01
The covariant spectator (or Gross) equations for the bound state of three identical spin 1/2 particles, in which two of the three interacting particles are always on shell, are developed and reduced to a form suitable for numerical solution. The equations are first written in operator form and compared to the Bethe-Salpeter equation, then expanded into plane wave momentum states, and finally expanded into partial waves using the three-body helicity formalism first introduced by Wick. In order to solve the equations, the two-body scattering amplitudes must be boosted from the overall three-body rest frame to their individual two-body rest frames, and all effects which arise from these boosts, including Wigner rotations and p-spin decomposition of the shell-particle, are treated exactly. In their final form, the equations reduce to a coupled set of Faddeev-like double integral equations with additional channels arising from the negative p-spin states of the off-shell particle
Meson spectra from two-body dirac equations with minimal interactions
Crater, H.W.; Becker, R.L.; Wong, C.Y.
1991-01-01
Many authors have used two-body relativistic wave equations with spin in nonperturbative numerical quark model calculations of the meson spectrum. Usually, they adopt a truncation of the Bethe-Salpeter equation of QED and/or scalar. QED and replace the static Coulomb interactions of those field theories with a semiphenomenological Q bar Q potential whose insertion in the Breit terms give the corresponding spin corrections. However, the successes of these wave equations in QED have invariably depended on perturbative treatment of the terms in each beyond the Coulomb terms. There have been no successful nonperturbative numerical test of two-body quantum wave equations in QED, because in most equations the effective potentials beyond the Coulomb are singular and can only be treated perturbatively. This is a glaring omission that we rectify here for the case of the two-body Dirac equations of constraint dynamics. We show in this paper that a nonperturbative numerical treatment of these equations for QED yields the same spectral results as a perturbative treatment of them which in turn agrees with the standard spectral results for positronium and muonium. This establishes that the vector and scalar interaction structures of our equations accurately incorporate field theoretic interactions in a bone fide relativistic wave equation. The last portion of this work will report recent quark model calculations using these equations with the Adler-Piran static Q bar Q potential
Relativistic two-and three-particle scattering equations using instant and light-front dynamics
Adhikari, S.K.; Tomio, L.; Frederico, T.
1992-01-01
Starting from the Bethe-Salpeter equation for two particles in the ladder approximation and integrating over the time component of momentum we derive three dimensional scattering integral equations satisfying constraints of unitarity and relativity, both employing the light-front and instant-form variables. The equations we arrive at are those first derived by Weinberg and by Blankenbecler and Sugar, and are shown to be related by a transformation of variables. Hence we show how to perform and relate identical dynamical calculation using these two equations. We extends this procedure to the case of three particles interacting via two-particle separable potentials. Using light-front and instant form variables we suggest a couple of three dimensional three-particle scattering equations satisfying constraints of two and three-particle unitarity and relativity. The three-particle light-front equation is shown to be approximately related by a transformation of variables to one of the instant-form three-particle equations. (author)
Gauge-invariant, nonperturbative approach to the infrared-finite bound-state problem in QCD
Gogokhia, V.Sh.
1989-09-01
Gauge invariant, nonperturbative approach to the bound state problem within the infrared finite Bethe-Salpeter equation is presented. Condition of cancellation of the nonperturbative infrared divergences is derived. Solutions for the quark propagator and corresponding quark gluon vertex function are written down which can be directly applied to the Bethe-Salpeter equation, in particular to the 'generalized ladder' approximation of this equation. (author) 18 refs.; 3 figs
Mainland, G.B.
1988-01-01
Zero four-momentum, helicity eigenstates of the Bethe--Salpeter equation are found for a composite system consisting of a charged, spin-0 constituent and a charged, spin- 1/2 constituent bound by minimal electrodynamics. The form of the Bethe--Salpeter equation used to describe the bound state includes the contributions from both single photon exchange (ladder approximation) and the ''seagull'' diagram. Attention is restricted to zero orbital angular momentum states since these appear to be the most interesting physically
QCD bound states at finite temperature and baryon number
Kalinovsky, Yu.L.; Muenchow, L.
1991-04-01
Quark-antiquark bound states are described within the Bethe-Salpeter equation for a class of quark models with instantaneous 4-quark interaction at finite temperature. Thereby decompositions of the Bethe-Salpeter vertex and wave functions according to their Lorentz structures and the particles content are used. As an application of general scheme, we determine the mass spectrum of low-lying mesons for a special Nambu-Jona-Lasinio model inspired by QCD for hadrons. (orig.)
Nonperturbative calculation of the shear viscosity in hot $\\phi^{4}$ theory in real time
Wang, E; Wang, Enke; Heinz, Ulrich
1999-01-01
Starting from the Kubo formula we calculate the shear viscosity in hot phi**4 theory nonperturbatively by resumming ladders with a real-time version of the Bethe-Salpeter equation at finite temperature. In the weak coupling limit, the generalized Fluctuation-Dissipation Theorem is shown to decouple the Bethe-Salpeter equations for the different real-time components of the 4-point function. The resulting scalar integral equation is identical with the one obtained by Jeon using diagrammatic ``cutting rules'' in the Imaginary Time Formalism.
Quasi-potential approach to the problem of bound states in quantum electrodynamics
Rizov, V A; Todorov, I T [Bylgarska Akademiya na Naukite, Sofia. Inst. za Yadrena Izsledvaniya i Yadrena Energetika
1975-07-01
The paper reviews two types of quasipotential equations. An equation with a non-local potential is derived from the equations of motion of quantum electrodynamics. It is also related to a Bethe-Salpeter type of equation for the retarded Green function. Most of the paper is devoted to a systematic study of a local version of the Logunov-Tavkhelidze quasipotential approach.
Covariant meson-baryon scattering with chiral and large Nc constraints
Lutz, M.F.M.; Kolomeitsev, E.E.
2001-05-01
We give a review of recent progress on the application of the relativistic chiral SU(3) Lagrangian to meson-baryon scattering. It is shown that a combined chiral and 1/N c expansion of the Bethe-Salpeter interaction kernel leads to a good description of the kaon-nucleon, antikaon-nucleon and pion-nucleon scattering data typically up to laboratory momenta of p lab ≅ 500 MeV. We solve the covariant coupled channel Bethe-Salpeter equation with the interaction kernel truncated to chiral order Q 3 where we include only those terms which are leading in the large N c limit of QCD. (orig.)
A structure preserving Lanczos algorithm for computing the optical absorption spectrum
Shao, Meiyue [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Div.; Jornada, Felipe H. da [Univ. of California, Berkeley, CA (United States). Dept. of Physics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Materials Science Div.; Lin, Lin [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Div.; Univ. of California, Berkeley, CA (United States). Dept. of Mathematics; Yang, Chao [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Div.; Deslippe, Jack [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Louie, Steven G. [Univ. of California, Berkeley, CA (United States). Dept. of Physics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Materials Science Div.
2016-11-16
We present a new structure preserving Lanczos algorithm for approximating the optical absorption spectrum in the context of solving full Bethe-Salpeter equation without Tamm-Dancoff approximation. The new algorithm is based on a structure preserving Lanczos procedure, which exploits the special block structure of Bethe-Salpeter Hamiltonian matrices. A recently developed technique of generalized averaged Gauss quadrature is incorporated to accelerate the convergence. We also establish the connection between our structure preserving Lanczos procedure with several existing Lanczos procedures developed in different contexts. Numerical examples are presented to demonstrate the effectiveness of our Lanczos algorithm.
On the Relativistic Separable Functions for the Breakup Reactions
Bondarenko, Serge G.; Burov, Valery V.; Rogochaya, Elena P.
2018-02-01
In the paper the so-called modified Yamaguchi function for the Bethe-Salpeter equation with a separable kernel is discussed. The type of the functions is defined by the analytic stucture of the hadron current with breakup - the reactions with interacting nucleon-nucleon pair in the final state (electro-, photo-, and nucleon-disintegration of the deuteron).
Pennington, Michael
2011-01-01
The Schwinger-Dyson, Bethe-Salpeter system of equations are the link between coloured quarks and gluons, and colourless hadrons and their properties. This talk reviews some aspects of these studies from the infrared behavior of ghosts to the prediction of electromagnetic form-factors.
Fermion-boson scattering in ladder approximation
Jafarov, R.G.; Hadjiev, S.A.
1992-10-01
A method of calculation of forward scattering amplitude for fermions and scalar bosons with exchanging of scalar particle is suggested. The Bethe-Salpeter ladder equation for the imaginary part of the amplitude is constructed and a solution in Regge asymptotical form is found and the corrections to the amplitude due to the exit from mass shell are calculated. (author). 8 refs
Eikonal multiple scattering model within the framework of Feynman's positron theory
Tekou, A.
1986-07-01
The Bethe Salpeter equation for nucleon-nucleon, nucleon-nucleus and nucleus-nucleus scattering is eikonalized. Multiple scattering series is obtained. Contributions of three body interations are included. The model presented below may be used to investigate atomic collisions. (author)
Elastic proton-deuteron backward scattering: relativistic effects and polarization observables
Kaptari, L.P.; Semikh, S.S.
1997-10-01
The elastic proton-deuteron backward reaction is analyzed within a covariant approach based on the Bethe-Salpeter equation with 000. Lorentz boost and other relativistic effects in the cross section and spin correlation observables, like tensor analyzing power and polarization transfer etc., are investigated in explicit form. Results of numerical calculations for a complete set of polarization observables are presented. (orig.)
The relativistic harmonic oscillator reconsidered
Hofsaess, T.
1978-01-01
The bound states of scalar quarks interacting through a scalar harmonic oscillator are investigated. In the presence of this interaction the dressed quark propagator differs substantially from the free one. This leads to a Bethe Salpeter equation which does not allow for any stable bound states of positive mass. (orig.) [de
On the proton exchange contribution to electron-hydrogen atom elastic scattering
Mignaco, J.A.; Tort, A.C.
1979-05-01
It is shown that the exchange contribution to the electron-proton potential Born term in elastic electron-hydrogen atom scattering arises as the non relativistic limit from the exchange of a proton between the two participant electrons - calculated from quantum electrodynamics including properly bound states (as solution of Bethe - Salpeter equation). (Author) [pt
Communication: Strong excitonic and vibronic effects determine the optical properties of Li₂O₂
García Lastra, Juan Maria; Bass, J. D.; Thygesen, Kristian Sommer
2011-01-01
The band structure and optical absorption spectrum of lithium peroxide (Li2O2) is calculated from first-principles using the G0W0 approximation and the Bethe-Salpeter equation, respectively. A strongly localized (Frenkel type) exciton corresponding to the π*→σ* transition on the O2 −2 peroxide ion...
Relativistic treatment of fermion-antifermion bound states
Lucha, W.; Rupprecht, H.; Schoeberl, F.F.
1990-01-01
We discuss the relativistic treatment of fermion-antifermion bound states by an effective-Hamiltonian method which imitates their description in terms of nonrelativistic potential models: the effective interaction potential, to be used in a Schroedinger equation which incorporates relativistic kinematics, is derived from the underlying quantum field theory. This approach is equivalent to the instantaneous approximation to the Bethe-Salpeter equation called Salpeter equation but comes closer to physical intuition than the latter one. (Author) 14 refs
Nonperturbative Aspects of Axial Vector Vertex
ZONG Hong-Shi; CHEN Xiang-Song; WANG Fan; CHANG Chao-Hsi; ZHAO En-Guang
2002-01-01
It is shown how the axial vector current of current quarks is related to that of constituent quarks within the framework of the global color symmetry model.Gluon dressing of the axial vector vertex and the quark self-energy functions are described by the inhomogeneous Bethe-Salpeter equation in the ladder approximation and the Schwinger Dyson equation in the rainbow approximation,respectively.
Determinantal method for complex angular momenta in potential scattering
Lee, B. W. [University of Pennsylvania, Philadelphia, PA (United States)
1963-01-15
In this paper I would like do describe a formulation of the complex angular momenta in potential scattering based on the Lippmann-Schwinger integral equation rather than on the Schrödinger differential equation. This is intended as a preliminary to the paper by SAWYER on the Regge poles and high energy limits in field theory (Bethe-Salpeter amplitudes), where the integral formulation is definitely more advantageous than the differential formulation.
Possible retardation effects of quark confinement on the meson spectrum
Qiao, C.; Huang, H.; Chao, K.
1996-01-01
The reduced Bethe-Salpeter equation with scalar confinement and vector gluon exchange is applied to quark-antiquark bound states. The so-called intrinsic flaw of the Salpeter equation with static scalar confinement is investigated. The notorious problem of narrow level spacings is found to be remedied by taking into consideration the retardation effect of scalar confinement. A good fit for the mass spectrum of both heavy and light quarkonium states is then obtained. copyright 1996 The American Physical Society
Tarighi Ahmadpour, Mahdi; Rostamnejadi, Ali; Hashemifar, S. Javad
2018-04-01
We study the electronic structure and optical properties of a body-centered tetragonal phase of carbon (bct-C4) within the framework of time-dependent density functional theory and Bethe-Salpeter equation. The results indicate that the optical properties of bct-C4 are strongly affected by the electron-hole interaction. It is demonstrated that the long-range corrected exchange-correlation kernels could fairly reproduce the Bethe-Salpeter equation results. The effective carrier number reveals that at energies above 30 eV, the excitonic effects are not dominant any more and that the optical transitions originate mainly from electronic excitations. The emerged peaks in the calculated electron energy loss spectra are discussed in terms of plasmon excitations and interband transitions. The results of the research indicate that bct-C4 is an indirect wide-band-gap semiconductor, which is transparent in the visible region and opaque in the ultraviolet spectral range.
Hyperon interaction in free space and nuclear matter
Dhar, Madhumita [Justus-Liebig University Giessen (Germany); Lenske, Horst [Justus-Liebig University Giessen (Germany); GSI, Darmstadt (Germany)
2016-07-01
A new approach to the SU(3) flavour symmetric meson-exchange model is introduced to describe free space baryon-baryon interaction. The Bethe-Salpeter equations are solved in a 3-D reduction scheme. The coupling of the various channels of total strangeness S and conserved total charge Q is studied in detail. Special attention is paid to the physical thresholds. The derived vacuum interaction has then been used to derive nuclear medium effect by employing the Pauli projector operator in 3-D reduced Bethe-Salpeter equation. The in-medium properties of the interaction are clearly seen in the variation of the in-medium low-energy parameters as a function of density.
Relativistic three-body approach to NN scattering at intermediate energies
van Faassen, E.; Tjon, J.A.
1986-01-01
The Bethe-Salpeter equation for coupled-channel N-Δ scattering is extended to satisfy unitarity in the NN and NNπ sectors. The procedure eliminates the unitarity violations characteristic of the standard ladder Bethe-Salpeter equation in the inelastic region, and improves the description of pion production near threshold. Results are presented for the NN phase shift and a number of observables up to 1 GeV. In particular, the 1D 2 inelasticity is found to be considerably smaller than found from phase shift analysis. In this context, the importance of the pion deuteron channel for the inelasticity parameter of is pointed out. 33 refs., 16 figs., 4 tabs
Some issues linked to the description of systems in strong interaction
Theussl, L.
2001-06-01
In the first part of this work we have dealt with some issues that are relevant in the area of nucleonic resonances within different constituent quark models. In this context we have concentrated on the theoretical description of Pi and Nu decays for N and Delta resonances. The results obtained point to the necessity of a more microscopic description of the dynamics which is at the same time responsible for the binding of quarks inside baryons and the decay of the latter ones. In the second part we have contributed to the study of crossed two-boson exchanges in the Bethe-Salpeter equation as well as to the investigation of different three-dimensional approaches that follow from the Bethe-Salpeter equation in a certain non-relativistic reduction scheme. These one include in particular an equation whose interaction depends on the total energy of the system. It was shown that such an equation is able to account for a certain number of properties of Bethe-Salpeter equation, in particular, that there also arise abnormal solutions in such an approach. (author)
Muonium hyperfine structure : An analytical solution to perturbative calculations
Wotzasek, C.J.; Gregorio, M.A.; Reinecke, S.
1982-01-01
The purely coulombian contribution to the terms of order E sub(F) (α 2 m sub(e)/m sub(μ))ln α - 1 of the hyperfine splitting of muonium is computed. Results agree with those of other authors. The goal of the work was twofold: first, to confirm that contribution; second, and perhaps more important, to check the analytic solution of the relativistic coulombian problem of the Bethe-Salpeter equation with instantaneous kernel. (Author) [pt
Microscopic Fermi liquid approach to disordered narrow band systems
Kolley, E.; Kolley, W.
1977-01-01
A Fermi liquid approach to tightly bound electrons in disordered systems is proposed to evaluate two-particle correlation functions L at T=0 deg K. Starting with a random Hubbard model and using a local ladder approximation in the particle-particle channel the irreducible particle-hole vertex is derived, being the kernel of the Bethe-Salpeter equation for L. CPA vertex corrections to the electrical conductivity and, for the ordered case, the correlation-enhanced paramagnetic susceptibility are calculated
The chiral Ward-Takahashi identity in the ladder approximation
Kugo, Taichiro; Mitchard, M.G.
1992-01-01
We show that the ladder approximation to the Schwinger-Dyson and Bethe-Salpeter equations preserves the Ward-Takahashi identity for the axial vector vertex if and only if we use the gluon momentum as the argument of the running coupling constant. However, in the usual Landau gauge this is inconsistent with the vector Ward identity. We propose a new method for making the ladder approximation scheme consistent with both vector and axial vector Ward identities. (orig.)
Aspects of open-flavour mesons in a comprehensive DSBSE study
Hilger, T. [University of Graz, NAWI Graz, Institute of Physics, Graz (Austria); Austrian Academy of Sciences, Institute of High Energy Physics, Vienna (Austria); Gomez-Rocha, M. [ECT*, Villazzano (Trento) (Italy); Krassnigg, A. [University of Graz, NAWI Graz, Institute of Physics, Graz (Austria); Lucha, W. [Austrian Academy of Sciences, Institute of High Energy Physics, Vienna (Austria)
2017-10-15
Open-flavour meson studies are the necessary completion to any comprehensive investigation of quarkonia. We extend recent studies of quarkonia in the Dyson-Schwinger-Bethe-Salpeter equation approach to explore their results for all possible flavour combinations. Within the inherent limitations of the setup, we present the most comprehensive results for meson masses and leptonic decay constants currently available and put them in perspective with respect to experiment and other approaches. (orig.)
On the dynamic polarizability of atoms
Nuroh, K.; Zaremba, E.
1989-04-01
The positive frequency dependent polarizability of atoms is discussed in terms of the particle-hole polarization propagator. It is considered in the simplest approximation defined by the Bethe-Salpeter equation which includes a subset of particle-hole interactions to all orders in the Coulomb potential. Its solution is used to show the relationship between different formulations of atomic photoabsorption via the effective dipole matrix element (Fermi's 'golden rule'), the TDLDA and the reaction matrix. (author). 21 refs, 7 figs
Electrical conductivity in random alloys
Mookerjee, A.; Yussouff, M.
1983-06-01
Starting from the augmented space formalism by one of us, and the use of the Ward identity and Bethe Salpeter equation, a complete formalism for the calculation of the electrical conductivity in tight-binding models of random binary alloys has been developed. The formalism is practical in the sense that viable calculations may be carried out with its help for realistics models of alloy systems. (author)
Squeezed condensate and confinement in a scalar model
Blaschke, D.; Pavel, H.P.; Roepke, G.; Peradze, G.; Pervushin, V.N.
1996-01-01
The generating functional of a free scalar field theory is generalized to the case of a squeezed vacuum. The squeezed vacuum is prepared by macroscopically populating the original vacuum with pairs of zero energy particles. It is shown that the corresponding quark propagator has no poles on the real-k 2 axis which can be interpreted as quark confinement. In contrast, a scalar meson-like bound state exists as solution of the corresponding Bethe-Salpeter equation. 20 refs
Cunningham, Brian; Grüning, Myrta; Azarhoosh, Pooya; Pashov, Dimitar; van Schilfgaarde, Mark
2018-03-01
We present an approach to calculate the optical absorption spectra that combines the quasiparticle self-consistent GW method [Phys. Rev. B 76, 165106 (2007), 10.1103/PhysRevB.76.165106] for the electronic structure with the solution of the ladder approximation to the Bethe-Salpeter equation for the macroscopic dielectric function. The solution of the Bethe-Salpeter equation has been implemented within an all-electron framework, using a linear muffin-tin orbital basis set, with the contribution from the nonlocal self-energy to the transition dipole moments (in the optical limit) evaluated explicitly. This approach addresses those systems whose electronic structure is poorly described within the standard perturbative GW approaches with density-functional theory calculations as a starting point. The merits of this approach have been exemplified by calculating optical absorption spectra of a strongly correlated transition metal oxide, NiO, and a narrow gap semiconductor, Ge. In both cases, the calculated spectrum is in good agreement with the experiment. It is also shown that for systems whose electronic structure is well-described within the standard perturbative GW , such as Si, LiF, and h -BN , the performance of the present approach is in general comparable to the standard GW plus Bethe-Salpeter equation. It is argued that both vertex corrections to the electronic screening and the electron-phonon interaction are responsible for the observed systematic overestimation of the fundamental band gap and spectrum onset.
ON THE USE OF FIELD THEORETICAL MODELS IN STRONG INTERACTION PHYSICS
Fubini, Sergio
1963-06-15
The effects of the short-range behavior in potential scattering upon the asymptotic behavior of the stronginteraction scattering amplitude and upon the validity of the methods of solution are discussed, using models. In particular, it is found that for certain singular potentials, the bound-state problem cannot be solved by a plane-wave expansion. For these singular potentials, an irtegral equation must be set up by means of an expansion in terms of eigenfunctions having the correct behavior at small distances. The study makes use of both the Schroedinger and Bethe-Salpeter equations. (T.F.H.)
Static correlation beyond the random phase approximation
Olsen, Thomas; Thygesen, Kristian Sommer
2014-01-01
derived from Hedin's equations (Random Phase Approximation (RPA), Time-dependent Hartree-Fock (TDHF), Bethe-Salpeter equation (BSE), and Time-Dependent GW) all reproduce the correct dissociation limit. We also show that the BSE improves the correlation energies obtained within RPA and TDHF significantly...... and confirms that BSE greatly improves the RPA and TDHF results despite the fact that the BSE excitation spectrum breaks down in the dissociation limit. In contrast, second order screened exchange gives a poor description of the dissociation limit, which can be attributed to the fact that it cannot be derived...
Non-perturbative QCD and hadron physics
Cobos-Martínez, J J
2016-01-01
A brief exposition of contemporary non-perturbative methods based on the Schwinger-Dyson (SDE) and Bethe-Salpeter equations (BSE) of Quantum Chromodynamics (QCD) and their application to hadron physics is given. These equations provide a non-perturbative continuum formulation of QCD and are a powerful and promising tool for the study of hadron physics. Results on some properties of hadrons based on this approach, with particular attention to the pion distribution amplitude, elastic, and transition electromagnetic form factors, and their comparison to experimental data are presented. (paper)
Two-photon processes of π0, η, η', ηc and ηb
Klabucar, D.
1997-01-01
Two-photon processes of π 0 , η, η', η c and η b are studied in the consistently coupled Schwinger-Dyson (SD) and Bethe-Salpeter (BS) approach, where dynamical chiral symmetry breaking (DχSB) is obtained through the SD equation for the quark propagator which is then used in the BS equation. It is shown that the coupled SD-BS approach is similarly successful in the description of two-photon processes of pseudoscalar mesons over a wide range of masses. (K.A.)
General QED/QCD aspects of simple systems
Telegdi, V.L.; Brodsky, S.J.
1989-09-01
This paper discusses the following topics: renormalization theory; the Kinoshita-Lee-Nauenberg theorem; the Yennie-Frautschi-Suura relation; scale invariance at large momentum transfer; scaling and scaling violation at large momentum transfers; low-energy theorem in Compton scattering; does the perturbation series in QED converge; renormalization of the weak angle Θ w ; the Nambu-Bethe-Salpeter (NBS) equation; the decay rate of 3 S, positronium; radiative corrections to QCD Born cross section; and progress on the relativistic 2-body equation
On one estimate of glueball mass
Boos, E.E.
1986-01-01
The Bethe-Salpeter equation for the wave function of the bound state of two gluons is considered. The mass of the glueball 0 ++ , (M gl ∼ 1.3 GeV), is estimated using some expansions in the equation kernel in the spirit of those made in the QCD sum rules method. In the leading approximation, the masses of the glueballs 0 ++ and 2 ++ appear to be degenerate. A possibility to improve the accuracy of estimating the mass by using the expansion in 1/N c is discussed
'Relativistic' quark model for mesons with flavour-independent potential
Kroesen, G.
1987-01-01
On the base of the Bethe-Salpeter equation in instantaneous approximation a unified model for the mass spectrum of the mesons was designed. The 'relativistic' structure of the Bethe-Salpeter equation allows a natural inclusion of the spin dependences and an extension of the model to small quark masses. The model contains as essential property two potential contributions where one represents the one-gluon exchange while the other represents the confinement potential. The annihilation of qanti q into gluons was not regarded. The spectrum and the amplitudes of the Bethe-Salpeter equation were solved approximatively in numerical way for the lowest states. The free parameters of the model were determined by a fit of the spectrum to a wellknown part of the meson spectrum. The results yield even at small quark masses a quantitatively good picture for all meson families. The result shows that the spectra of the heavy and light mesons can be described by a flavor-independent potential which contains 5 free parameters. Both the internal spin dependent structure and the absolute position of the families can so correctly be described. Especially the position of the D, D s , and B states and the position of the uanti u, danti d states can be simultaneously described by a constant C in the long-range part of the potential. The constant C is thereby essentially determined by the splitting between the Υ family and the B family repectively Ψ and D family. The 3 S 1- 3 D 1 respectively the 3 P 2 - 3 F 2 configuration mixing was regarded. The results show that this mixing is negligibly small. (orig./HSI) [de
A gauge-invariant chiral unitary framework for kaon photo- and electroproduction on the proton
Borasoy, B.; Bruns, P.C.; Nissler, R.; Meissner, U.G.
2007-01-01
We present a gauge-invariant approach to photoproduction of mesons on nucleons within a chiral unitary framework. The interaction kernel for meson-baryon scattering is derived from the chiral effective Lagrangian and iterated in a Bethe-Salpeter equation. Within the leading-order approximation to the interaction kernel, data on kaon photoproduction from SAPHIR, CLAS and CBELSA/TAPS are analyzed in the threshold region. The importance of gauge invariance and the precision of various approximations in the interaction kernel utilized in earlier works are discussed. (orig.)
Confinement, diquarks and goldstone's theorem
Roberts, C.D.
1996-01-01
Determinations of the gluon propagator in the continuum and in lattice simulations are compared. A systematic truncation procedure for the quark Dyson-Schwinger and bound state Bethe-Salpeter equations is described. The procedure ensures the flavor-octet axial- vector Ward identity is satisfied order-by-order, thereby guaranteeing the preservation of Goldstone's theorem; and identifies a mechanism that simultaneously ensures the absence of diquarks in QCD and their presence in QCD N c =2 , where the color singlet diquark is the ''baryon'' of the theory
Relativistic description of quark-antiquark bound states. Spin-independent treatment
Gara, A.; Durand, B.; Durand, L.; Nickisch, L.J.
1989-01-01
We present the results of a detailed study of light- and heavy-quark--antiquark bound states in the context of the reduced Bethe-Salpeter equation with static vector and scalar interactions. In the present paper, we consider the spin-averaged spectra. Spin effects are considered in a separate paper. We find that this approach, although apparently successful for the heavy-quark b bar b and c bar c states, fails for the s bar s, l bar l, and light-heavy states. The reasons for the failure are intrinsic to the method, as we discuss. Difficulties are already evident for the c bar c states
Relativistic quarkonium dynamics
Sazdjian, H.
1985-06-01
We present, in the framework of relativistic quantum mechanics of two interacting particles, a general model for quarkonium systems satisfying the following four requirements: confinement, spontaneous breakdown of chiral symmetry, soft explicit chiral symmetry breaking, short distance interactions of the vector type. The model is characterized by two arbitrary scalar functions entering in the large and short distance interaction potentials, respectively. Using relationships with corresponding quantities of the Bethe-Salpeter equation, we also present the normalization condition of the wave functions, as well as the expressions of the meson decay coupling constants. The quark masses appear in this model as free parameters
Gomez, Cesar; Gunnesson, Johan; Hernandez, Rafael
2008-01-01
We extract from the double logarithmic contributions to DGLAP anomalous dimensions for twist-two operators up to three-loops the magnon dispersion relation for planar N = 4 supersymmetric Yang-Mills. Perturbatively the magnon dispersion relation agrees with the expansion of the anomalous dimension for spin-one as well as with the non-collinear double logarithmic contributions to the BFKL anomalous dimensions analytically extended to negative spin. The all-loop expression for the magnon dispersion relation is determined by the double logarithmic resummation of the corresponding Bethe-Salpeter equation. A potential map relating the spin chain magnon to BFKL eigenfunctions in the double logarithm approximation is suggested.
Three-body unitarity with isobars revisited
Mai, M.; Hu, B. [The George Washington University, Washington, DC (United States); Doering, M. [The George Washington University, Washington, DC (United States); Thomas Jefferson National Accelerator Facility, Newport News, VA (United States); Pilloni, A. [Thomas Jefferson National Accelerator Facility, Newport News, VA (United States); Szczepaniak, A. [Thomas Jefferson National Accelerator Facility, Newport News, VA (United States); Indiana University, Center for Exploration of Energy and Matter, Bloomington, IN (United States); Indiana University, Physics Department, Bloomington, IN (United States)
2017-09-15
The particle exchange model of hadron interactions can be used to describe three-body scattering under the isobar assumption. In this study we start from the 3 → 3 scattering amplitude for spinless particles, which contains an isobar-spectator scattering amplitude. Using a Bethe-Salpeter Ansatz for the latter, we derive a relativistic three-dimensional scattering equation that manifestly fulfills three-body unitarity and two-body unitarity for the sub-amplitudes. This property holds for energies above breakup and also in the presence of resonances in the sub-amplitudes. (orig.)
High energy production of gluons in a quasi-multi-Regge kinematics
Fadin, V.S.; Lipatov, L.N.
1989-01-01
Inelastic gluon-gluon scattering amplitudes in the Born approximation for the quasi-multi-Regge kinematics are calculated, starting with the Veneziano-type expression for the inelastic amplitude of the gluon-tachyon scattering with its subsequent simplification in the region of large energies and the Regge slope α'→0. Results obtained allow one to determine the high order corrections to the gluon Regge trajectory, the reggeon-particle vertices and to the integral kernel of the Bethe-Salpeter equation for the vacuum t-channel partial waves. 10 refs.; 7 figs
Techniques for calculations with nPI effective actions
Carrington M.E.
2015-01-01
Full Text Available We consider a symmetric scalar theory with quartic coupling in 2- and 3- dimensions and compare the self-consistent 4-point vertex obtained from the 4PI effective action with the Bethe-Salpeter 4-vertex from 2PI effective action. We show that when the coupling is large the contributions from the higher order effective action are large. We also show that one can solve the 2PI equations of motion in 4-dimensions, without introducing counter-terms, using a renormalization group method. This method provides a promising starting point to study the renormalization of higher order nPI theories.
Dynamic retardation corrections to the mass spectrum of heavy quarkonia
Kopalejshvili, T.; Rusetskij, A.
1996-01-01
In the framework of the Logunov-Tavkhelidze quasipotential approach the first-order retardation corrections to the heavy quarkonia mass spectrum are calculated with the use of the stationary wave boundary condition in the covariant kernel of the Bethe-Salpeter equation. As has been expected, these corrections turn out to be small for all low-lying heavy meson states and vanish in the heavy quark limit (m Q →∞). The comparison of the suggested approach to the calculation of retardation corrections with others, known in literature, is carried out. 22 refs., 1 tab
Kurachi, Masafumi; Shrock, Robert
2006-01-01
We consider a vectorial, confining SU(N) gauge theory with a variable number, N f , of massless fermions transforming according to the fundamental representation. Using the Schwinger-Dyson and Bethe-Salpeter equations, we calculate the S parameter in terms of the current-current correlation functions. We focus on values of N f such that the theory is in the crossover region between the regimes of walking behavior and QCD-like (nonwalking) behavior. Our calculations indicate that the contribution to S from a given fermion decreases as one moves from the QCD-like to the walking regimes. The implications of this result for technicolor theories are discussed
Covariant solution of the three-quark problem in quantum ﬁeld theory: the nucleon
Nicmorus D.
2010-04-01
Full Text Available We provide details on a recent solution of the nucleon’s covariant Faddeev equation in an explicit three-quark approach. The full Poincaré-covariant structure of the three-quark amplitude is implemented through an orthogonal basis obtained from a partial-wave decomposition. We employ a rainbow-ladder gluon exchange kernel which allows for a comparison with meson Bethe-Salpeter and baryon quark-diquark studies. We describe the construction of the three-quark amplitude in full detail and compare it to a notation widespread in recent publications. Finally, we discuss ﬁrst numerical results for the nucleon’s amplitude.
Baryons in and beyond the quark-diquark model
Eichmann, G.; Alkofer, R.; Krassnigg, A.; Fischer, C. S.; Nicmorus, D.
2011-01-01
We examine the nucleon's electromagnetic form factors in a Poincare-covariant Faddeev framework. The three-quark core contributions to the form factors are obtained by employing a quark-diquark approximation. We implement the self-consistent solution for the quark-photon vertex from its inhomogeneous Bethe-Salpeter equation. We find that the resulting transverse parts which add to the Ball-Chiu vertex have no significant impact on nucleon magnetic moments. The current-quark mass evolution of the form factors agrees with results from lattice QCD.
Ab initio calculation of the electronic and optical properties of solid pentacene
Tiago, Murilo L.; Northrup, John E.; Louie, Steve G.
2002-01-01
The optical and electronic properties of crystalline pentacene are studied, using a first-principles Green's-function approach. The quasiparticle energies are calculated within the GW approximation and the electron-hole excitations are computed by solving the Bethe-Salpeter equation. We investigate the role of polymorphism on the electronic energy gap and linear optical spectrum by studying two different crystalline phases: the solution-phase structure and the vapor-phase structure. charge-transfer excitons are found to dominate the optical spectrum. Excitons with sizable binding energies are predicted for both phases
Dipole moments of the rho meson
Hecht, M.B.; McKellar, B.H.P.
1997-04-01
The electric and magnetic dipole moments (EDM) of the rho meson are calculated using the propagators and vertices derived from the quantum chromodynamics Dyson-Schwinger equations. Results obtained from using the Bethe-Salpeter amplitude studied by Chappell, Mitchell et. al., and Pichowsky and Lee, are compared. The rho meson EDM is generated through the inclusion of a quark electric dipole moment, which is left as a free variable. These results are compared to the perturbative results to obtain a measure of the effects of quark interactions and confinement. The two dipole moments are also calculated using the phenomenological MIT bag model to provide a further basis for comparison
Conformal operator product expansion in the Yukawa model
Prati, M.C.
1983-01-01
Conformal techniques are applied to the Yukawa model, as an example of a theory with spinor fields. It is written the partial-wave analysis of the 4-point function of two scalars and two spinors in the channel phi psi → phi psi in terms of spinor tensor representations of the conformal group. Using this conformal expansion, it is diagonalized the Bethe-Salpeter equation, which is reduced to algebraic relations among the partial waves. It is shown that in the γ 5 -invariant model, but not in the general case, it is possible to derive dynamically from the expansions of the 4-point function the vacuum operator product phi psi>
Sun, Bao-Xi; Wan, Da-Ming; Zhao, Si-Yu
2018-05-01
The {{{D}}\\bar{{{D}}}}{{* }} interaction via a ρ or ω exchange is constructed within an extended hidden gauge symmetry approach, where the strange quark is replaced by the charm quark in the SU(3) flavor space. With this {{{D}}\\bar{{{D}}}}{{* }} interaction, a bound state slightly lower than the {{{D}}\\bar{{{D}}}}{{* }} threshold is generated dynamically in the isospin zero sector by solving the Bethe-Salpeter equation in the coupled-channel approximation, which might correspond to the X(3872) particle announced by many collaborations. This formulism is also used to study the {{{B}}\\bar{{{B}}}}{{* }} interaction, and a {{{B}}\\bar{{{B}}}}{{* }} bound state with isospin zero is generated dynamically, which has no counterpart listed in the review of the Particle Data Group. Furthermore, the one-pion exchange between the D meson and the {\\bar{{{D}}}}{{* }} is analyzed precisely, and we do not think the one-pion exchange potential need be considered when the Bethe-Salpeter equation is solved.
Evidence of ghost suppression in gluon mass scale dynamics
Aguilar, A. C.; Binosi, D.; Figueiredo, C. T.; Papavassiliou, J.
2018-03-01
In this work we study the impact that the ghost sector of pure Yang-Mills theories may have on the generation of a dynamical gauge boson mass scale, which hinges on the appearance of massless poles in the fundamental vertices of the theory, and the subsequent realization of the well-known Schwinger mechanism. The process responsible for the formation of such structures is itself dynamical in nature, and is governed by a set of Bethe-Salpeter type of integral equations. While in previous studies the presence of massless poles was assumed to be exclusively associated with the background-gauge three-gluon vertex, in the present analysis we allow them to appear also in the corresponding ghost-gluon vertex. The full analysis of the resulting Bethe-Salpeter system reveals that the contribution of the poles associated with the ghost-gluon vertex are particularly suppressed, their sole discernible effect being a slight modification in the running of the gluon mass scale, for momenta larger than a few GeV. In addition, we examine the behavior of the (background-gauge) ghost-gluon vertex in the limit of vanishing ghost momentum, and derive the corresponding version of Taylor's theorem. These considerations, together with a suitable Ansatz, permit us the full reconstruction of the pole sector of the two vertices involved.
Fermionic bound states in Minkowski space. Light-cone singularities and structure
Paula, Wayne de; Frederico, Tobias; Pimentel, Rafael [Instituto Tecnologico de Aeronautica, DCTA, Dept. de Fisica, Sao Jose dos Campos, Sao Paulo (Brazil); Salme, Giovanni [Istituto Nazionale di Fisica Nucleare, Rome (Italy); Viviani, Michele [Istituto Nazionale di Fisica Nucleare, Pisa (Italy)
2017-11-15
The Bethe-Salpeter equation for two-body bound system with spin 1/2 constituent is addressed directly in the Minkowski space. In order to accomplish this aim we use the Nakanishi integral representation of the Bethe-Salpeter amplitude and exploit the formal tool represented by the exact projection onto the null-plane. This formal step allows one (i) to deal with end-point singularities one meets and (ii) to find stable results, up to strongly relativistic regimes, which settle in strongly bound systems. We apply this technique to obtain the numerical dependence of the binding energies upon the coupling constants and the light-front amplitudes for a fermion-fermion 0{sup +} state with interaction kernels, in ladder approximation, corresponding to scalar-, pseudoscalar- and vector-boson exchanges, respectively. After completing the numerical survey of the previous cases, we extend our approach to a quark-antiquark system in 0{sup -} state, taking both constituent-fermion and exchanged-boson masses, from lattice calculations. Interestingly, the calculated light-front amplitudes for such a mock pion show peculiar signatures of the spin degrees of freedom. (orig.)
Light-front Ward-Takahashi identity for two-fermion systems
Marinho, J. A. O.; Frederico, T.; Pace, E.; Salme, G.; Sauer, P. U.
2008-01-01
We propose a three-dimensional electromagnetic current operator within light-front dynamics that satisfies a light-front Ward-Takahashi identity for two-fermion systems. The light-front current operator is obtained by a quasipotential reduction of the four-dimensional current operator and acts on the light-front valence component of bound or scattering states. A relation between the light-front valence wave function and the four-dimensional Bethe-Salpeter amplitude both for bound or scattering states is also derived, such that the matrix elements of the four-dimensional current operator can be fully recovered from the corresponding light-front ones. The light-front current operator can be perturbatively calculated through a quasipotential expansion, and the divergence of the proposed current satisfies a Ward-Takahashi identity at any given order of the expansion. In the quasipotential expansion the instantaneous terms of the fermion propagator are accounted for by the effective interaction and two-body currents. We exemplify our theoretical construction in the Yukawa model in the ladder approximation, investigating in detail the current operator at the lowest nontrivial order of the quasipotential expansion of the Bethe-Salpeter equation. The explicit realization of the light-front form of the Ward-Takahashi identity is verified. We also show the relevance of instantaneous terms and of the pair contribution to the two-body current and the Ward-Takahashi identity
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
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.
Yannouleas, C.; Jang, S.
1986-01-01
The zero-temperature second RPA is generalized to finite temperatures through the use of the method of linearization of the equations of motion. After elimination of the quadruples, for low enough temperatures and within the subspace spanned by the doubles, a proper symmetrization yields an eigenvalue equation which exhibits formal properties like the simple RPA. From this second RPA eigenvalue equation, a closed formula for the spreading width of an isolated collective state is extracted. The second RPA can be recast in the form of a generalized collision term and be compared with the method of the Bethe-Salpeter equation for the two-body Green function. However, the second RPA method (and results) contrasts with the approach (and corresponding results) of the Boltzmann collision term, which is usually viewed as the appropriate agent for nuclear dissipation. (orig.)
External gauge invariance and anomaly in BS vertices and boundstates
Bando, Masako; Harada, Masayasu; Kugo, Taichiro
1994-01-01
A systematic method is given for obtaining consistent approximations to the Schwinger-Dyson (SD) and Bethe-Salpeter (BS) equations which maintain the external gauge invariance. We show that for any order of approximation to the SD equation there is a corresponding approximation to the BS equations such that the solutions to those equations satisfy the Ward-Takahashi identities of the external gauge symmetry. This formulation also clarifies the way how we can calculate the Green functions of current operators in a consistent manner with the gauge invariance and the axial anomaly. We show which type of diagrams for the π 0 → γγ amplitude using the pion BS amplitude give result consistent with the low-energy theorem. An interesting phenomenon is observed in the ladder approximation that the low-energy theorem is saturated by the zeroth order terms in the external momenta of the pseudoscalar BS amplitude and the vector vertex functions. (author)
Quark-anti-quark potential in N=4 SYM
Gromov, Nikolay [Mathematics Department, King’s College London,The Strand, London WC2R 2LS (United Kingdom); St. Petersburg INP,Gatchina, 188 300, St.Petersburg (Russian Federation); Levkovich-Maslyuk, Fedor [Mathematics Department, King’s College London,The Strand, London WC2R 2LS (United Kingdom); Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm (Sweden)
2016-12-22
We construct a closed system of equations describing the quark-anti-quark potential at any coupling in planar N=4 supersymmetric Yang-Mills theory. It is based on the Quantum Spectral Curve method supplemented with a novel type of asymptotics. We present a high precision numerical solution reproducing the classical and one-loop string predictions very accurately. We also analytically compute the first 7 nontrivial orders of the weak coupling expansion. Moreover, we study analytically the generalized quark-anti-quark potential in the limit of large imaginary twist to all orders in perturbation theory. We demonstrate how the QSC reduces in this case to a one-dimensional Schrodinger equation. In the process we establish a link between the Q-functions and the solution of the Bethe-Salpeter equation.
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.
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
On the absence of pentaquark states from dynamics in strongly coupled lattice QCD
Anjos, Petrus Henrique Ribeiro dos [Universidade Federal de Goias (UFG), Goiania, GO (Brazil); Veiga, Paulo Afonso Faria da; O' Carroll, Michael [Universidade de Sao Paulo (USP), SP (Brazil); Francisco Neto, Antonio [Universidade Federal de Ouro Preto (UFOP), MG (Brazil)
2011-07-01
Full text: We consider an imaginary time functional integral formulation of a two-flavor, 3 + 1 lattice QCD model with Wilson's action and in the strong coupling regime (with a small hopping parameter, {kappa}0, and a much smaller plaquette coupling, {beta} = 1/g{sub 0}{sup 2}, so that the quarks and glueballs are heavy). The model has local SU(3){sub c} gauge and global SU(2){sub f} flavor symmetries, and incorporates the corresponding part of the eightfold way particles: baryons (mesons) of asymptotic mass -3ln{kappa}(-2 ln {kappa}). We search for pentaquark states as meson-baryon bound states in the energy-momentum spectrum of the model, using a lattice Bethe-Salpeter equation. This equation is solved within a ladder approximation, given by the lowest nonvanishing order in {kappa} and {beta} of the Bethe-Salpeter kernel. It includes order 2 contributions with a q-barq exchange potential together with a contribution that is a local-in-space, energy-dependent potential. The attractive or repulsive nature of the exchange interaction depends on the spin of the meson-baryon states. The Bethe-Salpeter equation presents integrable singularities, forcing the couplings to be above a threshold value for the meson and the baryon to bind in a pentaquark. We analyzed all the total isospin sectors, I = 1/2/3/2/ 5/2, for the system. For all I, the net attraction resulting from the two sources of interaction is not strong enough for the meson and the baryon to bind. Thus, within our approximation, these pentaquark states are not present up to near the free meson-baryon energy threshold of - 5 ln{kappa}. This result is to be contrasted with the spinless case for which our method detects meson-baryon bound states, as well as for Yukawa effective baryon and meson field models. A physical interpretation of our results emerges from an approximate correspondence between meson-baryon bound states and negative energy states of a one-particle lattice Schroedinger Hamiltonian
Instabilities of the chiral-symmetry-breaking ground state in a truncation-free expansion
Rembiesa, P.
1988-01-01
We use the composite-field effective-action method to examine the stability of the chiral-symmetry-breaking vacua in a QED-like model of interacting fermion fields. Unlike most of the existing approaches, ours does not rely on the truncated Baker-Johnson-Willey expansion. Instead, we break the hierarchy of the Dyson-Schwinger equations by the requirement that the vertex function is dominated by the contributions from the vicinity of the mass shell of the exchanged gluon and that it explicitly satisfies the Ward identity. The composite-field effective potential is then expanded in terms of the eigenfunctions of the Bethe-Salpeter equation. The signature of the second derivatives of the effective potential shows that the broken-symmetry vacua are unstable
Nutt, W.T.
1976-01-01
A meson-theoretic model of the intermediate range nucleon-nucleon potential is presented with emphasis placed on the two-pion exchange contribution. The Bethe-Salpeter equation is reduced, by the Blankenbecler-Sugar technique, to a Lippmann-Schwinger equation, from which an approximate nonlocal, energy-dependent potential is obtained. The nucleon-antinucleon pair contribution, which plagues meson-theoretical two-pion calculations, is suppressed by the complex poles of the one-nucleon Green's function. The importance of the retention of the explicit energy dependence of the potential is demonstrated by calculating the off-shell scattering matrices. The potential is presented in a linearized (in energy) form with the core region adjusted to produce a fit to low energy data
Exotic states in the S=1 N-pi-K system and low-lying 1/2+ S=-1 resonances
Oset E.
2010-04-01
Full Text Available In this manuscript we discuss about our study of the $N pi ar{K}$ and the NπK systems made by solving the Faddeev equations with the two-body t-matrices obtained by solving the Bethe-Salpeter equations with the potentials obtained from chiral dynamics. In the strangeness = -1 case, we found that all the Λ and Σ resonances listed by the particle data group, with spin-parity 1/2+ , in the 1550-1800 MeV region get generated due to the involved three-body dynamics. This motivated us to study the strangeness =1 three-body system, i.e., NπK , where we did not ﬁnd any evidence for the Θ+ (1542 but found a broad bump around 1700 MeV which has a κ(800N structure.
Phenomenological dynamics in QCD at large distances
Gogohia, V.Sh.; Kluge, Gy.
1991-07-01
A gauge-invariant, nonperturbative approach to QCD at large distances in the context of the Schwinger-Dyson equations and corresponding Slavnov-Taylor identities in the quark sector is presented. Making only one widely accepted assumption that the full gluon propagator becomes an infrared singular like (q 2 ) -2 in the covariant gauge, we find three and only three confinement-type solutions for the quark propagator (quark confinement theorem.) The approach is free from ghost complications. Also show that multiplication by the quark infrared renormalization constant only, would make all the Green's functions infrared finite (multiplicative renormalizability). The bound-state problem in framework of Bethe-Salpeter equation is discussed as well. Some basic physical parameters of chiral QCD as pion decay constant and quark condensate, have been calculated within our approach. (author) 75 refs.; 14 figs
Schmidt, M.; Janke, T.; Redmer, R.
1989-01-01
Within a model calculation the influence of the Pauli exclusion principle on the electrical conductivity of a fully ionized and degenerate hydrogen plasma is investigated. Basing on a quantum kinetic equation solved with the relaxation time ansatz, the electron-ion contribution to the resistivity is calculated. The thermodynamical T-matrix for electron-ion scattering processes is evaluated under special account for the Pauli blocking of the intermediate scattering states. The corresponding Bethe-Salpeter equation is solved analytically using a separable approximation of the statically screened potential. The Pauli exclusion principle has been found to give rise for a considerable enhancement of the transport cross section near the Fermi energy. Thus, degeneracy effects tend to diminish the electrical conductivity in the density-temperature region considered here. (author)
Spectrum of Charmonia within a Contact Interaction
Bedolla, Marco Antonio
2016-01-01
For the flavour-singlet heavy quark system of charmonia, we compute the masses of the ground state mesons in four different channels: pseudo-scalar (η c (1 S )), vector ( J /ψ(1 S )), scalar (χ s0 (1 P )) and axial vector (χ c1 (1 P )), as well as the weak decay constants of the η c (1S) and J/ψ(1 S ). The framework for this analysis is provided by a symmetry-preserving Schwinger- Dyson equation (SDEs) treatment of a vector x vector contact interaction (CI). The results found for the meson masses and the weak decay constants, for the spin-spin combinations studied, are in fairly good agreement with experimental data and earlier model calculations based upon Schwinger-Dyson and Bethe-Salpeter equations (BSEs) involving sophisticated interaction kernels. (paper)
Meson exchange currents in nuclei; the triton beta decay as an example
Jaus, W.
1976-01-01
The method used to reduce the four-dimensional Bethe-Salpeter equation to the three-dimensional Schroedinger equation, thus defining a potential in terms of the field theoretic interaction, can be generalized to define a consistent exchange by considering the relativistic interaction of a current with a bound state of nucleons. This covariant approach allows a unified treatment of exchange current effects, renormalization of the nuclear wave function due to meson exchange, relativistic corrections and negative energy contributions to the wave function and it is discussed in detail how these effects influence the Gamow-Teller matrix element for the decay 3 H→ 3 He + e + antiγ. One and two-meson exchange processes are calculated including nucleon resonances in intermediate states, and good agreement of theoretical and experimental predictions for the GT matrix element is found. (Auth.)
Relativistic description of quark-antiquark bound states. II. Spin-dependent treatment
Gara, A.; Durand, B.; Durand, L.
1990-01-01
We present the results of a study of light- and heavy-quark--antiquark bound states in the context of the reduced Bethe-Salpeter equation, including the full spin dependence. We obtain good fits to the observed spin splittings in the b bar b and c bar c systems using a short-distance single-gluon-exchange interaction, and a long-distance scalar confining interaction. However, we cannot obtain satisfactory fits to the centers of gravity of the b bar b and c bar c spin multiplets at the same time, and the splittings calculated for q bar Q mesons containing the lighter quarks are very poor. The difficulty appears to be intrinsic to the reduced Salpeter equation for reasons which we discuss
Hyperon interaction in free space and nuclear matter
Dhar, Madhumita; Lenske, Horst [Institute for Theoretical Physics, Justus- Liebig-University Giessen (Germany)
2015-07-01
Baryon-baryon interactions within the SU(3)-octet are investigated in free space and nuclear matter.A meson exchange model based on SU(3) symmetry is used for determining the interaction. The Bethe-Salpeter equations are solved in a 3-D reduction scheme. In-medium effect has been incorporated by including a two particle Pauli projector operator in the scattering equation. The coupling of the various channels of total strangeness S and conserved total charge is studied in detail. Special attention is paid to the physical thresholds. The density dependence of interaction is clearly seen in the variation of the in-medium low-energy parameters. The approach is compared to descriptions derived from chiral-EFT and other meson-exchange models e.g. the Nijmegen and the Juelich model.
Quarkonia and heavy-light mesons in a covariant quark model
Leitão Sofia
2016-01-01
Full Text Available Preliminary calculations using the Covariant Spectator Theory (CST employed a scalar linear confining interaction and an additional constant vector potential to compute the mesonic mass spectra. In this work we generalize the confining interaction to include more general structures, in particular a vector and also a pseudoscalar part, as suggested by a recent study [1]. A one-gluon-exchange kernel is also implemented to describe the short-range part of the interaction. We solve the simplest CST approximation to the complete Bethe-Salpeter equation, the one-channel spectator equation, using a numerical technique that eliminates all singularities from the kernel. The parameters of the model are determined through a fit to the experimental pseudoscalar meson spectra, with a good agreement for both quarkonia and heavy-light states.
Differential Equations Compatible with KZ Equations
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
Scaling Universality between Band Gap and Exciton Binding Energy of Two-Dimensional Semiconductors
Jiang, Zeyu; Liu, Zhirong; Li, Yuanchang; Duan, Wenhui
2017-06-01
Using first-principles G W Bethe-Salpeter equation calculations and the k .p theory, we unambiguously show that for two-dimensional (2D) semiconductors, there exists a robust linear scaling law between the quasiparticle band gap (Eg) and the exciton binding energy (Eb), namely, Eb≈Eg/4 , regardless of their lattice configuration, bonding characteristic, as well as the topological property. Such a parameter-free universality is never observed in their three-dimensional counterparts. By deriving a simple expression for the 2D polarizability merely with respect to Eg, and adopting the screened hydrogen model for Eb, the linear scaling law can be deduced analytically. This work provides an opportunity to better understand the fantastic consequence of the 2D nature for materials, and thus offers valuable guidance for their property modulation and performance control.
The effect of meson wave function on heavy-quark fragmentation function
Moosavi Nejad, S.M. [Yazd University, Faculty of Physics (Iran, Islamic Republic of); Institute for Research in Fundamental Sciences (IPM), School of Particles and Accelerators, Tehran (Iran, Islamic Republic of)
2016-05-15
We calculate the process-independent fragmentation functions (FFs) for a heavy quark to fragment into heavy mesons considering the effects of meson wave function. In all previous works, where the FFs of heavy mesons or heavy baryons were calculated, a delta function form was approximated for the wave function of hadrons. Here, for the first time, we consider a typical mesonic wave function which is different from the delta function and is the nonrelativistic limit of the solution of Bethe-Salpeter equation with the QCD kernel. We present our numerical results for the heavy FFs and show how the proposed wave function improves the previous results. As an example, we focus on the fragmentation function for c-quark to split into S-wave D{sup 0} -meson and compare our results with experimental data from BELLE and CLEO. (orig.)
Topics in dual models and extended solutions
Roth, R.S.
1977-01-01
Two main topics are explored. The first deals with the infinities arising from the one loop planar string diagram of the standard dual model. It is shown that for the number of dimensions d = 25 or 26, these infinities lead to a renormalization of the slope of the Regge trajectories, in addition to a renormalization of the coupling constant. The second topic deals with the propagator for a confined particle (monopole) in a field theory. When summed to all orders, this propagator is altogether free of singularities in the finite momentum plane, and an attempt is made to illustrate this. The Bethe-Salpeter equation is examined and it is shown that ladder diagrams are not sufficient to obtain this result. However, in a nonrelativistic approximation confinement is obtained and all poles disappear
Theory of Raman scattering in coupled electron-phonon systems
Itai, K.
1992-01-01
The Raman spectrum is calculated for a coupled conduction-electron-phonon system in the zero-momentum-transfer limit. The Raman scattering is due to electron-hole excitations and phonons as well. The phonons of those branches that contribute to the electron self-energy and the correction of the electron-phonon vertex are assumed to have flat energy dispersion (the Einstein phonons). The effect of electron-impurity scattering is also incorporated. Both the electron-phonon interaction and the electron-impurity interaction cause the fluctuation of the electron distribution between different parts of the Fermi surface, which results in overdamped zero-sound modes of various symmetries. The scattering cross section is obtained by solving the Bethe-Salpeter equation. The spectrum shows a lower threshold at the smallest Einstein phonon energy when only the electron-phonon interaction is taken into consideration. When impurities are also taken into consideration, the threshold disappears.
Defect properties of Sn- and Ge-doped ZnTe: suitability for intermediate-band solar cells
Flores, Mauricio A.
2018-01-01
We investigate the electronic structure and defect properties of Sn- and Ge- doped ZnTe by first-principles calculations within the DFT+GW formalism. We find that ({{{Sn}}}{{Zn}}) and ({{{Ge}}}{{Zn}}) introduce isolated energy levels deep in the band gap of ZnTe, derived from Sn-5s and Ge-4s states, respectively. Moreover, the incorporation of Sn and Ge on the Zn site is favored in p-type ZnTe, in both Zn-rich and Te-rich environments. The optical absorption spectra obtained by solving the Bethe-Salpeter equation reveals that sub-bandgap absorptance is greatly enhanced due to the formation of the intermediate band. Our results suggest that Sn- and Ge-doped ZnTe would be a suitable material for the development of intermediate-band solar cells, which have the potential to achieve efficiencies beyond the single-junction limit.
Quasiparticle and optical properties of strained stanene and stanane.
Lu, Pengfei; Wu, Liyuan; Yang, Chuanghua; Liang, Dan; Quhe, Ruge; Guan, Pengfei; Wang, Shumin
2017-06-20
Quasiparticle band structures and optical properties of two dimensional stanene and stanane (fully hydrogenated stanene) are studied by the GW and GW plus Bethe-Salpeter equation (GW-BSE) approaches, with inclusion of the spin-orbit coupling (SOC). The SOC effect is significant for the electronic and optical properties in both stanene and stanane, compared with their group IV-enes and IV-anes counterparts. Stanene is a semiconductor with a quasiparticle band gap of 0.10 eV. Stanane has a sizable band gap of 1.63 eV and strongly binding exciton with binding energy of 0.10 eV. Under strain, the quasiparticle band gap and optical spectrum of both stanene and stanane are tunable.
Quasi-particle energies and optical excitations of hydrogenated and fluorinated germanene.
Shu, Huabing; Li, Yunhai; Wang, Shudong; Wang, Jinlan
2015-02-14
Using density functional theory, the G0W0 method and Bethe-Salpeter equation calculations, we systematically explore the structural, electronic and optical properties of hydrogenated and fluorinated germanene. The hydrogenated/fluorinated germanene tends to form chair and zigzag-line configurations and its electronic and optical properties show close geometry dependence. The chair hydrogenated/fluorinated and zigzag-line fluorinated germanene are direct band-gap semiconductors, while the zigzag-line hydrogenated germanene owns an indirect band-gap. Moreover, the quasi-particle corrections are significant and strong excitonic effects with large exciton binding energies are observed. Moreover, the zigzag-line hydrogenated/fluorinated germanene shows highly anisotropic optical responses, which may be used as a good optical linear polarizer.
Relativistic few quark dynamics for hadrons
Mitra, A.N.
1983-07-01
A microscopic confinement approach is presented to a few quarks systems through an effective (harmonic) kernel inserted at the level of q-q-bar and q-q pairs, using the vehicle of the Bethe-Salpeter equation for each such system. The formalism, which is realistic for light quark systems (which require an intrinsically relativistic treatment), has been developed in a simple enough form so as to be applicable in practice to a large class of phenomena amenable to experimental test. The comparison over a wide range of hadronic properties (from mass spectra to current matrix elements), all within a single integrated framework, would seem to strongly support the ansatz of universality of the reduced spring constant (ω-tilde) which plays a role analogous to the bag radius, but at a far more microscopic level
Hadron-quark vertex function. Interconnection between 3D and 4D wave function
Mitra, A.N.; Bhatnagar, S.
1990-01-01
Interconnection between 3D and 4D forms of Bethe-Salpeter equation (EBS) with a kernel depending on relative momenta is used to derive hadron-quark vertex function in Lorentz invariance form. The vertex function which is directly related to a 4D wave function satisfying a corresponding EBS determines the natural continuation outside mass surface for the entire momentum space and serves the basis for computing amplitudes of transitions through appropriate loop quark diagrams. Two applications (f p values for P→ll-bar and F π for n 0 +yy) are discussed briefly to illustrate this formalism. An attention is paid to the problem of complex amplitudes for quark loops with a larger number of external hadrons.A possible solution of the problem is proposed. 29 refs
Bhatnagar, S.; Mahecha, J.
2008-09-01
We have employed the framework of Bethe-Salpeter equation under Covariant Instantaneous Ansatz to calculate the leptonic decay constants of unequal mass pseudoscalar mesons. In the Dirac structure of BS wave function, the covariants are incorporated from their complete set in accordance with a recently proposed power counting rule, order-by-order in powers of inverse of meson mass. The decay constants are calculated incorporating both Leading Order (LO) as well as Next-to-leading Order (NLO) Dirac covariants. The contribution of both LO as well as NLO covariants to decay constants are studied in detail in this paper. The results are found to improve dramatically, and hence validating the power counting rule which also provides a practical means of incorporating Dirac covariants in the BS wave function of a hadron. (author)
Manipulating Light with Transition Metal Clusters, Organic Dyes, and Metal Organic Frameworks
Ogut, Serdar [Univ. of Illinois, Chicago, IL (United States)
2017-09-11
The primary goals of our research program is to develop and apply state-of-the-art first-principles methods to predict electronic and optical properties of three systems of significant scientific and technological interest: transition metal clusters, organic dyes, and metal-organic frameworks. These systems offer great opportunities to manipulate light for a wide ranging list of energy-related scientific problems and applications. During this grant period, we focused our investigations on the development, implementation, and benchmarking of many-body Green’s function methods (GW approximation and the Bethe-Salpeter equation) to examine excited-state properties of transition metal/transition-metal-oxide clusters and organic molecules that comprise the building blocks of dyes and metal-organic frameworks.
Barionic spectroscopy masses and hyperfine structure
Vale, M.A.B. do.
1986-01-01
Using the Bethe-Salpeter equation in QCD, we obtain, in the nonrelativistic approximation, a quark-antiquark interaction potential. We include, in a phenomenological way, a confining term in the potential (V(qq-bar) = V QCD (qq-bar) + V sub (conf) (qq-bar)). Assuming that the three-quark interaction can be described in terms of pair interactions, and that the quark-quark interaction is related to the quark-antiquark interaction (v (qq)= 1/2 V(qq-bar)), we evaluate the baryon masses as three-quark bound states. We also calculate the relativistic corrections coming from the spin-spin interaction. Finally, our results are compared to the available experimental data. (author) [pt
On the gluonic correction to lepton-pair decays in a relativistic quarkonium model
Ito, Hitoshi
1987-01-01
The gluonic correction to the leptonic decay of the heavy vector meson is investigated by using the perturbation theory to the order α s . The on-mass-shell approximation is assumed for the constituent quarks so that we assure the gauge independence of the correction. The decay rates in the model based on the Bethe-Salpeter equation are also shown, in which the gluonic correction with a high-momentum cutoff is calculated for the off-shell quarks. It is shown that the static approximation to the correction factor (1 - 16α s /3π) is not adequate and the gluonic correction does not suppress but enhance the decay rates of the ground states for the c anti c and b anti b systems. (author)
Haber, Jonah; Refaely-Abramson, Sivan; da Jornada, Felipe H.; Louie, Steven G.; Neaton, Jeffrey B.
Multi-exciton generation processes, in which multiple charge carriers are generated from a single photon, are mechanisms of significant interest for achieving efficiencies beyond the Shockley-Queisser limit of conventional p-n junction solar cells. One well-studied multiexciton process is singlet fission, whereby a singlet decays into two spin-correlated triplet excitons. Here, we use a newly developed computational approach to calculate singlet-fission coupling terms and rates with an ab initio Green's function formalism based on many-body perturbation theory (MBPT) within the GW approximation and the Bethe-Salpeter equation approach. We compare results for crystalline pentacene and TIPS-pentacene and explore the effect of molecular packing on the singlet fission mechanism. This work is supported by the Department of Energy.
Fields, particles and analyticity: recent results or 30 goldberg (ER) variations on B.A.C.H
Bros, J.
1991-01-01
As it is known, Axiomatic Field Theory (A) implies double analyticity of the η-point functions in space-time and energy-momentum Complex Variables (C), with various interconnections by Fourier-Laplace analysis. When the latter is replaced by. Harmonic Analysis (H) on spheres and hyperboloids, a new kind of double analyticity results from (A) (i.e. from locality, spectral condition, temperateness and invariance): complex angular momentum is thereby introduced (a missing chapter in (A)). Exploitation of Asymptotic Completeness via Bethe-Salpeter-type equations (B) leads to new developments of the previous theme on (A, C, H) (complex angular momentum) and of other themes on (A,C) (crossing, Haag-Swieca property etc...). Various aspects of (A) + (B) have been implemented in Constructive Field Theory (composite spectrum, asymptotic properties etc...) by a combination of specific techniques and of model-independent methods
Fields, particles and analyticity: recent results or 30 goldberg (ER) variations on B.A.C.H
Bros, J
1992-12-31
As it is known, Axiomatic Field Theory (A) implies double analyticity of the {eta}-point functions in space-time and energy-momentum Complex Variables (C), with various interconnections by Fourier-Laplace analysis. When the latter is replaced by. Harmonic Analysis (H) on spheres and hyperboloids, a new kind of double analyticity results from (A) (i.e. from locality, spectral condition, temperateness and invariance): complex angular momentum is thereby introduced (a missing chapter in (A)). Exploitation of Asymptotic Completeness via Bethe-Salpeter-type equations (B) leads to new developments of the previous theme on (A, C, H) (complex angular momentum) and of other themes on (A,C) (crossing, Haag-Swieca property etc...). Various aspects of (A) + (B) have been implemented in Constructive Field Theory (composite spectrum, asymptotic properties etc...) by a combination of specific techniques and of model-independent methods.
Spectral properties of nuclear matter
Bozek, P
2006-01-01
We review self-consistent spectral methods for nuclear matter calculations. The in-medium T-matrix approach is conserving and thermodynamically consistent. It gives both the global and the single-particle properties the system. The T-matrix approximation allows to address the pairing phenomenon in cold nuclear matter. A generalization of nuclear matter calculations to the super.uid phase is discussed and numerical results are presented for this case. The linear response of a correlated system going beyond the Hartree-Fock+ Random-Phase-Approximation (RPA) scheme is studied. The polarization is obtained by solving a consistent Bethe-Salpeter (BS) equation for the coupling of dressed nucleons to an external field. We find that multipair contributions are important for the spin(isospin) response when the interaction is spin(isospin) dependent
Meson-meson bound state in a 2+1 lattice QCD model with two flavors and strong coupling
Faria da Veiga, Paulo A.; O'Carroll, Michael; Neto, Antonio Francisco
2005-01-01
We consider the existence of bound states of two mesons in an imaginary-time formulation of lattice QCD. We analyze an SU(3) theory with two flavors in 2+1 dimensions and two-dimensional spin matrices. For a small hopping parameter and a sufficiently large glueball mass, as a preliminary, we show the existence of isoscalar and isovector mesonlike particles that have isolated dispersion curves (upper gap up to near the two-particle threshold ∼-4lnκ). The corresponding meson masses are equal up to and including O(κ 3 ) and are asymptotically of order -2lnκ-κ 2 . Considering the zero total isospin sector, we show that there is a meson-meson bound state solution to the Bethe-Salpeter equation in a ladder approximation, below the two-meson threshold, and with binding energy of order bκ 2 ≅0.02359κ 2 . In the context of the strong coupling expansion in κ, we show that there are two sources of meson-meson attraction. One comes from a quark-antiquark exchange. This is not a meson exchange, as the spin indices are not those of the meson particle, and we refer to this as a quasimeson exchange. The other arises from gauge field correlations of four overlapping bonds, two positively oriented and two of opposite orientation. Although the exchange part gives rise to a space range-one attractive potential, the main mechanism for the formation of the bound state comes from the gauge contribution. In our lattice Bethe-Salpeter equation approach, this mechanism is manifested by an attractive distance-zero energy-dependent potential. We recall that no bound state appeared in the one-flavor case, where the repulsive effect of Pauli exclusion is stronger
Sb2Te3 crystal a potential absorber material for broadband photodetector: A first-principles study
Abdullahi Lawal
Full Text Available Antimony telluride (Sb2Te3, a layered semiconductor material, is considered a promising absorbing material for a high-performance optoelectronic device within broadband wavelengths because of remarkable features like strong optical absorbance and the narrow direct band gap. In this work, based on the first-principles approach, we investigate in detail the structural, electronic and optical properties of the hexagonal Sb2Te3 compound. The structural and electronic properties were computed using the first-principles approach, treating exchange–correlation potential with generalized gradient approximation (GGA within density functional theory (DFT. Furthermore, for accurate prediction of the band gap, we go beyond DFT and calculated band structure using GW correction. The optical properties, namely, imaginary and real parts of complex dielectric function, absorption coefficient, refractive index, reflectivity, extinction coefficient, electron energy loss function and optical conductivity are performed by quasi-particle many-body perturbation theory (MBPT via Bethe-Salpeter equation (BSE. The computed structural parameters are in good agreement with available experimental data. The obtained quasi-particle (GW correction band structure show the semiconducting character of Sb2Te3 material with a direct band gap Eg of 0.221 eV, in agreement with previously reported value (Eg = 0.210 eV while the projected density of states indicates (PDOS that the p-orbital of Sb and Te atoms are responsible for material properties near the Fermi level. To our knowledge, our first reported calculations of optical properties, with the inclusion of electron-hole effects are consistent with available experimental measurements. Consistencies of our findings with experimental data validate the effectiveness of electron-hole interaction for theoretical investigation of optical properties. Keywords: DFT, Quasi-particle many-body perturbation theory, Bethe-Salpeter
Alkofer, Reinhard; von Smekal, Lorenz
2001-11-01
Recent studies of QCD Green's functions and their applications in hadronic physics are reviewed. We discuss the definition of the generating functional in gauge theories, in particular, the rôle of redundant degrees of freedom, possibilities of a complete gauge fixing versus gauge fixing in presence of Gribov copies, BRS invariance and positivity. The apparent contradiction between positivity and colour antiscreening in combination with BRS invariance in QCD is considered. Evidence for the violation of positivity by quarks and transverse gluons in the covariant gauge is collected, and it is argued that this is one manifestation of confinement. We summarise the derivation of the Dyson-Schwinger equations (DSEs) of QED and QCD. For the latter, the implications of BRS invariance on the Green's functions are explored. The possible influence of instantons on DSEs is discussed in a two-dimensional model. In QED in (2+1) and (3+1) dimensions, the solutions for Green's functions provide tests of truncation schemes which can under certain circumstances be extended to the DSEs of QCD. We discuss some limitations of such extensions and assess the validity of assumptions for QCD as motivated from studies in QED. Truncation schemes for DSEs are discussed in axial and related gauges, as well as in the Landau gauge. Furthermore, we review the available results from a systematic non-perturbative expansion scheme established for Landau gauge QCD. Comparisons to related lattice results, where available, are presented. The applications of QCD Green's functions to hadron physics are summarised. Properties of ground state mesons are discussed on the basis of the ladder Bethe-Salpeter equation for quarks and antiquarks. The Goldstone nature of pseudoscalar mesons and a mechanism for diquark confinement beyond the ladder approximation are reviewed. We discuss some properties of ground state baryons based on their description as Bethe-Salpeter/Faddeev bound states of quark
Shore, B.W.
1981-01-01
The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
Equating error in observed-score equating
van der Linden, Willem J.
2006-01-01
Traditionally, error in equating observed scores on two versions of a test is defined as the difference between the transformations that equate the quantiles of their distributions in the sample and population of test takers. But it is argued that if the goal of equating is to adjust the scores of
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
Handbook of integral equations
Polyanin, Andrei D
2008-01-01
This handbook contains over 2,500 integral equations with solutions as well as analytical and numerical methods for solving linear and nonlinear equations. It explores Volterra, Fredholm, WienerHopf, Hammerstein, Uryson, and other equations that arise in mathematics, physics, engineering, the sciences, and economics. This second edition includes new chapters on mixed multidimensional equations and methods of integral equations for ODEs and PDEs, along with over 400 new equations with exact solutions. With many examples added for illustrative purposes, it presents new material on Volterra, Fredholm, singular, hypersingular, dual, and nonlinear integral equations, integral transforms, and special functions.
Introduction to differential equations
Taylor, Michael E
2011-01-01
The mathematical formulations of problems in physics, economics, biology, and other sciences are usually embodied in differential equations. The analysis of the resulting equations then provides new insight into the original problems. This book describes the tools for performing that analysis. The first chapter treats single differential equations, emphasizing linear and nonlinear first order equations, linear second order equations, and a class of nonlinear second order equations arising from Newton's laws. The first order linear theory starts with a self-contained presentation of the exponen
Uraltseva, N N
1995-01-01
This collection focuses on nonlinear problems in partial differential equations. Most of the papers are based on lectures presented at the seminar on partial differential equations and mathematical physics at St. Petersburg University. Among the topics explored are the existence and properties of solutions of various classes of nonlinear evolution equations, nonlinear imbedding theorems, bifurcations of solutions, and equations of mathematical physics (Navier-Stokes type equations and the nonlinear Schrödinger equation). The book will be useful to researchers and graduate students working in p
Lebedev, D.R.
1979-01-01
Benney's equations of motion of incompressible nonviscous fluid with free surface in the approximation of long waves are analyzed. The connection between the Lie algebra of Hamilton plane vector fields and the Benney's momentum equations is shown
Fractional Schroedinger equation
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
Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Beginning partial differential equations
O'Neil, Peter V
2014-01-01
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible,combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is or
Ichiguchi, Katsuji
1998-01-01
A new reduced set of resistive MHD equations is derived by averaging the full MHD equations on specified flux coordinates, which is consistent with 3D equilibria. It is confirmed that the total energy is conserved and the linearized equations for ideal modes are self-adjoint. (author)
Singular stochastic differential equations
Cherny, Alexander S
2005-01-01
The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) arise often in theory and in applications. However, known conditions for the existence and uniqueness of a solution typically fail for such equations. The book concentrates on the study of the existence, the uniqueness, and, what is most important, on the qualitative behaviour of solutions of singular stochastic differential equations. This is done by providing a qualitative classification of isolated singular points, into 48 possible types.
Light tetraquarks and mesons in a DSE/BSE approach
Heupel, Walter
2015-07-01
Bound states and their properties are an inherent non-perturbative feature of QCD. Moreover, QCD is a confining theory so that instead of the elementary quarks and gluons themselves, only colourless bound states formed of these elementary particles are directly measurable. One non-perturbative framework to describe QCD are the Dyson-Schwinger equations, which interrelate all Green functions of the theory by an infinite tower of integral equations, and the corresponding Bethe-Salpeter equations that define the bound states of the theory. To reduce the infinite tower to a tractable form, the equations have to be truncated. In this thesis the so-called rainbow ladder' truncation was used that reduces the quark-gluon vertex to the bare vertex and replaces the gluon by an effective modeled one so that the only Green function that has to be solved, is the quark propagator. This truncation preserves the important axial Ward-Takahashi-identity and the Gell-Mann-Oakes-Renner relation. For the effective gluon the Maris-Tandy interaction was used, modeled to reproduce the pion mass and decay constant. Starting from this well-established truncation, the four-body tetraquark Bethe-Salpeter equation was constructed. To solve the tetraquark Bethe-Salpeter equation, a fully covariant basis for the tetraquark amplitude is necessary. Additionally, the basis has to reflect the quantum numbers of the tetraquark and has to fulfill the Pauli principle. The construction of such a basis was performed for all parts of the amplitude: The Dirac-tensor structure, the phase space, the colour and the flavour tensor structure. Upon solving the tetraquark bound state equation, dynamical pion poles in the tetraquark amplitude phase space appeared, reflecting the actual physics that determines the tetraquark: The tetraquark is dominated by two-body correlations which manifest themselves as poles in the phase space. It is especially noteworthy that these two-body correlations in form of poles
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
Functional equations with causal operators
Corduneanu, C
2003-01-01
Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with causal operators. Functional Equations with Causal Operators explains the connection between equations with causal operators and the classical types of functional equations encountered by mathematicians and engineers. It details the fundamentals of linear equations and stability theory and provides several applications and examples.
Partial differential equations
Evans, Lawrence C
2010-01-01
This text gives a comprehensive survey of modern techniques in the theoretical study of partial differential equations (PDEs) with particular emphasis on nonlinear equations. The exposition is divided into three parts: representation formulas for solutions; theory for linear partial differential equations; and theory for nonlinear partial differential equations. Included are complete treatments of the method of characteristics; energy methods within Sobolev spaces; regularity for second-order elliptic, parabolic, and hyperbolic equations; maximum principles; the multidimensional calculus of variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and entropy criteria for conservation laws; and, much more.The author summarizes the relevant mathematics required to understand current research in PDEs, especially nonlinear PDEs. While he has reworked and simplified much of the classical theory (particularly the method of characteristics), he primarily emphasizes the modern interplay between funct...
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Degenerate nonlinear diffusion equations
Favini, Angelo
2012-01-01
The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asympt...
Application of the N-quantum approximation method to bound state problems
Raychaudhuri, A.
1977-01-01
The N-quantum approximation (NQA) method is examined in the light of its application to bound state problems. Bound state wave functions are obtained as expansion coefficients in a truncated Haag expansion. From the equations of motion for the Heisenberg field and the NQA expansion, an equation satisfied by the wave function is derived. Two different bound state systems are considered. In one case, the bound state problem of two identical scalars by scalar exchange is analyzed using the NQA. An integral equation satisfied by the wave function is derived. In the nonrelativistic limit, the equation is shown to reduce to the Schroedinger equation. The equation is solved numerically, and the results compared with those obtained for this system by other methods. The NQA method is also applied to the bound state of two spin 1/2 particles with electromagnetic interaction. The integral equation for the wave function is shown to agree with the corresponding Bethe Salpeter equation in the nonrelativistic limit. Using the Dirac (4 x 4) matrices the wave function is expanded in terms of structure functions and the equation for the wave function is reduced to two disjoint sets of coupled equation for the structure functions
K. Banoo
1998-01-01
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.
Solving Ordinary Differential Equations
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Reactimeter dispersion equation
A.G. Yuferov
2016-01-01
The aim of this work is to derive and analyze a reactimeter metrological model in the form of the dispersion equation which connects reactimeter input/output signal dispersions with superimposed random noise at the inlet. It is proposed to standardize the reactimeter equation form, presenting the main reactimeter computing unit by a convolution equation. Hence, the reactimeter metrological characteristics are completely determined by this unit hardware function which represents a transient re...
Differential equations I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.
Laenen, E.
1995-01-01
We propose a new evolution equation for the gluon density relevant for the region of small x B . It generalizes the GLR equation and allows deeper penetration in dense parton systems than the GLR equation does. This generalization consists of taking shadowing effects more comprehensively into account by including multigluon correlations, and allowing for an arbitrary initial gluon distribution in a hadron. We solve the new equation for fixed α s . We find that the effects of multigluon correlations on the deep-inelastic structure function are small. (orig.)
Manca, V.; Salibra, A.; Scollo, Giuseppe
1990-01-01
Equational type logic is an extension of (conditional) equational logic, that enables one to deal in a single, unified framework with diverse phenomena such as partiality, type polymorphism and dependent types. In this logic, terms may denote types as well as elements, and atomic formulae are either
Alternative equations of gravitation
Pinto Neto, N.
1983-01-01
It is shown, trough a new formalism, that the quantum fluctuation effects of the gravitational field in Einstein's equations are analogs to the effects of a continuum medium in Maxwell's Electrodynamics. Following, a real example of the applications of these equations is studied. Qunatum fluctuations effects as perturbation sources in Minkowski and Friedmann Universes are examined. (L.C.) [pt
Yagi, M. [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Horton, W. [Texas Univ., Austin, TX (United States). Inst. for Fusion Studies
1993-11-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite {beta} that we solve the perpendicular component of Ohm`s law to conserve the physical energy while ensuring the relation {del} {center_dot} j = 0.
Yagi, M.; Horton, W.
1993-11-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that we solve the perpendicular component of Ohm's law to conserve the physical energy while ensuring the relation ∇ · j = 0
Yagi, M.; Horton, W.
1994-01-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that the perpendicular component of Ohm's law be solved to ensure ∇·j=0 for energy conservation
The currently proposed model compaction equation was derived from data sourced from the. Niger Delta and it relates porosity to depth for sandstones under hydrostatic pressure condition. The equation is useful in predicting porosity and compaction trend in hydrostatic sands of the. Niger Delta. GEOLOGICAL SETTING OF ...
M. Hazewinkel (Michiel)
1995-01-01
textabstractDedication: I dedicate this paper to Prof. P.C. Baayen, at the occasion of his retirement on 20 December 1994. The beautiful equation which forms the subject matter of this paper was invented by Wouthuysen after he retired. The four complex variable Wouthuysen equation arises from an
The generalized Fermat equation
Beukers, F.
2006-01-01
This article will be devoted to generalisations of Fermat’s equation xn + yn = zn. Very soon after the Wiles and Taylor proof of Fermat’s Last Theorem, it was wondered what would happen if the exponents in the three term equation would be chosen differently. Or if coefficients other than 1 would
Applied partial differential equations
Logan, J David
2004-01-01
This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Wu Zhuo Qun; Li Hui Lai; Zhao Jun Ning
2001-01-01
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which
Differential equations problem solver
Arterburn, David R
2012-01-01
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and
Local instant conservation equations
Delaje, Dzh.
1984-01-01
Local instant conservation equations for two-phase flow are derived. Derivation of the equation starts from the recording of integral laws of conservation for a fixed reference volume, containing both phases. Transformation of the laws, using the Leibniz rule and Gauss theory permits to obtain the sum of two integrals as to the volume and integral as to the surface. Integrals as to the volume result in local instant differential equations, in particular derivatives for each phase, and integrals as to the surface reflect local instant conditions of a jump on interface surface
Beginning partial differential equations
O'Neil, Peter V
2011-01-01
A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres
Ordinary differential equations
Miller, Richard K
1982-01-01
Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,
Uncertain differential equations
Yao, Kai
2016-01-01
This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.
Applied partial differential equations
Logan, J David
2015-01-01
This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Tsintsadze, Nodar L.; Tsintsadze, Levan N.
2008-01-01
A general derivation of the charging equation of a dust grain is presented, and indicated where and when it can be used. A problem of linear fluctuations of charges on the surface of the dust grain is discussed.
Equations For Rotary Transformers
Salomon, Phil M.; Wiktor, Peter J.; Marchetto, Carl A.
1988-01-01
Equations derived for input impedance, input power, and ratio of secondary current to primary current of rotary transformer. Used for quick analysis of transformer designs. Circuit model commonly used in textbooks on theory of ac circuits.
Problems in differential equations
Brenner, J L
2013-01-01
More than 900 problems and answers explore applications of differential equations to vibrations, electrical engineering, mechanics, and physics. Problem types include both routine and nonroutine, and stars indicate advanced problems. 1963 edition.
Applied partial differential equations
DuChateau, Paul
2012-01-01
Book focuses mainly on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included.
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics
Saaty, Thomas L
1981-01-01
Covers major types of classical equations: operator, functional, difference, integro-differential, and more. Suitable for graduate students as well as scientists, technologists, and mathematicians. "A welcome contribution." - Math Reviews. 1964 edition.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
Structural Equations and Causation
Hall, Ned
2007-01-01
Structural equations have become increasingly popular in recent years as tools for understanding causation. But standard structural equations approaches to causation face deep problems. The most philosophically interesting of these consists in their failure to incorporate a distinction between default states of an object or system, and deviations therefrom. Exploring this problem, and how to fix it, helps to illuminate the central role this distinction plays in our causal thinking.
Equations of radiation hydrodynamics
Mihalas, D.
1982-01-01
The purpose of this paper is to give an overview of the role of radiation in the transport of energy and momentum in a combined matter-radiation fluid. The transport equation for a moving radiating fluid is presented in both a fully Eulerian and a fully Lagrangian formulation, along with conservation equations describing the dynamics of the fluid. Special attention is paid to the problem of deriving equations that are mutually consistent in each frame, and between frames, to 0(v/c). A detailed analysis is made to show that in situations of broad interest, terms that are formally of 0(v/c) actually dominate the solution, demonstrating that it is esential (1) to pay scrupulous attention to the question of the frame dependence in formulating the equations; and (2) to solve the equations to 0(v/c) in quite general circumstances. These points are illustrated in the context of the nonequilibrium radiation diffusion limit, and a sketch of how the Lagrangian equations are to be solved will be presented
Quantum linear Boltzmann equation
Vacchini, Bassano; Hornberger, Klaus
2009-01-01
We review the quantum version of the linear Boltzmann equation, which describes in a non-perturbative fashion, by means of scattering theory, how the quantum motion of a single test particle is affected by collisions with an ideal background gas. A heuristic derivation of this Lindblad master equation is presented, based on the requirement of translation-covariance and on the relation to the classical linear Boltzmann equation. After analyzing its general symmetry properties and the associated relaxation dynamics, we discuss a quantum Monte Carlo method for its numerical solution. We then review important limiting forms of the quantum linear Boltzmann equation, such as the case of quantum Brownian motion and pure collisional decoherence, as well as the application to matter wave optics. Finally, we point to the incorporation of quantum degeneracies and self-interactions in the gas by relating the equation to the dynamic structure factor of the ambient medium, and we provide an extension of the equation to include internal degrees of freedom.
Covariant field equations in supergravity
Vanhecke, Bram [KU Leuven, Institute for Theoretical Physics, Leuven (Belgium); Ghent University, Faculty of Physics, Gent (Belgium); Proeyen, Antoine van [KU Leuven, Institute for Theoretical Physics, Leuven (Belgium)
2017-12-15
Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Covariant field equations in supergravity
Vanhecke, Bram; Proeyen, Antoine van
2017-01-01
Covariance is a useful property for handling supergravity theories. In this paper, we prove a covariance property of supergravity field equations: under reasonable conditions, field equations of supergravity are covariant modulo other field equations. We prove that for any supergravity there exist such covariant equations of motion, other than the regular equations of motion, that are equivalent to the latter. The relations that we find between field equations and their covariant form can be used to obtain multiplets of field equations. In practice, the covariant field equations are easily found by simply covariantizing the ordinary field equations. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Differential Equation over Banach Algebra
Kleyn, Aleks
2018-01-01
In the book, I considered differential equations of order $1$ over Banach $D$-algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. In noncommutative Banach algebra, initial value problem for linear homogeneous equation has infinitely many solutions.
Transport equation solving methods
Granjean, P.M.
1984-06-01
This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method [fr
Introduction to partial differential equations
Greenspan, Donald
2000-01-01
Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.
Quadratic Diophantine equations
Andreescu, Titu
2015-01-01
This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems, and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory. The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis.
Stochastic porous media equations
Barbu, Viorel; Röckner, Michael
2016-01-01
Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Boussinesq evolution equations
Bredmose, Henrik; Schaffer, H.; Madsen, Per A.
2004-01-01
This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic...... Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over...... a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave...
Equations of mathematical physics
Tikhonov, A N
2011-01-01
Mathematical physics plays an important role in the study of many physical processes - hydrodynamics, elasticity, and electrodynamics, to name just a few. Because of the enormous range and variety of problems dealt with by mathematical physics, this thorough advanced-undergraduate or graduate-level text considers only those problems leading to partial differential equations. The authors - two well-known Russian mathematicians - have focused on typical physical processes and the principal types of equations deailing with them. Special attention is paid throughout to mathematical formulation, ri
Iteration of adjoint equations
Lewins, J.D.
1994-01-01
Adjoint functions are the basis of variational methods and now widely used for perturbation theory and its extension to higher order theory as used, for example, in modelling fuel burnup and optimization. In such models, the adjoint equation is to be solved in a critical system with an adjoint source distribution that is not zero but has special properties related to ratios of interest in critical systems. Consequently the methods of solving equations by iteration and accumulation are reviewed to show how conventional methods may be utilized in these circumstances with adequate accuracy. (author). 3 refs., 6 figs., 3 tabs
Systematic Equation Formulation
Lindberg, Erik
2007-01-01
A tutorial giving a very simple introduction to the set-up of the equations used as a model for an electrical/electronic circuit. The aim is to find a method which is as simple and general as possible with respect to implementation in a computer program. The “Modified Nodal Approach”, MNA, and th......, and the “Controlled Source Approach”, CSA, for systematic equation formulation are investigated. It is suggested that the kernel of the P Spice program based on MNA is reprogrammed....
Partial differential equations
Agranovich, M S
2002-01-01
Mark Vishik's Partial Differential Equations seminar held at Moscow State University was one of the world's leading seminars in PDEs for over 40 years. This book celebrates Vishik's eightieth birthday. It comprises new results and survey papers written by many renowned specialists who actively participated over the years in Vishik's seminars. Contributions include original developments and methods in PDEs and related fields, such as mathematical physics, tomography, and symplectic geometry. Papers discuss linear and nonlinear equations, particularly linear elliptic problems in angles and gener
Generalized estimating equations
Hardin, James W
2002-01-01
Although powerful and flexible, the method of generalized linear models (GLM) is limited in its ability to accurately deal with longitudinal and clustered data. Developed specifically to accommodate these data types, the method of Generalized Estimating Equations (GEE) extends the GLM algorithm to accommodate the correlated data encountered in health research, social science, biology, and other related fields.Generalized Estimating Equations provides the first complete treatment of GEE methodology in all of its variations. After introducing the subject and reviewing GLM, the authors examine th
Li, Tatsien
2017-01-01
This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.
High energy asymptotics of perturbative multi-color QCD
Lipatov, L.N.
1993-01-01
The structure functions of deep-inelastic scattering at small-x satisfy two different equations in the leading logarithmic approximation (LLA). The first one -- the GLAP equation, describes the Q 2 -evolution of partonic distributions h i (x). The second one -- the BFKL, equation determines the x-dependence of parton densities H i (x, k perpendicular ). Analogous equations for matrix elements of higher twist operators were constructed in Refs. 3 and 4. Here the author discusses the possibility of finding an exact solution for multi-gluon compound states in LLA for the color group SU(N), in the limit N → ∞. The contributions of diagrams with many reggeized gluons are important for the unitarization of the perturbative Pomeron in QCD. It is shown that the Bethe-Salpeter equations for compound states of many reggeized gluons are conformally invariant in the two-dimensional impact parameter space. Their solutions can be written in holomorphically factorized form and there is a differential operator commuting with the holomorphic part of the corresponding Hamiltonian
Analysis of wave equation in electromagnetic field by Proca equation
Pamungkas, Oky Rio; Soeparmi; Cari
2017-01-01
This research is aimed to analyze wave equation for the electric and magnetic field, vector and scalar potential, and continuity equation using Proca equation. Then, also analyze comparison of the solution on Maxwell and Proca equation for scalar potential and electric field, both as a function of distance and constant wave number. (paper)
Comparison of Kernel Equating and Item Response Theory Equating Methods
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
Test equating methods and practices
Kolen, Michael J
1995-01-01
In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...
The Raychaudhuri equation is central to the understanding of gravitational attraction in ... of K Gödel on the ideas of shear and vorticity in cosmology (he defines the shear. (eq. (8) in [1]) .... which follows from the definition of the scale factor l.
Generalized reduced magnetohydrodynamic equations
Kruger, S.E.
1999-01-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics
Calculus & ordinary differential equations
Pearson, David
1995-01-01
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
research, teaching and practice related to the analysis and design ... its variants, are present in a large number of ma- chines used in daily ... with advanced electronics, sensors, control systems and computing ... ted perfectly well with the rapidly developing comput- .... velopment of the Freudenstein equation using Figure 3.
Differential Equation of Equilibrium
user
ABSTRACT. Analysis of underground circular cylindrical shell is carried out in this work. The forth order differential equation of equilibrium, comparable to that of beam on elastic foundation, was derived from static principles on the assumptions of P. L Pasternak. Laplace transformation was used to solve the governing ...
Equational binary decision diagrams
J.F. Groote (Jan Friso); J.C. van de Pol (Jaco)
2000-01-01
textabstractWe incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and
Hatem Mejjaoli
2008-12-01
Full Text Available We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.
Structural Equation Model Trees
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
ANTHROPOMETRIC PREDICTIVE EQUATIONS FOR ...
Keywords: Anthropometry, Predictive Equations, Percentage Body Fat, Nigerian Women, Bioelectric Impedance ... such as Asians and Indians (Pranav et al., 2009), ... size (n) of at least 3o is adjudged as sufficient for the ..... of people, gender and age (Vogel eta/., 1984). .... Fish Sold at Ile-Ife Main Market, South West Nigeria.
However, one can associate the term with any solution of nonlinear partial differential equations (PDEs) which (i) represents a wave of permanent form, (ii) is localized ... In the past several decades, many methods have been proposed for solving nonlinear PDEs, such as ... space–time fractional derivative form of eq. (1) and ...
Fay, Temple H.
2010-01-01
Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…
Guiding center drift equations
Boozer, A.H.
1979-03-01
The quations for particle guiding center drift orbits are given in a new magnetic coordinate system. This form of the equations not only separates the fast motion along the lines from the slow motion across, but also requires less information about the magnetic field than many other formulations of the problem
dimensional nonlinear evolution equations
in real-life situations, it is important to find their exact solutions. Further, in ... But only little work is done on the high-dimensional equations. .... Similarly, to determine the values of d and q, we balance the linear term of the lowest order in eq.
Stochastic nonlinear beam equations
Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan
2005-01-01
Roč. 132, č. 1 (2005), s. 119-149 ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005
Savoy, L. G.
1988-01-01
Describes a study of students' ability to balance equations. Answers to a test on this topic were analyzed to determine the level of understanding and processes used by the students. Presented is a method to teach this skill to high school chemistry students. (CW)
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Quantum equations from Brownian motions
Rajput, B.S.
2011-01-01
Classical Schrodinger and Dirac equations have been derived from Brownian motions of a particle, it has been shown that the classical Schrodinger equation can be transformed to usual Schrodinger Quantum equation on applying Heisenberg uncertainty principle between position and momentum while Dirac Quantum equation follows it's classical counter part on applying Heisenberg uncertainly principle between energy and time without applying any analytical continuation. (author)
Elements of partial differential equations
Sneddon, Ian Naismith
1957-01-01
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st
On generalized fractional vibration equation
Dai, Hongzhe; Zheng, Zhibao; Wang, Wei
2017-01-01
Highlights: • The paper presents a generalized fractional vibration equation for arbitrary viscoelastically damped system. • Some classical vibration equations can be derived from the developed equation. • The analytic solution of developed equation is derived under some special cases. • The generalized equation is particularly useful for developing new fractional equivalent linearization method. - Abstract: In this paper, a generalized fractional vibration equation with multi-terms of fractional dissipation is developed to describe the dynamical response of an arbitrary viscoelastically damped system. It is shown that many classical equations of motion, e.g., the Bagley–Torvik equation, can be derived from the developed equation. The Laplace transform is utilized to solve the generalized equation and the analytic solution under some special cases is derived. Example demonstrates the generalized transfer function of an arbitrary viscoelastic system.
Covariant density functional theory for nuclear matter
Badarch, U.
2007-07-01
The present thesis is organized as follows. In Chapter 2 we study the Nucleon-Nucleon (NN) interaction in Dirac-Brueckner (DB) approach. We start by considering the NN interaction in free-space in terms of the Bethe-Salpeter (BS) equation to the meson exchange potential model. Then we present the DB approach for nuclear matter by extending the BS equation for the in-medium NN interaction. From the solution of the three-dimensional in-medium BS equation, we derive the DB self-energies and total binding energy which are the main results of the DB approach, which we later incorporate in the field theoretical calculation of the nuclear equation of state. In Chapter 3, we introduce the basic concepts of density functional theory in the context of Quantum Hadrodynamics (QHD-I). We reach the main point of this work in Chapter 4 where we introduce the DDRH approach. In the DDRH theory, the medium dependence of the meson-nucleon vertices is expressed as functionals of the baryon field operators. Because of the complexities of the operator-valued functionals we decide to use the mean-field approximation. In Chapter 5, we contrast microscopic and phenomenological approaches to extracting density dependent meson-baryon vertices. Chapter 6 gives the results of our studies of the EOS of infinite nuclear matter in detail. Using formulas derived in Chapters 4 and 5 we calculate the properties of symmetric and asymmetric nuclear matter and pure neutron matter. (orig.)
Covariant density functional theory for nuclear matter
Badarch, U.
2007-01-01
The present thesis is organized as follows. In Chapter 2 we study the Nucleon-Nucleon (NN) interaction in Dirac-Brueckner (DB) approach. We start by considering the NN interaction in free-space in terms of the Bethe-Salpeter (BS) equation to the meson exchange potential model. Then we present the DB approach for nuclear matter by extending the BS equation for the in-medium NN interaction. From the solution of the three-dimensional in-medium BS equation, we derive the DB self-energies and total binding energy which are the main results of the DB approach, which we later incorporate in the field theoretical calculation of the nuclear equation of state. In Chapter 3, we introduce the basic concepts of density functional theory in the context of Quantum Hadrodynamics (QHD-I). We reach the main point of this work in Chapter 4 where we introduce the DDRH approach. In the DDRH theory, the medium dependence of the meson-nucleon vertices is expressed as functionals of the baryon field operators. Because of the complexities of the operator-valued functionals we decide to use the mean-field approximation. In Chapter 5, we contrast microscopic and phenomenological approaches to extracting density dependent meson-baryon vertices. Chapter 6 gives the results of our studies of the EOS of infinite nuclear matter in detail. Using formulas derived in Chapters 4 and 5 we calculate the properties of symmetric and asymmetric nuclear matter and pure neutron matter. (orig.)
Methods for Equating Mental Tests.
1984-11-01
1983) compared conventional and IRT methods for equating the Test of English as a Foreign Language ( TOEFL ) after chaining. Three conventional and...three IRT equating methods were examined in this study; two sections of TOEFL were each (separately) equated. The IRT methods included the following: (a...group. A separate base form was established for each of the six equating methods. Instead of equating the base-form TOEFL to itself, the last (eighth
equateIRT: An R Package for IRT Test Equating
Michela Battauz
2015-12-01
Full Text Available The R package equateIRT implements item response theory (IRT methods for equating different forms composed of dichotomous items. In particular, the IRT models included are the three-parameter logistic model, the two-parameter logistic model, the one-parameter logistic model and the Rasch model. Forms can be equated when they present common items (direct equating or when they can be linked through a chain of forms that present common items in pairs (indirect or chain equating. When two forms can be equated through different paths, a single conversion can be obtained by averaging the equating coefficients. The package calculates direct and chain equating coefficients. The averaging of direct and chain coefficients that link the same two forms is performed through the bisector method. Furthermore, the package provides analytic standard errors of direct, chain and average equating coefficients.
Solórzano, S.; Mendoza, M.; Succi, S.; Herrmann, H. J.
2018-01-01
We present a numerical scheme to solve the Wigner equation, based on a lattice discretization of momentum space. The moments of the Wigner function are recovered exactly, up to the desired order given by the number of discrete momenta retained in the discretization, which also determines the accuracy of the method. The Wigner equation is equipped with an additional collision operator, designed in such a way as to ensure numerical stability without affecting the evolution of the relevant moments of the Wigner function. The lattice Wigner scheme is validated for the case of quantum harmonic and anharmonic potentials, showing good agreement with theoretical results. It is further applied to the study of the transport properties of one- and two-dimensional open quantum systems with potential barriers. Finally, the computational viability of the scheme for the case of three-dimensional open systems is also illustrated.
Dyre, Jeppe
1995-01-01
energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk modelthe energy master equation...... (EME)is arrived at. The EME is one dimensional and involves only energy; it is derived by arguing that percolation dominates the relaxational properties of the random-walk model at low temperatures. The approximate EME description of the random-walk model is expected to be valid at low temperatures...... of the random-walk model. The EME allows a calculation of the energy probability distribution at realistic laboratory time scales for an arbitrarily varying temperature as function of time. The EME is probably the only realistic equation available today with this property that is also explicitly consistent...
Classical Diophantine equations
1993-01-01
The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, ...
Flavored quantum Boltzmann equations
Cirigliano, Vincenzo; Lee, Christopher; Ramsey-Musolf, Michael J.; Tulin, Sean
2010-01-01
We derive from first principles, using nonequilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe. Working to leading nontrivial order in ratios of relevant time scales, we study in detail a toy model for weak-scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath. This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a nontrivial background. We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes.
Causal electromagnetic interaction equations
Zinoviev, Yury M.
2011-01-01
For the electromagnetic interaction of two particles the relativistic causal quantum mechanics equations are proposed. These equations are solved for the case when the second particle moves freely. The initial wave functions are supposed to be smooth and rapidly decreasing at the infinity. This condition is important for the convergence of the integrals similar to the integrals of quantum electrodynamics. We also consider the singular initial wave functions in the particular case when the second particle mass is equal to zero. The discrete energy spectrum of the first particle wave function is defined by the initial wave function of the free-moving second particle. Choosing the initial wave functions of the free-moving second particle it is possible to obtain a practically arbitrary discrete energy spectrum.
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Equations of multiparticle dynamics
Chao, A.W.
1987-01-01
The description of the motion of charged-particle beams in an accelerator proceeds in steps of increasing complexity. The first step is to consider a single-particle picture in which the beam is represented as a collection on non-interacting test particles moving in a prescribed external electromagnetic field. Knowing the external field, it is then possible to calculate the beam motion to a high accuracy. The real beam consists of a large number of particles, typically 10 11 per beam bunch. It is sometimes inconvenient, or even impossible, to treat the real beam behavior using the single particle approach. One way to approach this problem is to supplement the single particle by another qualitatively different picture. The commonly used tools in accelerator physics for this purpose are the Vlasov and the Fokker-Planck equations. These equations assume smooth beam distributions and are therefore strictly valid in the limit of infinite number of micro-particles, each carrying an infinitesimal charge. The hope is that by studying the two extremes -- the single particle picture and the picture of smooth beam distributions -- we will be able to describe the behavior of our 10 11 -particle system. As mentioned, the most notable use of the smooth distribution picture is the study of collective beam instabilities. However, the purpose of this lecture is not to address this more advanced subject. Rather, it has the limited goal to familiarize the reader with the analytical tools, namely the Vlasov and the Fokker-Planck equations, as a preparation for dealing with the more advanced problems at later times. We will first derive these equations and then illustrate their applications by several examples which allow exact solutions
Electroweak evolution equations
Ciafaloni, Paolo; Comelli, Denis
2005-01-01
Enlarging a previous analysis, where only fermions and transverse gauge bosons were taken into account, we write down infrared-collinear evolution equations for the Standard Model of electroweak interactions computing the full set of splitting functions. Due to the presence of double logs which are characteristic of electroweak interactions (Bloch-Nordsieck violation), new infrared singular splitting functions have to be introduced. We also include corrections related to the third generation Yukawa couplings
Differential equations with Mathematica
Abell, Martha L
2004-01-01
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica
Damped nonlinear Schrodinger equation
Nicholson, D.R.; Goldman, M.V.
1976-01-01
High frequency electrostatic plasma oscillations described by the nonlinear Schrodinger equation in the presence of damping, collisional or Landau, are considered. At early times, Landau damping of an initial soliton profile results in a broader, but smaller amplitude soliton, while collisional damping reduces the soliton size everywhere; soliton speeds at early times are unchanged by either kind of damping. For collisional damping, soliton speeds are unchanged for all time
Fun with Differential Equations
IAS Admin
tion of ® with ¼=2. One can use the uniqueness of solutions of differential equations to prove the addition formulae for sin(t1 +t2), etc. But instead of continuing with this thought process, let us do something more interesting. Now we shall consider another system. Fix 0 < < 1. I am looking for three real-valued functions x(t), ...
Mathematics and Maxwell's equations
Boozer, Allen H
2010-01-01
The universality of mathematics and Maxwell's equations is not shared by specific plasma models. Computations become more reliable, efficient and transparent if specific plasma models are used to obtain only the information that would otherwise be missing. Constraints of high universality, such as those from mathematics and Maxwell's equations, can be obscured or lost by integrated computations. Recognition of subtle constraints of high universality is important for (1) focusing the design of control systems for magnetic field errors in tokamaks from perturbations that have little effect on the plasma to those that do, (2) clarifying the limits of applicability to astrophysics of computations of magnetic reconnection in fields that have a double periodicity or have B-vector =0 on a surface, as in a Harris sheet. Both require a degree of symmetry not expected in natural systems. Mathematics and Maxwell's equations imply that neighboring magnetic field lines characteristically separate exponentially with distance along a line. This remarkably universal phenomenon has been largely ignored, though it defines a trigger for reconnection through a critical magnitude of exponentiation. These and other examples of the importance of making distinctions and understanding constraints of high universality are explained.
M. Paul Gough
2008-07-01
Full Text Available LandauerÃ¢Â€Â™s principle is applied to information in the universe. Once stars began forming there was a constant information energy density as the increasing proportion of matter at high stellar temperatures exactly compensated for the expanding universe. The information equation of state was close to the dark energy value, w = -1, for a wide range of redshifts, 10 > z > 0.8, over one half of cosmic time. A reasonable universe information bit content of only 1087 bits is sufficient for information energy to account for all dark energy. A time varying equation of state with a direct link between dark energy and matter, and linked to star formation in particular, is clearly relevant to the cosmic coincidence problem. In answering the Ã¢Â€Â˜Why now?Ã¢Â€Â™ question we wonder Ã¢Â€Â˜What next?Ã¢Â€Â™ as we expect the information equation of state to tend towards w = 0 in the future.c
Generalized reduced MHD equations
Kruger, S.E.; Hegna, C.C.; Callen, J.D.
1998-07-01
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson
Computing generalized Langevin equations and generalized Fokker-Planck equations.
Darve, Eric; Solomon, Jose; Kia, Amirali
2009-07-07
The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.
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Reduction operators of Burgers equation.
Pocheketa, Oleksandr A; Popovych, Roman O
2013-02-01
The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special "no-go" case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf-Cole transformation to a parameterized family of Lie reductions of the linear heat equation.
Multistrange Meson-Baryon Dynamics and Resonance Generation
Khemchandani, K. P.; Martínez Torres, A.; Hosaka, A.; Nagahiro, H.; Navarra, F. S.; Nielsen, M.
2018-05-01
In this talk I review our recent studies on meson-baryon systems with strangeness - 1 and - 2. The motivation of our works is to find resonances generated as a consequence of coupled channel meson-baryon interactions. The coupled channels are all meson-baryon systems formed by combining a pseudoscalar or a vector meson with an octet baryon such that the system has the strange quantum number equal to - 1 or - 2. The lowest order meson-baryon interaction amplitudes are obtained from Lagrangians based on the chiral and the hidden local symmetries related to the vector mesons working as the gauge bosons. These lowest order amplitudes are used as an input to solve the Bethe-Salpeter equation and a search for poles is made in the resulting amplitudes, in the complex plane. In case of systems with strangeness - 1, we find evidence for the existence of some hyperons such as: Λ(2000), Σ(1750), Σ(1940), Σ(2000). More recently, in the study of strangeness - 2 systems we have found two narrow resonances which can be related to Ξ (1690) and Ξ(2120). In this latter work, we have obtained the lowest order amplitudes relativistically as well as in the nonrelativistic approximation to solve the scattering equations. We find that the existence of the poles in the complex plane does not get affected by the computation of the scattering equation with the lowest order amplitudes obtained in the nonrelativistic approximation.
Relationship of field-theory based single-boson-exchange potentials to static ones
Amghar, A.; Desplanques, B.
2000-01-01
It is shown that field-theory based single-boson-exchange potentials cannot be identified to those of the Yukawa or Coulomb type that are currently inserted in the Schroedinger equation. The potential which is obtained rather correspond to this current single-boson-exchange potential corrected for the probability that the system under consideration is in a two-body component, therefore missing contributions due to the interaction of these two bodies while bosons are exchanged. The role of these contributions, which involve at least two-boson exchanges, is examined. The conditions that allow one to recover the usual single-boson-exchange potential are given. It is shown that the present results have some relation: (i) to the failure of the Bethe-Salpeter equation in reproducing the Dirac or Klein-Gordon equations in the limit where one of the constituents has a large mass, (ii) to the absence of corrections of relative order α log 1/α to a full calculation of the binding energy in the case of neutral massless bosons or (iii) to large corrections of wave-functions calculated perturbatively in some light-front approaches. Refs. 48 (author)
Color-suppression of non-planar diagrams in bosonic bound states
Alvarenga Nogueira, J. H.; Ji, Chueng-Ryong; Ydrefors, E.; Frederico, T.
2018-02-01
We study the suppression of non-planar diagrams in a scalar QCD model of a meson system in 3 + 1 space-time dimensions due to the inclusion of the color degrees of freedom. As a prototype of the color-singlet meson, we consider a flavor-nonsinglet system consisting of a scalar-quark and a scalar-antiquark with equal masses exchanging a scalar-gluon of a different mass, which is investigated within the framework of the homogeneous Bethe-Salpeter equation. The equation is solved by using the Nakanishi representation for the manifestly covariant bound-state amplitude and its light-front projection. The resulting non-singular integral equation is solved numerically. The damping of the impact of the cross-ladder kernel on the binding energies are studied in detail. The color-suppression of the cross-ladder effects on the light-front wave function and the elastic electromagnetic form factor are also discussed. As our results show, the suppression appears significantly large for Nc = 3, which supports the use of rainbow-ladder truncations in practical non-perturbative calculations within QCD.
Auxiliary equation method for solving nonlinear partial differential equations
Sirendaoreji,; Jiong, Sun
2003-01-01
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation
Evaluating Equating Results: Percent Relative Error for Chained Kernel Equating
Jiang, Yanlin; von Davier, Alina A.; Chen, Haiwen
2012-01-01
This article presents a method for evaluating equating results. Within the kernel equating framework, the percent relative error (PRE) for chained equipercentile equating was computed under the nonequivalent groups with anchor test (NEAT) design. The method was applied to two data sets to obtain the PRE, which can be used to measure equating…
Differential Equations as Actions
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....
Partial differential equations
Levine, Harold
1997-01-01
The subject matter, partial differential equations (PDEs), has a long history (dating from the 18th century) and an active contemporary phase. An early phase (with a separate focus on taut string vibrations and heat flow through solid bodies) stimulated developments of great importance for mathematical analysis, such as a wider concept of functions and integration and the existence of trigonometric or Fourier series representations. The direct relevance of PDEs to all manner of mathematical, physical and technical problems continues. This book presents a reasonably broad introductory account of the subject, with due regard for analytical detail, applications and historical matters.
Ordinary differential equations
Cox, William
1995-01-01
Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further
Partial differential equations
Sloan, D; Süli, E
2001-01-01
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight in
Elliptic partial differential equations
Han, Qing
2011-01-01
Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things abo
dimensional Jaulent–Miodek equations
(2+1)-dimensional Jaulent–Miodek equation; the first integral method; kinks; ... and effective method for solving nonlinear partial differential equations which can ... of the method employed and exact kink and soliton solutions are constructed ...
Equationally Noetherian property of Ershov algebras
Dvorzhetskiy, Yuriy
2014-01-01
This article is about equationally Noetherian and weak equationally Noetherian property of Ershov algebras. Here we show two canonical forms of the system of equations over Ershov algebras and two criteria of equationally Noetherian and weak equationally Noetherian properties.
Thaller, B.
1992-01-01
This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics
Sydoriak, S.G.
1976-01-01
Although criteria for cryostatic stability of superconducting magnets cooled by pool boiling of liquid helium have been widely discussed the same cannot be said for magnets cooled by natural convection or forced flow boiling in channels. Boiling in narrow channels is shown to be qualitatively superior to pool boiling because the recovery heat flux equals the breakaway flux for narrow channels, whereas the two are markedly different in pool boiling. A second advantage of channel boiling is that it is well understood and calculable; pool peak nucleate boiling heat flux has been adequately measured only for boiling from the top of an immersed heated body. Peak boiling from the bottom is much less and (probably) depends strongly on the extent of the bottom surface. Equations are presented by which one can calculate the critical boiling heat flux for parallel wall vertical channels subject to either natural convection or forced flow boiling, with one or both walls heated. The one-heated-wall forced flow equation is discussed with regard to design of a spiral wound solenoid (pancake magnet) having a slippery insulating tape between the windings
Solving Nonlinear Coupled Differential Equations
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
Completely integrable operator evolutionary equations
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.)
Kalinowski, M.W.; Szymanowski, L.
1982-03-01
A generalization of the Truesdell F-equations is proposed and some solutions to them - generalized Fox F-functions - are found. It is also shown that a non-linear difference-differential equation, which does not belong to the Truesdell class, nevertheless may be transformed into the standard F-equation. (author)
On the Saha Ionization Equation
Abstract. We revisit the Saha Ionization Equation in order to highlightthe rich interdisciplinary content of the equation thatstraddles distinct areas of spectroscopy, thermodynamics andchemical reactions. In a self-contained discussion, relegatedto an appendix, we delve further into the hidden message ofthe equation in terms ...
Differential equations extended to superspace
Torres, J. [Instituto de Fisica, Universidad de Guanajuato, A.P. E-143, Leon, Guanajuato (Mexico); Rosu, H.C. [Instituto Potosino de Investigacion Cientifica y Tecnologica, A.P. 3-74, Tangamanga, San Luis Potosi (Mexico)
2003-07-01
We present a simple SUSY Ns = 2 superspace extension of the differential equations in which the sought solutions are considered to be real superfields but maintaining the common derivative operators and the coefficients of the differential equations unaltered. In this way, we get self consistent systems of coupled differential equations for the components of the superfield. This procedure is applied to the Riccati equation, for which we obtain in addition the system of coupled equations corresponding to the components of the general superfield solution. (Author)
Reduction of infinite dimensional equations
Zhongding Li
2006-02-01
Full Text Available In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.
Differential equations extended to superspace
Torres, J.; Rosu, H.C.
2003-01-01
We present a simple SUSY Ns = 2 superspace extension of the differential equations in which the sought solutions are considered to be real superfields but maintaining the common derivative operators and the coefficients of the differential equations unaltered. In this way, we get self consistent systems of coupled differential equations for the components of the superfield. This procedure is applied to the Riccati equation, for which we obtain in addition the system of coupled equations corresponding to the components of the general superfield solution. (Author)
Taouil Hajer
2012-08-01
Full Text Available This paper is devoted to the helices processes, i.e. the solutions H : ℝ × Ω → ℝd, (t, ω ↦ H(t, ω of the helix equation egin{eqnarray} H(0,o=0; quad H(s+t,o= H(s,Phi(t,o +H(t,oonumber end{eqnarray} H ( 0 ,ω = 0 ; H ( s + t,ω = H ( s, Φ ( t,ω + H ( t,ω where Φ : ℝ × Ω → Ω, (t, ω ↦ Φ(t, ω is a dynamical system on a measurable space (Ω, ℱ. More precisely, we investigate dominated solutions and non differentiable solutions of the helix equation. For the last case, the Wiener helix plays a fundamental role. Moreover, some relations with the cocycle equation defined by Φ, are investigated. Ce papier est consacré aux hélices, c’est-à-dire les solutions H : ℝ × Ω → ℝd, (t, ω ↦ H(t, ω de l’équation fonctionnelle egin{eqnarray} H(0,o=0; quad H(s+t,o= H(s,Phi(t,o +H(t,o onumber end{eqnarray} H ( 0 ,ω = 0 ; H ( s + t,ω = H ( s, Φ ( t,ω + H ( t,ω où Φ : ℝ × Ω → Ω, (t, ω ↦ Φ(t, ω est un système dynamique défini sur un espace mesurable (Ω, ℱ. Plus présisément, nous déterminons d’abord les hélices dominées puis nous caractérisons les hélices non différentiables. Dans ce dernier cas, l’hélice de Wiener joue un rôle important. Nous précisons aussi quelques relations des hélices avec les cocycles définis par Φ.
Goldstone bosons in a crystalline chiral phase
Schramm, Marco
2017-07-24
The phase diagram of strong interaction matter is expected to exhibit a rich structure. Different models have shown, that crystalline phases with a spatially varying chiral condensate can occur in the regime of low temperatures and moderate densities, where they replace the first-order phase transition found for spatially constant order parameters. We investigate this inhomogeneous phase, where in addition to the chiral symmetry, translational and rotational symmetry are broken as well, in a two flavor Nambu--Jona-Lasinio model. The main goal of this work is to describe the Goldstone bosons in this phase, massless excitations that occur for spontaneously broken symmetries. We take one of the simplest possible modulations, the chiral density wave, and show how to derive the quark propagator of the theory analytically, by means of transformations in chiral and momentum space. We apply this to a test case for the gap equation. We show the derivation of Nambu-Goldstone modes in the inhomogeneous phase and find, that for our case only three different modes have to be taken into account. We proceed to calculate the Goldstone boson related to the breaking of spatial symmetry, which can be related to the neutral pion. By evaluating a Bethe-Salpeter equation, we can show, that we have indeed found a Goldstone boson and give its dispersion relation in terms of momenta perpendicular, as well as parallel to the mass modulation.
Goldstone bosons in a crystalline chiral phase
Schramm, Marco
2017-01-01
The phase diagram of strong interaction matter is expected to exhibit a rich structure. Different models have shown, that crystalline phases with a spatially varying chiral condensate can occur in the regime of low temperatures and moderate densities, where they replace the first-order phase transition found for spatially constant order parameters. We investigate this inhomogeneous phase, where in addition to the chiral symmetry, translational and rotational symmetry are broken as well, in a two flavor Nambu--Jona-Lasinio model. The main goal of this work is to describe the Goldstone bosons in this phase, massless excitations that occur for spontaneously broken symmetries. We take one of the simplest possible modulations, the chiral density wave, and show how to derive the quark propagator of the theory analytically, by means of transformations in chiral and momentum space. We apply this to a test case for the gap equation. We show the derivation of Nambu-Goldstone modes in the inhomogeneous phase and find, that for our case only three different modes have to be taken into account. We proceed to calculate the Goldstone boson related to the breaking of spatial symmetry, which can be related to the neutral pion. By evaluating a Bethe-Salpeter equation, we can show, that we have indeed found a Goldstone boson and give its dispersion relation in terms of momenta perpendicular, as well as parallel to the mass modulation.
Self-consistent model of confinement
Swift, A.R.
1988-01-01
A model of the large-spatial-distance, zero--three-momentum, limit of QCD is developed from the hypothesis that there is an infrared singularity. Single quarks and gluons do not propagate because they have infinite energy after renormalization. The Hamiltonian formulation of the path integral is used to quantize QCD with physical, nonpropagating fields. Perturbation theory in the infrared limit is simplified by the absence of self-energy insertions and by the suppression of large classes of diagrams due to vanishing propagators. Remaining terms in the perturbation series are resummed to produce a set of nonlinear, renormalizable integral equations which fix both the confining interaction and the physical propagators. Solutions demonstrate the self-consistency of the concepts of an infrared singularity and nonpropagating fields. The Wilson loop is calculated to provide a general proof of confinement. Bethe-Salpeter equations for quark-antiquark pairs and for two gluons have finite-energy solutions in the color-singlet channel. The choice of gauge is addressed in detail. Large classes of corrections to the model are discussed and shown to support self-consistency
t-Channel unitarity construction of small-x kernels
Coriano, C.; White, A.R.
1995-01-01
In the leading-log approximation, the small-x behavior of parton distributions in QCD is derived from the BFKL evolution equation. The authors describe the ion as a reggeon Bethe-Salpeter equation and discuss the use of reggeon diagrams to obtain 2-2 and 2-4 reggeon interactions at O(g 4 ). They then outline the dispersion theory basis of multiparticle j-plane analysis and describe how a gauge theory can be studied by combining Ward identity constraints with the group structure of reggeon interactions. Gluon reggeization, the O(g 2 ) BFKL kernel and O(g 4 ) corrections to it, are derived within this formalism. They give an explicit expression for the O(g 4 ) forward ''parton'' kernel in terms of logarithms and evaluate the eigenvalues. A separately infra-red finite component with a holomorphically factorizable spectrum is shown to be present and conjectured to be a new leading-order partial-wave amplitude. A comparison is made with Kirschner's discussion of O(g 4 ) contributions from the multi-Regge effective action
Electronic collective modes and instabilities on semiconductor surfaces. I
Muramatsu, A.; Hanke, W.
1984-01-01
A Green's-function theory of electronic collective modes is presented which leads to a practical scheme for a microscopic determination of surface elementary excitations in conducting as well as nonconducting solids. Particular emphasis is placed on semiconductor surfaces where the jellium approximation is not valid, due to the importance of density fluctuations on a microscopic scale (reflected in the local-field effects). Starting from the Bethe-Salpeter equation for the two-particle Green's function of the surface system, an equation of motion for the electron-hole pair is obtained. Its solutions determine the energy spectra, lifetimes, and amplitudes of the surface elementary excitations, i.e., surface plasmons, excitons, polaritons, and magnons. Exchange and correlation effects are taken into account through the random-phase and time-dependent Hartree-Fock (screened electron-hole attraction) approximations. The formalism is applied to the study of electronic (charge- and spin-density) instabilities at covalent semiconductor surfaces. Quantitative calculations for an eight-layer Si(111) slab display an instability of the ideal paramagnetic surface with respect to spin-density waves with wavelength nearly corresponding to (2 x 1) and (7 x 7) superstructures
Isgur-Wise function for Λb→Λc in the BS approach
Guo, X.; Muta, T.
1996-01-01
In the heavy quark limit, the heavy baryon Λ Q (Q=b or c) can be regarded as composed of a heavy quark and a scalar light diquark which has good spin and flavor quantum numbers. Based on this picture we establish the Bethe-Salpeter (BS) equation for Λ Q in the leading order of 1/m Q expansion. With the kernel containing both the scalar confinement and one-gluon-exchange terms we solve the BS equation numerically. The Isgur-Wise function for Λ b →Λ c is obtained numerically from our model. A comparison with other model calculations is also presented. It seems that the Isgur-Wise function for Λ b →Λ c drops faster than that for B→D. The differential and total decay widths for Λ b →Λ c l bar ν are given in the limit m b,c →∞. copyright 1996 The American Physical Society
Hyperon interactions in nuclear matter
Dhar, Madhumita; Lenske, Horst [Institut fuer Theoretische Physik, Universitaet Giessen (Germany)
2014-07-01
Baryon-baryon interactions within the SU(3)-octet are investigated in free space and nuclear matter. A meson exchange model is used for determining the interaction. The Bethe-Salpeter equations are solved in a 3-D reduction scheme. In-medium effects have been incorporated by including a two particle Pauli projection operator in the scattering equation. The coupling of the various channels of total strangeness S=-1,-2 and conserved total charge is studied in detail. Calculations and the corresponding results are compared for using the isospin and the particle basis. Matrix elements are compared in detail, in particular discussing mixing effects of different hyperon channels. Special attention is paid to the physical thresholds. The density dependence of interaction is clearly seen in the variation of the in-medium low-energy parameters. The approach is compared to descriptions derived from chiral-EFT and other meson-exchange models e.g. the Nijmegen and the Juelich model.
Kohn-Luttinger superconductivity in monolayer and bilayer semimetals with the Dirac spectrum
Kagan, M. Yu.; Mitskan, V. A.; Korovushkin, M. M.
2014-01-01
The effect of Coulomb interaction in an ensemble of Dirac fermions on the formation of superconducting pairing in monolayer and bilayer doped graphene is studied using the Kohn-Luttinger mechanism disregarding the Van der Waals potential of the substrate and impurities. The electronic structure of graphene is described using the Shubin-Vonsovsky model taking into account the intratomic, interatomic, and interlayer (in the case of bilayer graphene) Coulomb interactions between electrons. The Cooper instability is determined by solving the Bethe-Saltpeter integral equation. The renormalized scattering amplitude is obtained with allowance for the Kohn-Luttinger polarization contributions up to the second order of perturbation theory in the Coulomb interaction. It plays the role of effective interaction in the Bethe-Salpeter integral equation. It is shown that the allowance for the Kohn-Luttinger renormalizations as well as intersite Coulomb interaction noticeably affects the competition between the superconducting phases with the f-wave and d + id-wave symmetries of the order parameter. It is demonstrated that the superconducting transition temperature for an idealized graphene bilayer with significant interlayer Coulomb interaction between electrons is noticeably higher than in the monolayer case
A renormalizable extension of the NJL-model
Langfeld, K.; Kettner, C.; Reinhardt, H.
1996-01-01
The Nambu-Jona-Lasinio model is supplemented by the quark interaction generated by the one-gluon exchange. The employed gluon propagator exhibits the correct large-momentum behavior of QCD, whereas the Landau pole at low energies is screened. The emerging constituent quark model is one-loop renormalizable and interpolates between the phenomenologically successful Nambu-Jona-Lasinio model (modified by a transversal projector) at low energies and perturbative QCD at high momenta. Consequently, the momentum dependence of the quark self-energy at high energy coincides with the prediction from perturbative QCD. The chiral phase transition is studied in dependence on the low-energy four-quark interaction strength in the Dyson-Schwinger equation approach. The critical exponents of the quark self-energy and the quark condensate are obtained. The latter exponent deviates from the NJL-result. Pion properties are addressed by means of the Bethe-Salpeter equation. The validity of the Gell-Mann-Oakes-Renner relation is verified. Finally, we study the conditions under which the Nambu-Jona-Lasinio model is a decent approximation to our renormalizable theory as well as the shortcoming of the NJL-model due to its inherent non-renormalizability. (orig.)
Liang, Yufeng; Vinson, John; Pemmaraju, Sri; Drisdell, Walter S; Shirley, Eric L; Prendergast, David
2017-03-03
Constrained-occupancy delta-self-consistent-field (ΔSCF) methods and many-body perturbation theories (MBPT) are two strategies for obtaining electronic excitations from first principles. Using the two distinct approaches, we study the O 1s core excitations that have become increasingly important for characterizing transition-metal oxides and understanding strong electronic correlation. The ΔSCF approach, in its current single-particle form, systematically underestimates the pre-edge intensity for chosen oxides, despite its success in weakly correlated systems. By contrast, the Bethe-Salpeter equation within MBPT predicts much better line shapes. This motivates one to reexamine the many-electron dynamics of x-ray excitations. We find that the single-particle ΔSCF approach can be rectified by explicitly calculating many-electron transition amplitudes, producing x-ray spectra in excellent agreement with experiments. This study paves the way to accurately predict x-ray near-edge spectral fingerprints for physics and materials science beyond the Bethe-Salpether equation.
Nonperturbative aspects of the quark-photon vertex
Frank, M.R.
1994-01-01
The electromagnetic interaction with quarks is investigated through a relativistic, electromagnetic gauge-invariant treatment. Gluon dressing of the quark-photon vertex and the quark self-energy functions is described by the inhomogeneous Bethe-Salpeter equation in the ladder approximation and the Schwinger-Dyson equation in the rainbow approximation respectively. Results for the calculation of the quark-photon vertex are presented in both the time-like and space-like regions of photon momentum squared, however emphasis is placed on the space-like region relevant to electron scattering. The treatment presented here simultaneously addresses the role of dynamically generated q bar q vector bound states and the approach to asymptotic behavior. The resulting description is therefore applicable over the entire range of momentum transfers available in electron scattering experiments. Input parameters are limited to the model gluon two-point function which is chosen to reflect confinement and asymptotic freedom and are largely constrained by the obtained bound-state spectrum
On the glue content in heavy quarkonia
Gromes, D.
2003-01-01
Starting with two coupled Bethe-Salpeter equations for the quark-antiquark, and for the quark-glue-antiquark component of the quarkonium, we solve the bound state equations perturbatively. The resulting admixture of glue can be partially understood in a semiclassical way; one has, however, to take care of the different use of time ordered versus retarded Green functions. Subtle questions concerning the precise definition of the equal time wave function arise, because the wave function for the Coulomb gluon is discontinuous with respect to the relative time of the gluon. A striking feature is that a one loop non-abelian graph contributes to the same order as tree graphs, because the couplings of transverse gluons in the tree graphs are suppressed in the non-relativistic bound state, while the higher order loop graph can couple to quarks via non-suppressed Coulomb gluons. We also calculate the amplitude for quark and antiquark at zero distance in the quark-glue-antiquark component of the P-state. This quantity is of importance for annihilation decays of P-states. It shows a remarkable compensation between the tree graph and the non-abelian loop graph contribution. An extension of our results to include non-perturbative effects is possible. (orig.)
Large degeneracy of excited hadrons and quark models
Bicudo, P.
2007-01-01
The pattern of a large approximate degeneracy of the excited hadron spectra (larger than the chiral restoration degeneracy) is present in the recent experimental report of Bugg. Here we try to model this degeneracy with state of the art quark models. We review how the Coulomb Gauge chiral invariant and confining Bethe-Salpeter equation simplifies in the case of very excited quark-antiquark mesons, including angular or radial excitations, to a Salpeter equation with an ultrarelativistic kinetic energy with the spin-independent part of the potential. The resulting meson spectrum is solved, and the excited chiral restoration is recovered, for all mesons with J>0. Applying the ultrarelativistic simplification to a linear equal-time potential, linear Regge trajectories are obtained, for both angular and radial excitations. The spectrum is also compared with the semiclassical Bohr-Sommerfeld quantization relation. However, the excited angular and radial spectra do not coincide exactly. We then search, with the classical Bertrand theorem, for central potentials producing always classical closed orbits with the ultrarelativistic kinetic energy. We find that no such potential exists, and this implies that no exact larger degeneracy can be obtained in our equal-time framework, with a single principal quantum number comparable to the nonrelativistic Coulomb or harmonic oscillator potentials. Nevertheless we find it plausible that the large experimental approximate degeneracy will be modeled in the future by quark models beyond the present state of the art
p-Euler equations and p-Navier-Stokes equations
Li, Lei; Liu, Jian-Guo
2018-04-01
We propose in this work new systems of equations which we call p-Euler equations and p-Navier-Stokes equations. p-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-p distances, with incompressibility constraint. p-Euler equations have similar structures with the usual Euler equations but the 'momentum' is the signed (p - 1)-th power of the velocity. In the 2D case, the p-Euler equations have streamfunction-vorticity formulation, where the vorticity is given by the p-Laplacian of the streamfunction. By adding diffusion presented by γ-Laplacian of the velocity, we obtain what we call p-Navier-Stokes equations. If γ = p, the a priori energy estimates for the velocity and momentum have dual symmetries. Using these energy estimates and a time-shift estimate, we show the global existence of weak solutions for the p-Navier-Stokes equations in Rd for γ = p and p ≥ d ≥ 2 through a compactness criterion.
Generalized quantal equation of motion
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)
Alternatives to the Dirac equation
Girvin, S.M.; Brownstein, K.R.
1975-01-01
Recent work by Biedenharn, Han, and van Dam (BHvD) has questioned the uniqueness of the Dirac equation. BHvD have obtained a two-component equation as an alternate to the Dirac equation. Although they later show their alternative to be unitarily equivalent to the Dirac equation, certain physical differences were claimed. BHvD attribute the existence of this alternate equation to the fact that their factorizing matrices were position-dependent. To investigate this, we factor the Klein-Gordon equation in spherical coordinates allowing the factorizing matrices to depend arbitrarily upon theta and phi. It is shown that despite this additional freedom, and without involving any relativistic covariance, the conventional four-component Dirac equation is the only possibility
Wave Partial Differential Equation
Szöllös, Alexandr
2009-01-01
Práce se zabývá diferenciálními rovnicemi, jejich využitím při analýze vedení, experimenty s vedením a možnou akcelerací výpočtu v GPU s využitím prostředí nVidia CUDA. This work deals with diffrential equations, with the possibility of using them for analysis of the line and the possibility of accelerating the computations in GPU using nVidia CUDA. C
Gomez, Humberto
2016-06-01
The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter Λ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting Λ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the Λ algorithm.
Scaling of differential equations
Langtangen, Hans Petter
2016-01-01
The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and exam...
Parabolized stability equations
Herbert, Thorwald
1994-01-01
The parabolized stability equations (PSE) are a new approach to analyze the streamwise evolution of single or interacting Fourier modes in weakly nonparallel flows such as boundary layers. The concept rests on the decomposition of every mode into a slowly varying amplitude function and a wave function with slowly varying wave number. The neglect of the small second derivatives of the slowly varying functions with respect to the streamwise variable leads to an initial boundary-value problem that can be solved by numerical marching procedures. The PSE approach is valid in convectively unstable flows. The equations for a single mode are closely related to those of the traditional eigenvalue problems for linear stability analysis. However, the PSE approach does not exploit the homogeneity of the problem and, therefore, can be utilized to analyze forced modes and the nonlinear growth and interaction of an initial disturbance field. In contrast to the traditional patching of local solutions, the PSE provide the spatial evolution of modes with proper account for their history. The PSE approach allows studies of secondary instabilities without the constraints of the Floquet analysis and reproduces the established experimental, theoretical, and computational benchmark results on transition up to the breakdown stage. The method matches or exceeds the demonstrated capabilities of current spatial Navier-Stokes solvers at a small fraction of their computational cost. Recent applications include studies on localized or distributed receptivity and prediction of transition in model environments for realistic engineering problems. This report describes the basis, intricacies, and some applications of the PSE methodology.
Pomeau, Yves; Piasecki, Jarosław
2017-11-01
The existence of atoms has been long predicted by philosophers and scientists. The development of thermodynamics and of the statistical interpretation of its concepts at the end of the nineteenth century and in the early years of the twentieth century made it possible to bridge the gap of scales between the macroscopic world and the world of atoms. Einstein and Smoluchowski showed in 1905 and 1906 that the Brownian motion of particles of measurable size is a manifestation of the motion of atoms in fluids. Their derivation was completely different from each other. Langevin showed in 1908 how to put in a coherent framework the subtle effect of the randomness of the atomic world, responsible for the fluctuating force driving the motion of the Brownian particle and the viscosity of the "macroscopic" flow taking place around the same Brownian particle. Whereas viscous forces were already well understood at this time, the "Langevin" force appears there for the first time: it represents the fluctuating part of the interaction between the Brownian particle and the surrounding fluid. We discuss the derivation by Einstein and Smoluchowski as well as a previous paper by Sutherland on the diffusion coefficient of large spheres. Next we present Langevin's short note and explain the fundamental splitting into a random force and a macroscopic viscous force. This brings us to discuss various points, like the kind of constraints on Langevin-like equations. We insist in particular on the one arising from the time-reversal symmetry of the equilibrium fluctuations. Moreover, we discuss another constraint, raised first by Lorentz, which implies that, if the Brownian particle is not very heavy, the viscous force cannot be taken as the standard Stokes drag on an object moving at uniform speed. Lastly, we examine the so-called Langevin-Heisenberg and/or Langevin-Schrödinger equation used in quantum mechanics.
Introduction to partial differential equations
Borthwick, David
2016-01-01
This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise.Within each section the author creates a narrative that answers the five questions: (1) What is the scientific problem we are trying to understand? (2) How do we model that with PDE? (3) What techniques can we use to analyze the PDE? (4) How do those techniques apply to this equation? (5) What information or insight did we obtain by developing and analyzing the PDE? The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.
Analytic solutions of hydrodynamics equations
Coggeshall, S.V.
1991-01-01
Many similarity solutions have been found for the equations of one-dimensional (1-D) hydrodynamics. These special combinations of variables allow the partial differential equations to be reduced to ordinary differential equations, which must then be solved to determine the physical solutions. Usually, these reduced ordinary differential equations are solved numerically. In some cases it is possible to solve these reduced equations analytically to obtain explicit solutions. In this work a collection of analytic solutions of the 1-D hydrodynamics equations is presented. These can be used for a variety of purposes, including (i) numerical benchmark problems, (ii) as a basis for analytic models, and (iii) to provide insight into more complicated solutions
On matrix fractional differential equations
Adem Kılıçman
2017-01-01
Full Text Available The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objective of this article is to discuss the Laplace transform method based on operational matrices of fractional derivatives for solving several kinds of linear fractional differential equations. Moreover, we present the operational matrices of fractional derivatives with Laplace transform in many applications of various engineering systems as control system. We present the analytical technique for solving fractional-order, multi-term fractional differential equation. In other words, we propose an efficient algorithm for solving fractional matrix equation.
Differential equations methods and applications
Said-Houari, Belkacem
2015-01-01
This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .
Integral equations and their applications
Rahman, M
2007-01-01
For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of integral equations governing the physical situation of the problems. It also contains elegant analytical and numerical methods, and an important topic of the variational principles. Primarily intended for senior undergraduate students and first year postgraduate students of engineering and science courses, students of mathematical and physical sciences will also find many sections of direct relevance. The book contains eig...
Stochastic partial differential equations
Lototsky, Sergey V
2017-01-01
Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions. As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected ...
Menikoff, Ralph [Los Alamos National Laboratory
2015-12-15
The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.
Wetterich, C.
2018-06-01
We propose a closed gauge-invariant functional flow equation for Yang-Mills theories and quantum gravity that only involves one macroscopic gauge field or metric. It is based on a projection on physical and gauge fluctuations. Deriving this equation from a functional integral we employ the freedom in the precise choice of the macroscopic field and the effective average action in order to realize a closed and simple form of the flow equation.
The generalized Airy diffusion equation
Frank M. Cholewinski
2003-08-01
Full Text Available Solutions of a generalized Airy diffusion equation and an associated nonlinear partial differential equation are obtained. Trigonometric type functions are derived for a third order generalized radial Euler type operator. An associated complex variable theory and generalized Cauchy-Euler equations are obtained. Further, it is shown that the Airy expansions can be mapped onto the Bessel Calculus of Bochner, Cholewinski and Haimo.
Introduction to ordinary differential equations
Rabenstein, Albert L
1966-01-01
Introduction to Ordinary Differential Equations is a 12-chapter text that describes useful elementary methods of finding solutions using ordinary differential equations. This book starts with an introduction to the properties and complex variable of linear differential equations. Considerable chapters covered topics that are of particular interest in applications, including Laplace transforms, eigenvalue problems, special functions, Fourier series, and boundary-value problems of mathematical physics. Other chapters are devoted to some topics that are not directly concerned with finding solutio
On matrix fractional differential equations
Adem Kılıçman; Wasan Ajeel Ahmood
2017-01-01
The aim of this article is to study the matrix fractional differential equations and to find the exact solution for system of matrix fractional differential equations in terms of Riemann–Liouville using Laplace transform method and convolution product to the Riemann–Liouville fractional of matrices. Also, we show the theorem of non-homogeneous matrix fractional partial differential equation with some illustrative examples to demonstrate the effectiveness of the new methodology. The main objec...
Electronic representation of wave equation
Veigend, Petr; Kunovský, Jiří, E-mail: kunovsky@fit.vutbr.cz; Kocina, Filip; Nečasová, Gabriela; Valenta, Václav [University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic); Šátek, Václav [IT4Innovations, VŠB Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava-Poruba (Czech Republic); University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic)
2016-06-08
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Generalized Lorentz-Force equations
Yamaleev, R.M.
2001-01-01
Guided by Nambu (n+1)-dimensional phase space formalism we build a new system of dynamic equations. These equations describe a dynamic state of the corporeal system composed of n subsystems. The dynamic equations are formulated in terms of dynamic variables of the subsystems as well as in terms of dynamic variables of the corporeal system. These two sets of variables are related respectively as roots and coefficients of the n-degree polynomial equation. In the special n=2 case, this formalism reproduces relativistic dynamics for the charged spinning particles
The forced nonlinear Schroedinger equation
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.)
Correct Linearization of Einstein's Equations
Rabounski D.
2006-06-01
Full Text Available Regularly Einstein's equations can be reduced to a wave form (linearly dependent from the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel's symbols. As shown here, the origin of the problem is that one uses the general covariant theory of measurement. Here the wave form of Einstein's equations is obtained in the terms of Zelmanov's chronometric invariants (physically observable projections on the observer's time line and spatial section. The obtained equations depend on solely the second derivatives even if gravitation, the space rotation and Christoffel's symbols. The correct linearization proves: the Einstein equations are completely compatible with weak waves of the metric.
The Dirac equation for accountants
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
Difference equations theory, applications and advanced topics
Mickens, Ronald E
2015-01-01
THE DIFFERENCE CALCULUS GENESIS OF DIFFERENCE EQUATIONS DEFINITIONS DERIVATION OF DIFFERENCE EQUATIONS EXISTENCE AND UNIQUENESS THEOREM OPERATORS ∆ AND E ELEMENTARY DIFFERENCE OPERATORS FACTORIAL POLYNOMIALS OPERATOR ∆−1 AND THE SUM CALCULUS FIRST-ORDER DIFFERENCE EQUATIONS INTRODUCTION GENERAL LINEAR EQUATION CONTINUED FRACTIONS A GENERAL FIRST-ORDER EQUATION: GEOMETRICAL METHODS A GENERAL FIRST-ORDER EQUATION: EXPANSION TECHNIQUES LINEAR DIFFERENCE EQUATIONSINTRODUCTION LINEARLY INDEPENDENT FUNCTIONS FUNDAMENTAL THEOREMS FOR HOMOGENEOUS EQUATIONSINHOMOGENEOUS EQUATIONS SECOND-ORDER EQUATIONS STURM-LIOUVILLE DIFFERENCE EQUATIONS LINEAR DIFFERENCE EQUATIONS INTRODUCTION HOMOGENEOUS EQUATIONS CONSTRUCTION OF A DIFFERENCE EQUATION HAVING SPECIFIED SOLUTIONS RELATIONSHIP BETWEEN LINEAR DIFFERENCE AND DIFFERENTIAL EQUATIONS INHOMOGENEOUS EQUATIONS: METHOD OF UNDETERMINED COEFFICIENTS INHOMOGENEOUS EQUATIONS: OPERATOR METHODS z-TRANSFORM METHOD SYSTEMS OF DIFFERENCE EQUATIONS LINEAR PARTIAL DIFFERENCE EQUATI...
Differential equations a dynamical systems approach ordinary differential equations
Hubbard, John H
1991-01-01
This is a corrected third printing of the first part of the text Differential Equations: A Dynamical Systems Approach written by John Hubbard and Beverly West. The authors' main emphasis in this book is on ordinary differential equations. The book is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. Traditional courses on differential equations focus on techniques leading to solutions. Yet most differential equations do not admit solutions which can be written in elementary terms. The authors have taken the view that a differential equations defines functions; the object of the theory is to understand the behavior of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods. The companion software, MacMath, is designed to bring these notions to life.
Chiral symmetry breaking and the pion quark structure
Bernard, V.
1986-01-01
The mechanism of dynamical breaking of chiral symmetry in hadronic matter is first studied in the framework of the Nambu and Jona-Lasinio model on one hand and its generalisation to finite hadron size on the other hand. The analysis uses a variational procedure modelled after the BCS superconductor. Our study indicates for example, a great sensitivity of various quantities characterizing the breaking of symmetry to the shape of the interaction. Also the mechanism of breaking of chiral symmetry is essentially related to the mechanism of confinement. When a symmetry is spontaneously broken, there exists a Goldstone particle of zero mass. This is true in our model. This particle, the pion, is obtained as solution of a Bethe Salpeter equation for a qantiq bound state. This enables us to establish a connection between the pion as a Goldstone boson related to spontaneous symmetry breaking and the quark-antiquark structure of the pion. The finite mass of the physical pion is obtained with non zero current quark mass. Various properties of this particle are then studied in the RPA formalism. One important point of our model is the highly collective character of the pion. 85 refs [fr
Anomalous behavior of the excited state of the A exciton in bulk WS2
Jindal, Vishwas; Bhuyan, Sumi; Deilmann, Thorsten
2018-01-01
Results of optical spectroscopy studies on bulk 2H-WS2 at energies close to its direct band gap are presented. Reflectance and absorption measurements at low temperature show only one dominant feature due to the A exciton of bulk WS2 at similar to 2.02 eV. However, a laser-modulated photoreflecta......Results of optical spectroscopy studies on bulk 2H-WS2 at energies close to its direct band gap are presented. Reflectance and absorption measurements at low temperature show only one dominant feature due to the A exciton of bulk WS2 at similar to 2.02 eV. However, a laser....... The experimental results are analyzed by comparison with many-body perturbation theory calculations, including the solutions of the Bethe-Salpeter equation. A* is identified as the first excited state of the A exciton, that is, A(n = 2). The anomalous behavior of A* is explained by its distinct wave function...... spread along the c axis, the direction of weak van der Waals bonding, which makes it more susceptible to perturbations. Our ab initio calculations suggest that the A exciton in the ground state has a two-dimensional (2D) nature with a large binding energy E-b, in fair agreement with E-b similar to 90...
Strong excitonic interactions in the oxygen K-edge of perovskite oxides.
Tomita, Kota; Miyata, Tomohiro; Olovsson, Weine; Mizoguchi, Teruyasu
2017-07-01
Excitonic interactions of the oxygen K-edge electron energy-loss near-edge structure (ELNES) of perovskite oxides, CaTiO 3 , SrTiO 3 , and BaTiO 3 , together with reference oxides, MgO, CaO, SrO, BaO, and TiO 2 , were investigated using a first-principles Bethe-Salpeter equation calculation. Although the transition energy of oxygen K-edge is high, strong excitonic interactions were present in the oxygen K-edge ELNES of the perovskite oxides, whereas the excitonic interactions were negligible in the oxygen K-edge ELNES of the reference compounds. Detailed investigation of the electronic structure suggests that the strong excitonic interaction in the oxygen K-edge ELNES of the perovskite oxides is caused by the directionally confined, low-dimensional electronic structure at the Ti-O-Ti bonds. Copyright © 2016 Elsevier B.V. All rights reserved.
Electromagnetic properties of the pion as a composite Nambu-Goldstone boson
Ito, H.; Buck, W.W.; Gross, F.
1992-01-01
Motivated by the Nambu--Jona-Lasinio model of light mesons, we introduce a covariant separable interaction to model the structure of relativistic quark-antiquark systems. The Schwinger-Dyson equation for the quark self-energy is solved analytically, generating a dynamical quark mass through spontaneous breaking of chiral symmetry, and yielding a pion which has zero mass in the chiral limit. The Bethe-Salpeter vertex function for this q bar q pion, which has a momentum distribution and composite structure associated with the interaction, is obtained analytically. Using this vertex function, and a similar one for the ρ meson, we calculate the electromagnetic observables of this composite Nambu-Goldstone boson, including effects from ρ-meson dominance processes. Our calculation takes the composite structure of the mesons into account. The ρ-meson effects are found to be very small in the pion charge form factor, but substantial in the charge radius. Using the model, predictions are made for γ * π 0 →γ and ρπγ transition form factors
About the role of 2D screening in high temperature superconductivity
Vazquez-Ponce, Y.; Oliva Aguero, D.; Cabo Montes de Oca, D.
2005-09-01
The 2D screening is investigated in a simple single band square tight-binding model which qualitatively resembles the known electronic structure in high temperature superconductors. The Coulomb kernel for the two particle Bethe-Salpeter equation in the single loop (RPA) approximation for the polarization can be evaluated in a strong tight binding limit. The results indicate an intense screening of the Coulomb repulsion between the particles, which becomes stronger and anisotropic when the Fermi level approaches half filling (or equivalently, when the Fermi surface turns to be near the Van Hove singularities) and rapidly decreases away from it. The effect is also more pronounced for quasi-momenta regions near the corners of the Brillouin cell, which corresponds to dual spatial distances of the order of a few unit cells. Therefore, a possible mechanism is identified which could explain the existence of extremely small Cooper pairs in these materials, as bounded anisotropic composite particles joined by residual superexchange or phonon interactions. (author)
Quark model calculations of current correlators in the nonperturbative domain
Celenza, L.S.; Shakin, C.M.; Sun, W.D.
1995-01-01
The authors study the vector-isovector current correlator in this work, making use of a generalized Nambu-Jona-Lasinio (NJL) model. In their work, the original NJL model is extended to describe the coupling of the quark-antiquark states to the two-pion continuum. Further, a model for confinement is introduced that is seen to remove the nonphysical cuts that appear in various amplitudes when the quark and antiquark go on mass shell. Quite satisfactory results are obtained for the correlator. The authors also use the correlator to define a T-matrix for confined quarks and discuss a rho-dominance model for that T-matrix. It is also seen that the Bethe-Salpeter equation that determines the rho mass (in the absence of the coupling to the two-pion continuum) has more satisfactory behavior in the generalized model than in the model without confinement. That improved behavior is here related to the absence of the q bar q cut in the basic quark-loop integral of the generalized model. In this model, it is seen how one may work with both quark and hadron degrees of freedom, with only the hadrons appearing as physical particles. 12 refs., 16 figs., 1 tab
Structure of the vertex function in finite quantum electrodynamics
Mannheim, P.D.
1975-01-01
We study the structure of the renormalized electromagnetic current vertes, GAMMA-tilde/sub μ/(p,p+q,q), in finite quantum electrodynamics. Using conformal invariance we find that GAMMA-tilde/sub μ/(p,p,0) takes the simple form of Z 1 γ/sub μ/ when the external fermions are far off the mass shell. We interpret this result as an old theorem on the structure of the vertex function due to Gell--Mann and Zachariasen. We give the general structure of the vertex for arbitrary momentum transfer parametrically, and discuss how the Bethe--Salpeter equation and the Federbush--Johnson theorem are satisfied. We contrast the meaning of pointlike in a finite field theory with the meaning understood in the parton model. We discuss to what extent the condition Z 1 = 0, which may hold in conformal theories other than finite quantum electrodynamics, may be interpreted as a bootstrap condition. We show that the vanishing of Z 1 prevents their being bound states in the Migdal--Polyakov bootstrap
Baumeier, Björn; Andrienko, Denis; Rohlfing, Michael
2012-08-14
Excited states of donor-acceptor dimers are studied using many-body Green's functions theory within the GW approximation and the Bethe-Salpeter equation. For a series of prototypical small-molecule based pairs, this method predicts energies of local Frenkel and intermolecular charge-transfer excitations with the accuracy of tens of meV. Application to larger systems is possible and allowed us to analyze energy levels and binding energies of excitons in representative dimers of dicyanovinyl-substituted quarterthiophene and fullerene, a donor-acceptor pair used in state of the art organic solar cells. In these dimers, the transition from Frenkel to charge transfer excitons is endothermic and the binding energy of charge transfer excitons is still of the order of 1.5-2 eV. Hence, even such an accurate dimer-based description does not yield internal energetics favorable for the generation of free charges either by thermal energy or an external electric field. These results confirm that, for qualitative predictions of solar cell functionality, accounting for the explicit molecular environment is as important as the accurate knowledge of internal dimer energies.
Faria da Veiga, Paulo A.; O'Carroll, Michael
2006-01-01
We determine baryon-baryon bound states in (3+1)-dimensional SU(3) lattice QCD with two flavors, 4x4 spin matrices, and in an imaginary time formulation. For small hopping parameter κ>0 and large glueball mass (strong coupling), we show the existence of three-quark isospin 1/2 particles (proton and neutron) and isospin 3/2 baryons (delta particles), with asymptotic masses -3lnκ and isolated dispersion curves. Baryon-baryon bound states of isospin zero are found with binding energy of order κ 2 , using a ladder approximation to a lattice Bethe-Salpeter equation. The dominant baryon-baryon interaction is an energy-independent spatial range-one attractive potential with an O(κ 2 ) strength. There is also attraction arising from gauge field correlations associated with six overlapping bonds, but it is counterbalanced by Pauli repulsion to give a vanishing zero-range potential. The overall range-one potential results from a quark, antiquark exchange with no meson exchange interpretation; the repulsive or attractive nature of the interaction depends on the isospin and spin of the two-baryon state
Decreasing the electronic confinement in layered perovskites through intercalation.
Smith, Matthew D; Pedesseau, Laurent; Kepenekian, Mikaël; Smith, Ian C; Katan, Claudine; Even, Jacky; Karunadasa, Hemamala I
2017-03-01
We show that post-synthetic small-molecule intercalation can significantly reduce the electronic confinement of 2D hybrid perovskites. Using a combined experimental and theoretical approach, we explain structural, optical, and electronic effects of intercalating highly polarizable molecules in layered perovskites designed to stabilize the intercalants. Polarizable molecules in the organic layers substantially alter the optical and electronic properties of the inorganic layers. By calculating the spatially resolved dielectric profiles of the organic and inorganic layers within the hybrid structure, we show that the intercalants afford organic layers that are more polarizable than the inorganic layers. This strategy reduces the confinement of excitons generated in the inorganic layers and affords the lowest exciton binding energy for an n = 1 perovskite of which we are aware. We also demonstrate a method for computationally evaluating the exciton's binding energy by solving the Bethe-Salpeter equation for the exciton, which includes an ab initio determination of the material's dielectric profile across organic and inorganic layers. This new semi-empirical method goes beyond the imprecise phenomenological approximation of abrupt dielectric-constant changes at the organic-inorganic interfaces. This work shows that incorporation of polarizable molecules in the organic layers, through intercalation or covalent attachment, is a viable strategy for tuning 2D perovskites towards mimicking the reduced electronic confinement and isotropic light absorption of 3D perovskites while maintaining the greater synthetic tunability of the layered architecture.
Inquiry for the conversion of the (π+ - π-) bound state into two π0
Bunatyan, G.G.
1998-01-01
In the work presented, the decay of the pionium, that is the (π + π - ) bound state, into two π 0 is studied, the ππ-interaction causing this transition being described by the underlying Weinberg Lagrangian. The calculation with such a ππ-Lagrangian being carried out, the π-meson size r 0 emerges to be allowed for, and this quantity occurs in the final result. The bound (π + π - )-system itself is presumed to be due to the instantaneous Coulomb interaction and is treated consistently nonrelativistically, the Bethe-Salpeter equation being utilized. When calculating, the terms to the lowest order in the fine structure constant α and the terms ∼ ln (r 0 ) are retained. The obtained pionium lifetime τ is thought to be compatible with the conceivable future experimental data. The dependence of the results on the effective Lagrangian parameters is visualized. The investigation carried out persuades us that it is just the complete form of the genuine ππ-interaction that determines the pionium lifetime , but not much simply the ππ scattering lengths. The inquiry into pionium decaying promotes to specify the validity of the various ππ-interaction descriptions
He, Jun [Nanjing Normal University, Department of Physics and Institute of Theoretical Physics, Nanjing (China); Chen, Dian-Yong [Southeast University, School of Physics, Nanjing (China)
2017-06-15
Invoked by the recent observation of Y(4390) at BESIII, which is about 40 MeV below the D*(2010) anti D{sub 1}(2420) threshold, we investigate possible bound and resonance states from the D*(2010) anti D{sub 1}(2420) interaction with the one-boson-exchange model in a quasipotential Bethe-Salpeter equation approach. A bound state with quantum number 0{sup -}(1{sup --}) is produced at 4384 MeV from the D*(2010) anti D{sub 1}(2420) interaction, which can be related to experimentally observed Y(4390). Another state with quantum number 1{sup +}(1{sup +}) is also produced at 4461 + i39 MeV from this interaction. Different from the 0{sup -}(1{sup --}) state, the 1{sup +}(1{sup +}) state is a resonance state above the D*(2010) anti D{sub 1}(2420) threshold. This resonance state can be related to the first observed charged charmonium-like state Z(4430), which has a mass about 4475 MeV measured above the threshold as observed at Belle and LHCb. Our result suggests that Y(4390) is an isoscalar partner of the Z(4430) as a hadronic-molecular state from the D*(2010) anti D{sub 1}(2420) interaction. (orig.)
A hadron-quark vertex function
Mitra, A.N.; Bhatnagar, S.
1992-01-01
This paper reports that the interrelation between the 4D and 3D forms of the Bethe-Salpeter equation (BSE) with a kernel K(q,q') which depends on the relative four-momenta, q μ = q μ - P · qP μ /P 2 , orthogonal to P μ is exploited to obtain a hadron-quark vertex function of the Lorentz-invariant form Γ(q) = D(q 2 ) circle time φ(q). The denominator function D(q 2 ) is universal and controls the 3D BSE, which provides the mass spectra with the eigenfunctions φ(q). The vertex function, directly related to the 4D wave function Ψ which satisfies a corresponding BSE, defines a natural off-shell extension over the whole of four-momentum space, and provides the basis for the evaluation of transition amplitudes via appropriate quark-loop diagrams. The key role of the quantity q 2 in this formalism is clarified in relation to earlier approaches, in which the applications of this quantity had mostly been limited to the mass shell (q · P = 0). Two applications (f p values for P → ell bar ell and F π for π 0 → γγ) are sketched as illustrations of this formalism, and attention is drawn to the problem of complex amplitudes for bigger quark loops with more hadrons, together with the role of the D(q) function in overcoming this problem
Relativistic chiral SU(3) symmetry, large Nc sum rules and meson-baryon scattering
Lutz, M.F.M.; Kolomeitsev, E.E.
2001-05-01
The relativistic chiral SU(3) Lagrangian is used to describe kaon-nucleon scattering imposing constraints from the pion-nucleon sector and the axial-vector coupling constants of the baryon octet states. We solve the covariant coupled-channel Bethe-Salpeter equation with the interaction kernel truncated at chiral order Q 3 where we include only those terms which are leading in the large N c limit of QCD. The baryon decuplet states are an important explicit ingredient in our scheme, because together with the baryon octet states they form the large N c baryon ground states of QCD. Part of our technical developments is a minimal chiral subtraction scheme within dimensional regularization, which leads to a manifest realization of the covariant chiral counting rules. All SU(3) symmetry-breaking effects are well controlled by the combined chiral and large N c expansion, but still found to play a crucial role in understanding the empirical data. We achieve an excellent description of the data set typically up to laboratory momenta of p lab ≅ 500 MeV. (orig.)
Comparison of two Minkowski-space approaches to heavy quarkonia
Leitao, Sofia; Biernat, Elmar P. [Universidade de Lisboa, CFTP, Instituto Superior Tecnico, Lisbon (Portugal); Li, Yang [Iowa State University, Department of Physics and Astronomy, Ames, IA (United States); College of William and Mary, Department of Physics, Williamsburg, VA (United States); Maris, Pieter; Vary, James P. [Iowa State University, Department of Physics and Astronomy, Ames, IA (United States); Pena, M.T. [Universidade de Lisboa, CFTP, Instituto Superior Tecnico, Lisbon (Portugal); Universidade de Lisboa, Departamento de Fisica, Instituto Superior Tecnico, Lisbon (Portugal); Stadler, Alfred [Universidade de Lisboa, CFTP, Instituto Superior Tecnico, Lisbon (Portugal); Universidade de Evora, Departamento de Fisica, Evora (Portugal)
2017-10-15
In this work we compare mass spectra and decay constants obtained from two recent, independent, and fully relativistic approaches to the quarkonium bound-state problem: the Basis Light-Front Quantization approach, where light-front wave functions are naturally formulated; and, the Covariant Spectator Theory (CST), based on a reorganization of the Bethe-Salpeter equation. Even though conceptually different, both solutions are obtained in Minkowski space. Comparisons of decay constants for more than ten states of charmonium and bottomonium show favorable agreement between the two approaches as well as with experiment where available. We also apply the Brodsky-Huang-Lepage prescription to convert the CST amplitudes into functions of light-front variables. This provides an ideal opportunity to investigate the similarities and differences at the level of the wave functions. Several qualitative features are observed in remarkable agreement between the two approaches even for the rarely addressed excited states. Leading-twist distribution amplitudes as well as parton distribution functions of heavy quarkonia are also analyzed. (orig.)
Excitation spectrum and staggering transformations in lattice quantum models.
Faria da Veiga, Paulo A; O'Carroll, Michael; Schor, Ricardo
2002-08-01
We consider the energy-momentum excitation spectrum of diverse lattice Hamiltonian operators: the generator of the Markov semigroup of Ginzburg-Landau models with Langevin stochastic dynamics, the Hamiltonian of a scalar quantum field theory, and the Hamiltonian associated with the transfer matrix of a classical ferromagnetic spin system at high temperature. The low-lying spectrum consists of a one-particle state and a two-particle band. The two-particle spectrum is determined using a lattice version of the Bethe-Salpeter equation. In addition to the two-particle band, depending on the lattice dimension and on the attractive or repulsive character of the interaction between the particles of the system, there is, respectively, a bound state below or above the two-particle band. We show how the existence or nonexistence of these bound states can be understood in terms of a nonrelativistic single-particle lattice Schrödinger Hamiltonian with a delta potential. A staggering transformation relates the spectra of the attractive and the repulsive cases.
Optical absorption in disordered monolayer molybdenum disulfide
Ekuma, C. E.; Gunlycke, D.
2018-05-01
We explore the combined impact of sulfur vacancies and electronic interactions on the optical properties of monolayer MoS2. First, we present a generalized Anderson-Hubbard Hamiltonian that accounts for both randomly distributed sulfur vacancies and the presence of dielectric screening within the material. Second, we parametrize this energy-dependent Hamiltonian from first-principles calculations based on density functional theory and the Green's function and screened Coulomb (GW) method. Third, we apply a first-principles-based many-body typical medium method to determine the single-particle electronic structure. Fourth, we solve the Bethe-Salpeter equation to obtain the charge susceptibility χ with its imaginary part being related to the absorbance A . Our results show that an increased vacancy concentration leads to decreased absorption both in the band continuum and from exciton states within the band gap. We also observe increased absorption below the band-gap threshold and present an expression, which describes Lifshitz tails, in excellent qualitative agreement with our numerical calculations. This latter increased absorption in the 1.0 -2.5 eV range makes defect engineering of potential interest for solar cell applications.
Isostructural and heterostructural MgZnO and CdZnO alloys
Schleife, Andre; Roedl, Claudia; Bechstedt, Friedhelm [Institut fuer Festkoerpertheorie und -optik and European Theoretical Spectroscopy Facility, Friedrich-Schiller-Universitaet Jena, 07743 Jena (Germany)
2011-07-01
One important goal of materials design is to purposefully tailor the fundamental band gap. Recently, group-II oxides such as MgO, ZnO, and CdO are discussed to possibly supersede the nitrides for certain applications, e.g. in optoelectronics. Since the band gaps of the oxides cover a large spectral range from 1.8 eV (CdO) up to 7.7 eV (MgO), their alloys seem to be promising - unless the different equilibrium crystal structures of the end components (rocksalt-MgO and -CdO vs. wurtzite-ZnO) prevent their application. By employing three different cluster statistics within a cluster-expansion approach we investigate the impact of different growth conditions on the composition of isostructural and heterostructural Mg{sub x}Zn{sub 1-x}O and Cd{sub x}Zn{sub 1-x}O alloys. Our total-energy calculations are based on density-functional theory using a generalized-gradient approximation for exchange and correlation. We also compute quasiparticle energies using the HSE03+G{sub 0}W{sub 0} approach. This allows us to derive the bowings of fundamental band gaps, which reveal a strongly nonlinear behavior. Using the solution of the Bethe-Salpeter equation for the optical polarization function we investigate the influence of the preparation conditions on the peaks related to bound excitonic states at the absorption edge.
Nucleon scattering on one-hole nuclei in the framework of the continuum RPA
Abdi, S.M.
1979-01-01
It is shown that the scattering transition amplitude for nucleon-nucleus scattering on one-hole nuclei can be directly related to so-called linear response function. The theory of the linear response function is formulated in the framework of Greens functions. The linear response function obreys an equation of Bethe-Salpeter-type. From to the spectral decomposition of the response function one can deduce, that the knowledge of the response function in equivalent to the knowledge of the so-called 'particle-hole'- and 'hole-particle'-amplitudes, respectively, and the exitation energies of the compound A-particle system. In this work the binding- and resonance energies for 16 O and 16 N, respectively, have been calculated, which are needed for the scattering processes 15 N(p,n) 15 O and 15 N(n,n') 15 N, respectively. For low energies the agreement with experiment is not statisfactory, probably due to the coupling to 3p-3h-states, which was neglected in the treatment. (orig./HSI) [de
Strongly bound excitons in monolayer PtS2 and PtSe2
Sajjad, M.
2018-01-22
Based on first-principles calculations, the structural, electronic, and optical properties of monolayers PtS2 and PtSe2 are investigated. The bond stiffnesses and elastic moduli are determined by means of the spring constants and strain-energy relations, respectively. Dynamic stability is confirmed by calculating the phonon spectra, which shows excellent agreement with experimental reports for the frequencies of the Raman-active modes. The Heyd-Scuseria-Ernzerhof functional results in electronic bandgaps of 2.66 eV for monolayer PtS2 and 1.74 eV for monolayer PtSe2. G0W0 calculations combined with the Bethe-Salpeter equation are used to predict the optical spectra and exciton binding energies (0.78 eV for monolayer PtS2 and 0.60 eV for monolayer PtSe2). It turns out that the excitons are strongly bound and therefore very stable against external perturbations.
Dressing the nucleon propagator
Fishman, S.; Gersten, A.
1976-01-01
The nucleon propagator in the ''nested bubbles'' approximation is analyzed. The approximation is built from the minimal set of diagrams which is needed to maintain the unitarity condition under two-pion production threshold in the two-nucleon Bethe--Salpeter equation. Recursive formulas for subsets of ''nested bubbles'' diagrams calculated in the framework of the pseudoscalar interaction are obtained by the use of dispersion relations. We prove that the sum of all the ''nested bubbles'' diverges. Moreover, the successive iterations are plagued with ghost poles. We prove that the first approximation--which is the so-called chain approximation--has ghost poles for any nonvanishing coupling constant. In an earlier paper we have shown that ghost poles lead to ghost cuts. These cuts are present in the ''nested bubbles.'' Ghost elimination procedures are discussed. Modifications of the ''nested bubbles'' approximation are introduced in order to obtain convergence and in order to eliminate the ghost poles and ghost cuts. In a similar way as in the Lee model, cutoff functions are introduced in order to eliminate the ghost poles. The necessary and sufficient conditions for the absence of ghost poles are formulated and analyzed. The spectral functions of the modified ''nested bubbles'' are analyzed and computed. Finally, we present a theorem, similar in its form to Levinson's theorem in scattering theory, which enables one to compute in a simple way the number of ghost poles
Relativistic many-body theory a new field-theoretical approach
Lindgren, Ingvar
2016-01-01
This revised second edition of the author’s classic text offers readers a comprehensively updated review of relativistic atomic many-body theory, covering the many developments in the field since the publication of the original title. In particular, a new final section extends the scope to cover the evaluation of QED effects for dynamical processes. The treatment of the book is based upon quantum-field theory, and demonstrates that when the procedure is carried to all orders of perturbation theory, two-particle systems are fully compatible with the relativistically covariant Bethe-Salpeter equation. This procedure can be applied to arbitrary open-shell systems, in analogy with the standard many-body theory, and it is also applicable to systems with more than two particles. Presently existing theoretical procedures for treating atomic systems are, in several cases, insufficient to explain the accurate experimental data recently obtained, particularly for highly charged ions. The main text is divided into...
Nurhuda, Maryam; Aziz Majidi, Muhammad
2018-04-01
The role of excitons in semiconducting materials carries potential applications. Experimental results show that excitonic signals also appear in optical absorption spectra of semiconductor system with narrow gap, such as Gallium Arsenide (GaAs). While on the theoretical side, calculation of optical spectra based purely on Density Functional Theory (DFT) without taking electron-hole (e-h) interactions into account does not lead to the appearance of any excitonic signal. Meanwhile, existing DFT-based algorithms that include a full vertex correction through Bethe-Salpeter equation may reveal an excitonic signal, but the algorithm has not provided a way to analyze the excitonic signal further. Motivated to provide a way to isolate the excitonic effect in the optical response theoretically, we develop a method of calculation for the optical conductivity of a narrow band-gap semiconductor GaAs within the 8-band k.p model that includes electron-hole interactions through first-order electron-hole vertex correction. Our calculation confirms that the first-order e-h vertex correction reveals excitonic signal around 1.5 eV (the band gap edge), consistent with the experimental data.
Polarization of recoil deuteron in ed elastic scattering at medium energies
Bhalerao, R. S.
1981-12-01
Vector and tensor polarizations of the recoil deuteron in ed elastic scattering are calculated for THETA=0deg-180deg and q 2 2 . A longitudinally polarized electron beam is assumed to scatter off an unpolarized deuteron target. Calculations are made in the relativistic impulse approximation using a recently described approach based on the Bethe-Salpeter equation. Results are different, at high q 2 even qualitatively so, from those of a non-relativistic calculation, and a relativistic calculation which takes the spectator nucleon on-mass-shell. In the light of these results a recent suggestion that the polarization measurements would throw new light on the off-shell behavior and tensor force strength of the NN interaction are reexamined. Results are also presented for the three deuteron form factors Gsub(C), Gsub(Q), and Gsub(M), and the often-needed related quantities Ssub(S), Ssub(Q), and Ssub(M). The latter results may have an important implication in high-momentum transfer reactions involving deuteron. (author)
58. annual symposium of the Austrian Physical Society. Conference programme
Oswald, J.
2008-01-01
Full text: This annual conference consisted of a plenary session, oral and poster sessions on the research fields of: acoustics; atoms, quantum optics and plasma (doped helium droplets, biomolecules studies in super fluid helium droplets, quantum physics with neutrons); solid state physics (terahertz quantum-cascade lasers, semiconductors nanostructures, magnetic studies on steel pipeline tubes, magnetic characterization magnetic materials, spin properties of confined electrons); physics history; nuclear and particle physics (antiprotonic helium - hyperfine structure, pionic atoms (hydrogen), CMS experiment at LHC (level 1-trigger, super symmetry), vertex reconstruction toolkit RAVE, silicon strip detectors, chiral transition temperature, quantum physics - Bell theorem, Bethe-Salpeter equation, plane static magnetic field, low-lying eigen modes of the dirac operator, SU(3) potentials by thick-center-vortex-model); medical, bio - and environmental physics; neutrons and synchrotron radiation physics (neutron holography - advances, atomic diffusion by XPCS, micro-diffraction experiments, cold three-axis spectrometer - next generation, superconductive radio resonating cavities- roughness, neutron polarization); surfaces and thin films (carbon monoxide adsorption on metal surfaces, laser - assisted deposition, nanostructures (magnetic properties, semiconductors, electronic structure, erosion, crystal growth, adsorption, sputtering)); physics - industry - energy; besides a poster session on polymer physics and the Max Auwaerter symposium are included. Those contributions which are in the INIS subject scope are indexed individually. (nevyjel)
Raman scattering in a Heisenberg S = 1/2 antiferromagnet on the anisotropic triangular lattice
Perkins, Natalia; Brenig, Wolfram
2009-01-01
We investigate two-magnon Raman scattering from the S = 1/2 Heisenberg antiferromagnet on the triangular lattice (THAF), considering both isotropic and anisotropic exchange interactions. We find that the Raman intensity for the isotropic THAF is insensitive to the scattering geometry, while both the line profile and the intensity of the Raman response for the anisotropic THAF shows a strong dependence on the scattering geometry. For the isotropic case we present an analytical and numerical study of the Raman intensity including both the effect of renormalization of the one-magnon spectrum by 1 = S corrections and final-state magnonmagnon interactions. The bare Raman intensity displays two peaks related to one-magnon van-Hove singularities. We find that 1 = S self-energy corrections to the one-magnon spectrum strongly modify this intensity profile. The central Raman-peak is significantly enhanced due to plateaus in the magnon dispersion, the high frequency peak is suppressed due to magnon damping, and the overall spectral support narrows considerably. Additionally we investigate final-state interactions by solving the Bethe-Salpeter equation to O(1 = S). In contrast to collinear antiferromagnets, the non-collinear nature of the magnetic ground state leads to an irreducible magnon scattering which is retarded and non-separable already to lowest order. We show that final-state interactions lead to a rather broad Raman-continuum centered around approximately twice the 'roton'-energy.
Seiler, Christian; Evers, Ferdinand
2016-10-01
A formalism for electronic-structure calculations is presented that is based on the functional renormalization group (FRG). The traditional FRG has been formulated for systems that exhibit a translational symmetry with an associated Fermi surface, which can provide the organization principle for the renormalization group (RG) procedure. We here advance an alternative formulation, where the RG flow is organized in the energy-domain rather than in k space. This has the advantage that it can also be applied to inhomogeneous matter lacking a band structure, such as disordered metals or molecules. The energy-domain FRG (ɛ FRG) presented here accounts for Fermi-liquid corrections to quasiparticle energies and particle-hole excitations. It goes beyond the state of the art G W -BSE , because in ɛ FRG the Bethe-Salpeter equation (BSE) is solved in a self-consistent manner. An efficient implementation of the approach that has been tested against exact diagonalization calculations and calculations based on the density matrix renormalization group is presented. Similar to the conventional FRG, also the ɛ FRG is able to signalize the vicinity of an instability of the Fermi-liquid fixed point via runaway flow of the corresponding interaction vertex. Embarking upon this fact, in an application of ɛ FRG to the spinless disordered Hubbard model we calculate its phase boundary in the plane spanned by the interaction and disorder strength. Finally, an extension of the approach to finite temperatures and spin S =1 /2 is also given.
Interquark potential with finite quark mass from lattice QCD.
Kawanai, Taichi; Sasaki, Shoichi
2011-08-26
We present an investigation of the interquark potential determined from the q ̄q Bethe-Salpeter (BS) amplitude for heavy quarkonia in lattice QCD. The q ̄q potential at finite quark mass m(q) can be calculated from the equal-time and Coulomb gauge BS amplitude through the effective Schrödinger equation. The definition of the potential itself requires information about a kinetic mass of the quark. We then propose a self-consistent determination of the quark kinetic mass on the same footing. To verify the proposed method, we perform quenched lattice QCD simulations with a relativistic heavy-quark action at a lattice cutoff of 1/a≈2.1 GeV in a range 1.0≤m(q)≤3.6 GeV. Our numerical results show that the q ̄q potential in the m(q)→∞ limit is fairly consistent with the conventional one obtained from Wilson loops. The quark-mass dependence of the q ̄q potential and the spin-spin potential are also examined. © 2011 American Physical Society
Infrared Dielectric Screening Determines the Low Exciton Binding Energy of Metal-Halide Perovskites.
Umari, Paolo; Mosconi, Edoardo; De Angelis, Filippo
2018-02-01
The performance of lead-halide perovskites in optoelectronic devices is due to a unique combination of factors, including highly efficient generation, transport, and collection of photogenerated charge carriers. The mechanism behind efficient charge generation in lead-halide perovskites is still largely unknown. Here, we investigate the factors that influence the exciton binding energy (E b ) in a series of metal-halide perovskites using accurate first-principles calculations based on solution of the Bethe-Salpeter equation, coupled to ab initio molecular dynamics simulations. We find that E b is strongly modulated by screening from low-energy phonons, which account for a factor ∼2 E b reduction, while dynamic disorder and rotational motion of the organic cations play a minor role. We calculate E b = 15 meV for MAPbI 3 , in excellent agreement with recent experimental estimates. We then explore how different material combinations (e.g., replacing Pb → Pb:Sn→ Sn; and MA → FA → Cs) may lead to different E b values and highlight the mechanisms underlying E b tuning.
Mehmood, Faisal; Pachter, Ruth; Murphy, Neil R.; Johnson, Walter E.; Ramana, Chintalapalle V.
2016-12-01
In this work, we investigated theoretically the role of oxygen vacancies on the electronic and optical properties of cubic, γ-monoclinic, and tetragonal phases of tungsten oxide (WO3) thin films. Following the examination of structural properties and stability of the bulk tungsten oxide polymorphs, we analyzed band structures and optical properties, applying density functional theory (DFT) and GW (Green's (G) function approximation with screened Coulomb interaction (W)) methods. Careful benchmarking of calculated band gaps demonstrated the importance of using a range-separated functional, where results for the pristine room temperature γ-monoclinic structure indicated agreement with experiment. Further, modulation of the band gap for WO3 structures with oxygen vacancies was quantified. Dielectric functions for cubic WO3, calculated at both the single-particle, essentially time-dependent DFT, as well as many-body GW-Bethe-Salpeter equation levels, indicated agreement with experimental data for pristine WO3. Interestingly, we found that introducing oxygen vacancies caused appearance of lower energy absorptions. A smaller refractive index was indicated in the defective WO3 structures. These predictions could lead to further experiments aimed at tuning the optical properties of WO3 by introducing oxygen vacancies, particularly for the lower energy spectral region.
Relativistic Many-Body Theory A New Field-Theoretical Approach
Lindgren, Ingvar
2011-01-01
Relativistic Many-Body Theory treats — for the first time — the combination of relativistic atomic many-body theory with quantum-electrodynamics (QED) in a unified manner. This book can be regarded as a continuation of the book by Lindgren and Morrison, Atomic Many-Body Theory (Springer 1986), which deals with the non-relativistic theory of many-electron systems, describing several means of treating the electron correlation to essentially all orders of perturbation theory. The treatment of the present book is based upon quantum-field theory, and demonstrates that when the procedure is carried to all orders of perturbation theory, two-particle systems are fully compatible with the relativistically covariant Bethe-Salpeter equation. This procedure can be applied to arbitrary open-shell systems, in analogy with the standard many-body theory, and it is also applicable to systems with more than two particles. Presently existing theoretical procedures for treating atomic systems are, in several cases, insuffici...
Origins of Singlet Fission in Solid Pentacene from an ab initio Green's Function Approach
Refaely-Abramson, Sivan; da Jornada, Felipe H.; Louie, Steven G.; Neaton, Jeffrey B.
2017-12-01
We develop a new first-principles approach to predict and understand rates of singlet fission with an ab initio Green's-function formalism based on many-body perturbation theory. Starting with singlet and triplet excitons computed from a G W plus Bethe-Salpeter equation approach, we calculate the exciton-biexciton coupling to lowest order in the Coulomb interaction, assuming a final state consisting of two noninteracting spin-correlated triplets with finite center-of-mass momentum. For crystalline pentacene, symmetries dictate that the only purely Coulombic fission decay process from a bright singlet state requires a final state consisting of two inequivalent nearly degenerate triplets of nonzero, equal and opposite, center-of-mass momenta. For such a process, we predict a singlet lifetime of 30-70 fs, in very good agreement with experimental data, indicating that this process can dominate singlet fission in crystalline pentacene. Our approach is general and provides a framework for predicting and understanding multiexciton interactions in solids.
Spectra of heavy-light mesons in a relativistic model
Liu, Jing-Bin; Lue, Cai-Dian [Institute of High Energy Physics, Beijing (China)
2017-05-15
The spectra and wave functions of heavy-light mesons are calculated within a relativistic quark model which is based on a heavy-quark expansion of the instantaneous Bethe-Salpeter equation by applying the Foldy-Wouthuysen transformation. The kernel we choose is the standard combination of linear scalar and Coulombic vector. The effective Hamiltonian for heavy-light quark-antiquark system is calculated up to order 1/m{sub Q}{sup 2}. Our results are in good agreement with available experimental data except for the anomalous D{sub s0}{sup *}(2317) and D{sub s1}(2460) states. The newly observed heavy-light meson states can be accommodated successfully in the relativistic quark model with their assignments presented. The D{sub sJ}{sup *}(2860) can be interpreted as the vertical stroke 1{sup 3/2}D{sub 1} right angle and vertical stroke 1{sup 5/2}D{sub 3} right angle states being members of the 1D family with J{sup P} = 1{sup -} and 3{sup -}. (orig.)
Hadronic molecular states from the K anti K* interaction
Lue, Pei-Liang; He, Jun [Chinese Academy of Sciences, Theoretical Physics Division, Institute of Modern Physics, Lanzhou (China); Institute of Modern Physics of CAS and Lanzhou University, Research Center for Hadron and CSR Physics, Lanzhou (China)
2016-12-15
In this work, the K anti K* interaction is studied in a quasipotential Bethe-Salpeter equation approach combined with the one-boson-exchange model. With the help of the hidden-gauge Lagrangian, the exchanges of pseudoscalar mesons (π and η) and vector mesons (ρ, ω and φ) are considered to describe the K anti K* interaction. Besides the direct vector-meson exchange which can be related to the Weinberg-Tomozawa term, pseudoscalar-meson exchanges also play important roles in the mechanism of the K anti K* interaction. The poles of scattering amplitude are searched to find the molecular states produced from the K anti K* interaction. In the case of quantum number I{sup G}(J{sup PC}) = 0{sup +}(1{sup ++}), a pole is found with a reasonable cutoff, which can be related to the f{sub 1}(1285) in experiment. Another bound state with 0{sup -}(1{sup +-}) is also produced from the K anti K* interaction, which can be related to the h{sub 1}(1380). In the isovector sector, the interaction is much weaker and a bound state with 1{sup +}(1{sup +}) relevant to the b{sub 1}(1235) is produced but at a larger cutoff. Our results suggest that in the hadronic molecular state picture the f{sub 1}(1285) and b{sub 1}(1235) are the strange partners of the X(3872) and Z{sub c}(3900), respectively. (orig.)
Trevisanutto, Paolo E. [Graphene Research Centre and CA2DM, National University of Singapore, Singapore 117542, Singapore and Singapore Synchrotron Light Source, National University of Singapore, Singapore 117603 (Singapore); Vignale, Giovanni, E-mail: vignaleg@missouri.edu [Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211 (United States)
2016-05-28
Ab initio electronic structure calculations of two-dimensional layered structures are typically performed using codes that were developed for three-dimensional structures, which are periodic in all three directions. The introduction of a periodicity in the third direction (perpendicular to the layer) is completely artificial and may lead in some cases to spurious results and to difficulties in treating the action of external fields. In this paper we develop a new approach, which is “native” to quasi-2D materials, making use of basis function that are periodic in the plane, but atomic-like in the perpendicular direction. We show how some of the basic tools of ab initio electronic structure theory — density functional theory, GW approximation and Bethe-Salpeter equation — are implemented in the new basis. We argue that the new approach will be preferable to the conventional one in treating the peculiarities of layered materials, including the long range of the unscreened Coulomb interaction in insulators, and the effects of strain, corrugations, and external fields.
Optical properties of single-layer, double-layer, and bulk MoS2
Molina-Sanchez, Alejandro; Wirtz, Ludger [University of Luxembourg (Luxembourg); Hummer, Kerstin [University of Vienna, Vienna (Austria)
2013-07-01
The rise of graphene has brought attention also to other layered materials that can complement graphene or that can be an alternative in applications as transistors. Single-layer MoS{sub 2} has shown interesting electronic and optical properties such as as high electron mobility at room temperature and an optical bandgap of 1.8 eV. This makes the material suitable for transistors or optoelectronic devices. We present a theoretical study of the optical absorption and photoluminescence spectra of single-layer, double-layer and bulk MoS{sub 2}. The excitonic states have been calculated in the framework of the Bethe-Salpeter equation, taking into account the electron-hole interaction via the screened Coulomb potential. In addition to the step-function like behaviour that is typical for the joint-density of states of 2D materials with parabolic band dispersion, we find a bound excitonic peak that is dominating the luminescence spectra. The peak is split due to spin-orbit coupling for the single-layer and split due to layer-layer interaction for few-layer and bulk MoS{sub 2}. We discuss the changes of the optical bandgap and of the exciton binding energy with the number of layers, comparing our results with the reported experimental data.
Garcia-Recio, C.; Salcedo, L.L.; Gamermann, D.; Nieves, J.; Romanets, O.; Tolos, L.
2014-01-01
We study odd-parity baryonic resonances with one heavy and three light flavors, dynamically generated by meson-baryon interactions. Special attention is paid to Heavy Quark Spin Symmetry (HQSS), hence pseudoscalar and vector mesons and baryons with J π = 1/2 + and 3/2 + are considered as constituent hadrons. For the hidden-charm sector (N c = N c ¯ = 1), the meson-baryon Lagrangian with Heavy Flavor Symmetry is constructed by a minimal extension of the SU(3) Weinberg-Tomozawa (WT) Lagrangian to fulfill HQSS, such that not new parameters are needed. This interaction can be presented in different formal ways: as a Field Lagrangian, as Hadron creation-annihilation operators, as SU(6)×HQSS group projectors and as multichannel matrices. The multichannel Bethe-Salpeter equation is solved for odd-parity light baryons, hidden-charm N and Δ and Beauty Baryons (Λ b ). Results of calculations with this model are shown in comparison with other models and experimental values for baryonic resonances. (author)
Wang, Tianhong; Jiang, Yue; Wang, Guo-Li [Harbin Institute of Technology, Department of Physics, Harbin (China); Wang, Zhi-Hui [Beifang University of Nationalities, School of Electrical and Information Engineering, Yinchuan (China); Jiang, Libo [University of Pittsburgh, Department of Physics and Astronomy, Pittsburgh, PA (United States)
2017-01-15
In this paper, we study the OZI-allowed two-body strong decays of 3{sup -} heavy-light mesons. Experimentally the charmed D{sub 3}{sup *}(2760) and the charm-strange D{sub s3}{sup *}(2860) states with these quantum numbers have been discovered. For the bottomed B(5970) state, which was found by the CDF Collaboration recently, its quantum number has not been decided yet and we assume it is a 3{sup -} meson in this paper. The theoretical prediction for the strong decays of bottom-strange state B{sub s3}{sup *} is also given. The relativistic wave functions of 3{sup -} heavy mesons are constructed and their numerical values are obtained by solving the corresponding Bethe-Salpeter equation with instantaneous approximation. The transition matrix is calculated by using the PCAC and low energy theorem, following which the decay widths are obtained. For D{sub 3}{sup *}(2760) and D{sub s3}{sup *}(2860), the total strong decay widths are 72.6 and 47.6 MeV, respectively. For B{sub 3}{sup *} with M = 5978 MeV and B{sub s3}{sup *} with M = 6178 MeV, their strong decay widths are 22.9 and 40.8 MeV, respectively. (orig.)
Range-separated density-functional theory for molecular excitation energies
Rebolini, E.
2014-01-01
Linear-response time-dependent density-functional theory (TDDFT) is nowadays a method of choice to compute molecular excitation energies. However, within the usual adiabatic semi-local approximations, it is not able to describe properly Rydberg, charge-transfer or multiple excitations. Range separation of the electronic interaction allows one to mix rigorously density-functional methods at short range and wave function or Green's function methods at long range. When applied to the exchange functional, it already corrects most of these deficiencies but multiple excitations remain absent as they need a frequency-dependent kernel. In this thesis, the effects of range separation are first assessed on the excitation energies of a partially-interacting system in an analytic and numerical study in order to provide guidelines for future developments of range-separated methods for excitation energy calculations. It is then applied on the exchange and correlation TDDFT kernels in a single-determinant approximation in which the long-range part of the correlation kernel vanishes. A long-range frequency-dependent second-order correlation kernel is then derived from the Bethe-Salpeter equation and added perturbatively to the range-separated TDDFT kernel in order to take into account the effects of double excitations. (author)
Solutions to Arithmetic Convolution Equations
Glöckner, H.; Lucht, L.G.; Porubský, Štefan
2007-01-01
Roč. 135, č. 6 (2007), s. 1619-1629 ISSN 0002-9939 R&D Projects: GA ČR GA201/04/0381 Institutional research plan: CEZ:AV0Z10300504 Keywords : arithmetic functions * Dirichlet convolution * polynomial equations * analytic equations * topological algebras * holomorphic functional calculus Subject RIV: BA - General Mathematics Impact factor: 0.520, year: 2007
On Degenerate Partial Differential Equations
Chen, Gui-Qiang G.
2010-01-01
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, fundamental problems in fluid mechanics and differential geometry. The solution to these fundamental problems greatly requires a deep understanding of nonlinear degenerate parti...
Differential equations a concise course
Bear, H S
2011-01-01
Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complex-valued solutions, linear differential operators, inverse operators and variation of parameters method, the Laplace transform, Picard's existence theorem, and an exploration of various interpretations of systems of equations. Numerous clearly stated theorems and proofs, examples, and problems followed by solutions.
Differential equations and finite groups
Put, Marius van der; Ulmer, Felix
2000-01-01
The classical solution of the Riemann-Hilbert problem attaches to a given representation of the fundamental group a regular singular linear differential equation. We present a method to compute this differential equation in the case of a representation with finite image. The approach uses Galois
Saturation and linear transport equation
Kutak, K.
2009-03-01
We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term. (orig.)
Lie symmetries in differential equations
Pleitez, V.
1979-01-01
A study of ordinary and Partial Differential equations using the symmetries of Lie groups is made. Following such a study, an application to the Helmholtz, Line-Gordon, Korleweg-de Vries, Burguer, Benjamin-Bona-Mahony and wave equations is carried out [pt
Introduction to nonlinear dispersive equations
Linares, Felipe
2015-01-01
This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research. The second edition of Introdu...
Students' Understanding of Quadratic Equations
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-01-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…
Solving equations by topological methods
Lech Górniewicz
2005-01-01
Full Text Available In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.
Generalized Fermat equations: A miscellany
Bennett, M.A.; Chen, I.; Dahmen, S.R.; Yazdani, S.
2015-01-01
This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing
Equation with the many fathers
Kragh, Helge
1984-01-01
In this essay I discuss the origin and early development of the first relativistic wave equation, known as the Klein-Gordon equation. In 1926 several physicists, among them Klein, Fock, Schrödinger, and de Broglie, announced this equation as a candidate for a relativistic generalization of the us...... as electrodynamics. Although this ambitious attempt attracted some interest in 1926, its impact on the mainstream of development in quantum mechanics was virtually nil....... of the usual Schrödinger equation. In most of the early versions the Klein-Gordon equation was connected with the general theory of relativity. Klein and some other physicists attempted to express quantum mechanics within a five-dimensional unified theory, embracing general relativity as well...
The relativistic electron wave equation
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.)
Higher order field equations. II
Tolhoek, H.A.
1977-01-01
In a previous paper wave propagation was studied according to a sixth-order partial differential equation involving a complex mass M. The corresponding Yang-Feldman integral equations (indicated as SM-YF-equations), were formulated using modified Green's functions Gsub(R)sup(M)(x) and Gsub(A)sup(M)(x), which then incorporate the partial differential equation together with certain boundary conditions. In this paper certain limit properties of these modified Green's functions are derived: (a) It is shown that for mod(M)→infinity the Green's functions Gsub(R)sup(M)(x) and Gsub(A)sup(M)(x) approach the Green's functions Δsub(R)(x) and Δsub(A)(x) of the corresponding KG-equation (Klein-Gordon equation). (b) It is further shown that the asymptotic behaviour of Gsub(R)sup(M)(x) and Gsub(A)sup(M)(x) is the same as of Δsub(R)(x) and Δsub(A)(x)-and also the same as for Dsub(R)(x) and Dsub(A)(x) for t→+-infinity;, where Dsub(R) and Dsub(A) are the Green's functions for the KG-equation with mass zero. It is essential to take limits in the sense of distribution theory in both cases (a) and (b). The property (b) indicates that the wave propagation properties of the SM-YF-equations, the KG-equation with finite mass and the KG-equation with mass zero are closely related in an asymptotic sense. (Auth.)
Ozdemir, Burhanettin
2017-01-01
The purpose of this study is to equate Trends in International Mathematics and Science Study (TIMSS) mathematics subtest scores obtained from TIMSS 2011 to scores obtained from TIMSS 2007 form with different nonlinear observed score equating methods under Non-Equivalent Anchor Test (NEAT) design where common items are used to link two or more test…
Neoclassical MHD equations for tokamaks
Callen, J.D.; Shaing, K.C.
1986-03-01
The moment equation approach to neoclassical-type processes is used to derive the flows, currents and resistive MHD-like equations for studying equilibria and instabilities in axisymmetric tokamak plasmas operating in the banana-plateau collisionality regime (ν* approx. 1). The resultant ''neoclassical MHD'' equations differ from the usual reduced equations of resistive MHD primarily by the addition of the important viscous relaxation effects within a magnetic flux surface. The primary effects of the parallel (poloidal) viscous relaxation are: (1) Rapid (approx. ν/sub i/) damping of the poloidal ion flow so the residual flow is only toroidal; (2) addition of the bootstrap current contribution to Ohm's laws; and (3) an enhanced (by B 2 /B/sub theta/ 2 ) polarization drift type term and consequent enhancement of the perpendicular dielectric constant due to parallel flow inertia, which causes the equations to depend only on the poloidal magnetic field B/sub theta/. Gyroviscosity (or diamagnetic vfiscosity) effects are included to properly treat the diamagnetic flow effects. The nonlinear form of the neoclassical MHD equations is derived and shown to satisfy an energy conservation equation with dissipation arising from Joule and poloidal viscous heating, and transport due to classical and neoclassical diffusion
Approximate solutions to Mathieu's equation
Wilkinson, Samuel A.; Vogt, Nicolas; Golubev, Dmitry S.; Cole, Jared H.
2018-06-01
Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.
Soliton equations and Hamiltonian systems
Dickey, L A
2002-01-01
The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also becau
Galois theory of difference equations
Put, Marius
1997-01-01
This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.
Integral equation methods for electromagnetics
Volakis, John
2012-01-01
This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo
Bridging the Knowledge Gaps between Richards' Equation and Budyko Equation
Wang, D.
2017-12-01
The empirical Budyko equation represents the partitioning of mean annual precipitation into evaporation and runoff. Richards' equation, based on Darcy's law, represents the movement of water in unsaturated soils. The linkage between Richards' equation and Budyko equation is presented by invoking the empirical Soil Conservation Service curve number (SCS-CN) model for computing surface runoff at the event-scale. The basis of the SCS-CN method is the proportionality relationship, i.e., the ratio of continuing abstraction to its potential is equal to the ratio of surface runoff to its potential value. The proportionality relationship can be derived from the Richards' equation for computing infiltration excess and saturation excess models at the catchment scale. Meanwhile, the generalized proportionality relationship is demonstrated as the common basis of SCS-CN method, monthly "abcd" model, and Budyko equation. Therefore, the linkage between Darcy's law and the emergent pattern of mean annual water balance at the catchment scale is presented through the proportionality relationship.
Iterative Splitting Methods for Differential Equations
Geiser, Juergen
2011-01-01
Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations. In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential
Nonlinear integrodifferential equations as discrete systems
Tamizhmani, K. M.; Satsuma, J.; Grammaticos, B.; Ramani, A.
1999-06-01
We analyse a class of integrodifferential equations of the `intermediate long wave' (ILW) type. We show that these equations can be formally interpreted as discrete, differential-difference systems. This allows us to link equations of this type with previous results of ours involving differential-delay equations and, on the basis of this, propose new integrable equations of ILW type. Finally, we extend this approach to pure difference equations and propose ILW forms for the discrete lattice KdV equation.
Direct 'delay' reductions of the Toda equation
Joshi, Nalini
2009-01-01
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated with this equation is obtained, also by reduction. (fast track communication)
Integral equation for Coulomb problem
Sasakawa, T.
1986-01-01
For short range potentials an inhomogeneous (homogeneous) Lippmann-Schwinger integral equation of the Fredholm type yields the wave function of scattering (bound) state. For the Coulomb potential, this statement is no more valid. It has been felt difficult to express the Coulomb wave function in a form of an integral equation with the Coulomb potential as the perturbation. In the present paper, the author shows that an inhomogeneous integral equation of a Volterra type with the Coulomb potential as the perturbation can be constructed both for the scattering and the bound states. The equation yielding the binding energy is given in an integral form. The present treatment is easily extended to the coupled Coulomb problems
Geophysical interpretation using integral equations
Eskola, L
1992-01-01
Along with the general development of numerical methods in pure and applied to apply integral equations to geophysical modelling has sciences, the ability improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assu med to have a back...
Singularity: Raychaudhuri equation once again
Cosmology; Raychaudhuri equation; Universe; quantum gravity; loop quan- tum gravity ... than the observation verifying the prediction of theory. This gave .... which was now expanding, would have had a singular beginning in a hot Big Bang.
Kinetic equations in dirty superconductors
Kraehenbuehl, Y.
1981-01-01
Kinetic equations for superconductors in the dirty limit are derived using a method developed for superfluid systems, which allows a systematic expansion in small parameters; exact charge conservation is obeyed. (orig.)
Skyrme, T.H.R.
1994-01-01
In a model quantum theory of interacting mesons, the motion of certain conserved particle-like structures is discussed. It is shown how collective coordinates may be introduced to describe them, leading, in lowest approximation, to a Dirac equation. (author)
Solving Differential Equations in R
Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...
Wave-equation dispersion inversion
Li, Jing; Feng, Zongcai; Schuster, Gerard T.
2016-01-01
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained
Jannussis, A.; Streclas, A.; Sourlas, D.; Vlachos, K.
1977-01-01
Using the theorem of the derivative of a function of operators with respect to any parameter, we can find the equation of motion of a system in classical mechanics, in canonical as well as in non-canonical mechanics
Quantum-statistical kinetic equations
Loss, D.; Schoeller, H.
1989-01-01
Considering a homogeneous normal quantum fluid consisting of identical interacting fermions or bosons, the authors derive an exact quantum-statistical generalized kinetic equation with a collision operator given as explicit cluster series where exchange effects are included through renormalized Liouville operators. This new result is obtained by applying a recently developed superoperator formalism (Liouville operators, cluster expansions, symmetrized projectors, P q -rule, etc.) to nonequilibrium systems described by a density operator ρ(t) which obeys the von Neumann equation. By means of this formalism a factorization theorem is proven (being essential for obtaining closed equations), and partial resummations (leading to renormalized quantities) are performed. As an illustrative application, the quantum-statistical versions (including exchange effects due to Fermi-Dirac or Bose-Einstein statistics) of the homogeneous Boltzmann (binary collisions) and Choh-Uhlenbeck (triple collisions) equations are derived
Lorentz Covariance of Langevin Equation
Koide, T.; Denicol, G.S.; Kodama, T.
2008-01-01
Relativistic covariance of a Langevin type equation is discussed. The requirement of Lorentz invariance generates an entanglement between the force and noise terms so that the noise itself should not be a covariant quantity. (author)
Equational theories of tropical sernirings
Aceto, Luca; Esik, Zoltan; Ingolfsdottir, Anna
2003-01-01
examples of such structures are the (max,+) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each......This paper studies the equational theories of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime...... of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results. Udgivelsesdato: APR 11...
Wave equations for pulse propagation
Shore, B.W.
1987-01-01
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity. The memo discusses various ways of characterizing the polarization characteristics of plane waves, that is, of parameterizing a transverse unit vector, such as the Jones vector, the Stokes vector, and the Poincare sphere. It discusses the connection between macroscopically defined quantities, such as the intensity or, more generally, the Stokes parameters, and microscopic field amplitudes. The material presented here is a portion of a more extensive treatment of propagation to be presented separately. The equations presented here have been described in various books and articles. They are collected here as a summary and review of theory needed when treating pulse propagation
Feynman integrals and difference equations
Moch, S.; Schneider, C.
2007-09-01
We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space. The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called ΠΣ * -fields. We test the implementation of the Mathematica package Sigma on examples from recent higher order perturbative calculations in Quantum Chromodynamics. (orig.)
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.
Feynman integrals and difference equations
Moch, S. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Schneider, C. [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation
2007-09-15
We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space. The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called {pi}{sigma}{sup *}-fields. We test the implementation of the Mathematica package Sigma on examples from recent higher order perturbative calculations in Quantum Chromodynamics. (orig.)
Numerical solution of Boltzmann's equation
Sod, G.A.
1976-04-01
The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig
Computational partial differential equations using Matlab
Li, Jichun
2008-01-01
Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE
Linear determining equations for differential constraints
Kaptsov, O V
1998-01-01
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie operators. As applications of this approach equations of an ideal incompressible fluid and non-linear heat equations are discussed
Equationally Compact Acts : Coproducts / Peeter Normak
Normak, Peeter
1998-01-01
In this article equational compactness of acts and its generalizations are discussed. As equational compactness does not carry over to coproducts a slight generalization of c-equational campactness is introduced. It is proved that a coproduct of acts is c-equationally compact if and only if all components are c-equationally campact
Exact results for the Boltzmann equation and Smoluchowski's coagulation equation
Hendriks, E.M.
1983-01-01
Almost no analytical solutions have been found for realistic intermolecular forces, largely due to the complicated structure of the collision term which calls for the construction of simplified models, in which as many physical properties are maintained as possible. In the first three chapters of this thesis such model Boltzmann equations are studied. Only spatially homogeneous gases with isotropic distribution functions are considered. Chapter I considers transition kernels, chapter II persistent scattering models and chapter III very hard particles. The second part of this dissertation deals with Smoluchowski's coagulation equation for the size distribution function in a coagulating system, with chapters devoted to the following topics: kinetics of gelation and universality, coagulation equations with gelation and exactly soluble models of nucleation. (Auth./C.F.)
Abstract methods in partial differential equations
Carroll, Robert W
2012-01-01
Detailed, self-contained treatment examines modern abstract methods in partial differential equations, especially abstract evolution equations. Suitable for graduate students with some previous exposure to classical partial differential equations. 1969 edition.
Linear integral equations and soliton systems
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.)
ON THE EQUIVALENCE OF THE ABEL EQUATION
无
2006-01-01
This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations.
Exact solitary waves of the Fisher equation
Kudryashov, Nikolai A.
2005-01-01
New method is presented to search exact solutions of nonlinear differential equations. This approach is used to look for exact solutions of the Fisher equation. New exact solitary waves of the Fisher equation are given
How to obtain the covariant form of Maxwell's equations from the continuity equation
Heras, Jose A
2009-01-01
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations
How to obtain the covariant form of Maxwell's equations from the continuity equation
Heras, Jose A [Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Av. San Pablo No. 180, Col. Reynosa, 02200, Mexico D. F. (Mexico); Departamento de Fisica y Matematicas, Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Mexico D. F. 01210 (Mexico)
2009-07-15
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
Extraction of dynamical equations from chaotic data
Rowlands, G.; Sprott, J.C.
1991-02-01
A method is described for extracting from a chaotic time series a system of equations whose solution reproduces the general features of the original data even when these are contaminated with noise. The equations facilitate calculation of fractal dimension, Lyapunov exponents and short-term predictions. The method is applied to data derived from numerical solutions of the Logistic equation, the Henon equations, the Lorenz equations and the Roessler equations. 10 refs., 5 figs
First-order partial differential equations
Rhee, Hyun-Ku; Amundson, Neal R
2001-01-01
This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of mo
Differential equations, mechanics, and computation
Palais, Richard S
2009-01-01
This book provides a conceptual introduction to the theory of ordinary differential equations, concentrating on the initial value problem for equations of evolution and with applications to the calculus of variations and classical mechanics, along with a discussion of chaos theory and ecological models. It has a unified and visual introduction to the theory of numerical methods and a novel approach to the analysis of errors and stability of various numerical solution algorithms based on carefully chosen model problems. While the book would be suitable as a textbook for an undergraduate or elementary graduate course in ordinary differential equations, the authors have designed the text also to be useful for motivated students wishing to learn the material on their own or desiring to supplement an ODE textbook being used in a course they are taking with a text offering a more conceptual approach to the subject.
Generalized equations of gravitational field
Stanyukovich, K.P.; Borisova, L.B.
1985-01-01
Equations for gravitational fields are obtained on the basis of a generalized Lagrangian Z=f(R) (R is the scalar curvature). Such an approach permits to take into account the evolution of a gravitation ''constant''. An expression for the force Fsub(i) versus the field variability is obtained. Conservation laws are formulated differing from the standard ones by the fact that in the right part of new equations the value Fsub(i) is present that goes to zero at an ultimate passage to the standard Einstein theory. An equation of state is derived for cosmological metrics for a particular case, f=bRsup(1+α) (b=const, α=const)
Numerical optimization using flow equations
Punk, Matthias
2014-12-01
We develop a method for multidimensional optimization using flow equations. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimizing functional correspond to fixed points of the flow equation. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the additional step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow equation only as an initial condition. We demonstrate the applicability of this optimization method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.
Quantum Gross-Pitaevskii Equation
Jutho Haegeman, Damian Draxler, Vid Stojevic, J. Ignacio Cirac, Tobias J. Osborne, Frank Verstraete
2017-07-01
Full Text Available We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of many body system ---including entanglement and correlations--- and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for (quasi one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.
Introductory course on differential equations
Gorain, Ganesh C
2014-01-01
Introductory Course on DIFFERENTIAL EQUATIONS provides an excellent exposition of the fundamentals of ordinary and partial differential equations and is ideally suited for a first course of undergraduate students of mathematics, physics and engineering. The aim of this book is to present the elementary theories of differential equations in the forms suitable for use of those students whose main interest in the subject are based on simple mathematical ideas. KEY FEATURES: Discusses the subject in a systematic manner without sacrificing mathematical rigour. A variety of exercises drill the students in problem solving in view of the mathematical theories explained in the book. Worked out examples illustrated according to the theories developed in the book with possible alternatives. Exhaustive collection of problems and the simplicity of presentation differentiate this book from several others. Material contained will help teachers as well as aspiring students of different competitive examinations.
The respiratory system in equations
Maury, Bertrand
2013-01-01
The book proposes an introduction to the mathematical modeling of the respiratory system. A detailed introduction on the physiological aspects makes it accessible to a large audience without any prior knowledge on the lung. Different levels of description are proposed, from the lumped models with a small number of parameters (Ordinary Differential Equations), up to infinite dimensional models based on Partial Differential Equations. Besides these two types of differential equations, two chapters are dedicated to resistive networks, and to the way they can be used to investigate the dependence of the resistance of the lung upon geometrical characteristics. The theoretical analysis of the various models is provided, together with state-of-the-art techniques to compute approximate solutions, allowing comparisons with experimental measurements. The book contains several exercises, most of which are accessible to advanced undergraduate students.
Dynamics of partial differential equations
Wayne, C Eugene
2015-01-01
This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation. The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equ...
Evolution equations for Killing fields
Coll, B.
1977-01-01
The problem of finding necessary and sufficient conditions on the Cauchy data for Einstein equations which insure the existence of Killing fields in a neighborhood of an initial hypersurface has been considered recently by Berezdivin, Coll, and Moncrief. Nevertheless, it can be shown that the evolution equations obtained in all these cases are of nonstrictly hyperbolic type, and, thus, the Cauchy data must belong to a special class of functions. We prove here that, for the vacuum and Einstein--Maxwell space--times and in a coordinate independent way, one can always choose, as evolution equations for the Killing fields, a strictly hyperbolic system: The above theorems can be thus extended to all Cauchy data for which the Einstein evolution problem has been proved to be well set
Quasisymmetry equations for conventional stellarators
Pustovitov, V.D.
1994-11-01
General quasisymmetry condition, which demands the independence of B 2 on one of the angular Boozer coordinates, is reduced to two equations containing only geometrical characteristics and helical field of a stellarator. The analysis is performed for conventional stellarators with a planar circular axis using standard stellarator expansion. As a basis, the invariant quasisymmetry condition is used. The quasisymmetry equations for stellarators are obtained from this condition also in an invariant form. Simplified analogs of these equations are given for the case when averaged magnetic surfaces are circular shifted torii. It is shown that quasisymmetry condition can be satisfied, in principle, in a conventional stellarator by a proper choice of two satellite harmonics of the helical field in addition to the main harmonic. Besides, there appears a restriction on the shift of magnetic surfaces. Thus, in general, the problem is closely related with that of self-consistent description of a configuration. (author)
The generalized good cut equation
Adamo, T M; Newman, E T
2010-01-01
The properties of null geodesic congruences (NGCs) in Lorentzian manifolds are a topic of considerable importance. More specifically NGCs with the special property of being shear-free or asymptotically shear-free (as either infinity or a horizon is approached) have received a great deal of recent attention for a variety of reasons. Such congruences are most easily studied via solutions to what has been referred to as the 'good cut equation' or the 'generalization good cut equation'. It is the purpose of this paper to study these equations and show their relationship to each other. In particular we show how they all have a four-complex-dimensional manifold (known as H-space, or in a special case as complex Minkowski space) as a solution space.
Integration rules for scattering equations
Baadsgaard, Christian; Bjerrum-Bohr, N.E.J.; Bourjaily, Jacob L.; Damgaard, Poul H.
2015-01-01
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.
Coupled Higgs field equation and Hamiltonian amplitude equation ...
Home; Journals; Pramana – Journal of Physics; Volume 79; Issue 1. Coupled Higgs ﬁeld equation and ... School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, India; Department of Mathematics, Jaypee University of Information Technology, Waknaghat, Distt. Solan 173 234, India ...
Coupled Higgs field equation and Hamiltonian amplitude equation ...
the rational functions are obtained. Keywords. ... differential equations as is evident by the number of research papers, books and a new symbolic software .... Now using (2.11), (2.14) in (2.8) with C1 = 0 and integrating once we get. P. 2 = − β.
Kahana, S.
1986-01-01
The role of the nuclear equation of state in determining the fate of the collapsing cores of massive stars is examined in light of both recent theoretical advances in this subject and recent experimental measurements with relativistic heavy ions. The difficulties existing in attempts to bring the softer nuclear matter apparently required by the theory of Type II supernovae into consonance with the heavy ion data are discussed. Relativistic mean field theory is introduced as a candidate for derivation of the equation of state, and a simple form for the saturation compressibility is obtained. 28 refs., 4 figs., 1 tab
Kinetic equations with pairing correlations
Fauser, R.
1995-12-01
The Gorkov equations are derived for a general non-equilibrium system. The Gorkov factorization is generalized by the cumulant expansion of the 2-particle correlation and by a generalized Wick theorem in the case of a perturbation expansion. A stationary solution for the Green functions in the Schwinger-Keldysh formalism is presented taking into account pairing correlations. Especially the effects of collisional broadening on the spectral functions and Green functions is discussed. Kinetic equations are derived in the quasi-particle approximation and in the case of particles with width. Explicit expressions for the self-energies are given. (orig.)
Partial differential equations an introduction
Colton, David
2004-01-01
Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of
Geometric approach to soliton equations
Sasaki, R.
1979-09-01
A class of nonlinear equations that can be solved in terms of nxn scattering problem is investigated. A systematic geometric method of exploiting conservation laws and related equations, the so-called prolongation structure, is worked out. The nxn problem is reduced to nsub(n-1)x(n-1) problems and finally to 2x2 problems, which have been comprehensively investigated recently by the author. A general method of deriving the infinite numbers of polynomial conservation laws for an nxn problem is presented. The cases of 3x3 and 2x2 problems are discussed explicitly. (Auth.)
Sensitivity for the Smoluchowski equation
Bailleul, I F
2011-01-01
This paper investigates the question of sensitivity of the solutions μ λ t of the Smoluchowski equation on R + * with respect to the parameters λ in the interaction kernel K λ . It is proved that μ λ t is a C 1 function of (t, λ) with values in a good space of measures under the hypotheses K λ (x, y) ≤ ψ(x) ψ(y), for some sub-linear function ψ, and ∫ψ 4+ε (x) μ 0 (dx) < ∞, and that the derivative is the unique solution of a related equation.
Basic linear partial differential equations
Treves, Francois
1975-01-01
Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their
Kahana, S.
1986-01-01
The role of the nuclear equation of state in determining the fate of the collapsing cores of massive stars is examined in light of both recent theoretical advances in this subject and recent experimental measurements with relativistic heavy ions. The difficulties existing in attempts to bring the softer nuclear matter apparently required by the theory of Type II supernovae into consonance with the heavy ion data are discussed. Relativistic mean field theory is introduced as a candidate for derivation of the equation of state, and a simple form for the saturation compressibility is obtained. 28 refs., 4 figs., 1 tab.
Solution of the Baxter equation
Janik, R.A.
1996-01-01
We present a method of construction of a family of solutions of the Baxter equation arising in the Generalized Leading Logarithmic Approximation (GLLA) of the QCD pomeron. The details are given for the exchange of N = 2 reggeons but everything can be generalized in a straightforward way to arbitrary N. A specific choice of solutions is shown to reproduce the correct energy levels for half integral conformal weights. It is shown that the Baxter's equation must be supplemented by an additional condition on the solution. (author)
Fundamentals of equations of state
Eliezer, Shalom; Hora, Heinrich
2002-01-01
The equation of state was originally developed for ideal gases, and proved central to the development of early molecular and atomic physics. Increasingly sophisticated equations of state have been developed to take into account molecular interactions, quantization, relativistic effects, etc. Extreme conditions of matter are encountered both in nature and in the laboratory, for example in the centres of stars, in relativistic collisions of heavy nuclei, in inertial confinement fusion (where a temperature of 10 9 K and a pressure exceeding a billion atmospheres can be achieved). A sound knowledg
Nielsen number and differential equations
Andres Jan
2005-01-01
Full Text Available In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations, two main approaches are presented. The first is via Poincaré's translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics are indicated, jointly with some further consequences like the nontrivial -structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.
Applied analysis and differential equations
Cârj, Ovidiu
2007-01-01
This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.
Sequent Calculus and Equational Programming
Nicolas Guenot
2015-07-01
Full Text Available Proof assistants and programming languages based on type theories usually come in two flavours: one is based on the standard natural deduction presentation of type theory and involves eliminators, while the other provides a syntax in equational style. We show here that the equational approach corresponds to the use of a focused presentation of a type theory expressed as a sequent calculus. A typed functional language is presented, based on a sequent calculus, that we relate to the syntax and internal language of Agda. In particular, we discuss the use of patterns and case splittings, as well as rules implementing inductive reasoning and dependent products and sums.
Radar equations for modern radar
Barton, David K
2012-01-01
Based on the classic Radar Range-Performance Analysis from 1980, this practical volume extends that work to ensure applicability of radar equations to the design and analysis of modern radars. This unique book helps you identify what information on the radar and its environment is needed to predict detection range. Moreover, it provides equations and data to improve the accuracy of range calculations. You find detailed information on propagation effects, methods of range calculation in environments that include clutter, jamming and thermal noise, as well as loss factors that reduce radar perfo
Equating accelerometer estimates among youth
Brazendale, Keith; Beets, Michael W; Bornstein, Daniel B
2016-01-01
from one set of cutpoints into another. Bland Altman plots illustrate the agreement between actual MVPA and predicted MVPA values. RESULTS: Across the total sample, mean MVPA ranged from 29.7MVPAmind(-1) (Puyau) to 126.1MVPAmind(-1) (Freedson 3 METs). Across conversion equations, median absolute...
Variational linear algebraic equations method
Moiseiwitsch, B.L.
1982-01-01
A modification of the linear algebraic equations method is described which ensures a variational bound on the phaseshifts for potentials having a definite sign at all points. The method is illustrated by the elastic scattering of s-wave electrons by the static field of atomic hydrogen. (author)
Integrodifferential equation approach. Pt. 1
Oehm, W.; Sofianos, S.A.; Fiedeldey, H.; South Africa Univ., Pretoria. Dept. of Physics); Fabre de la Ripelle, M.; South Africa Univ., Pretoria. Dept. of Physics)
1990-02-01
A single integrodifferential equation in two variables, valid for A nucleons interacting by pure Wigner forces, which has previously only been solved in the extreme and uncoupled adiabatic approximations is now solved exactly for three- and four-nucleon systems. The results are in good agreement with the values obtained for the binding energies by means of an empirical interpolation formula. This validates all our previous conclusions, in particular that the omission of higher (than two) order correlations in our four-body equation only produces a rather small underbinding. The integrodifferential equation approach (IDEA) is here also extended to spin-dependent forces of the Malfliet-Tjon type, resulting in two coupled integrodifferential equations in two variables. The exact solution and the interpolated adiabatic approximation are again in good agreement. The inclusion of the hypercentral part of the two-body interaction in the definition of the Faddeev-type components again leads to substantial improvement for fully local potentials, acting in all partial waves. (orig.)
A generalized advection dispersion equation
This paper examines a possible effect of uncertainties, variability or heterogeneity of any dynamic system when being included in its evolution rule; the notion is illustrated with the advection dispersion equation, which describes the groundwater pollution model. An uncertain derivative is defined; some properties of.
Nonlocal higher order evolution equations
Rossi, Julio D.; Schö nlieb, Carola-Bibiane
2010-01-01
In this article, we study the asymptotic behaviour of solutions to the nonlocal operator ut(x, t)1/4(-1)n-1 (J*Id -1)n (u(x, t)), x ∈ ℝN, which is the nonlocal analogous to the higher order local evolution equation vt(-1)n-1(Δ)nv. We prove
Crowe, C.T.
1975-01-01
General features of a vapor-droplet flow are discussed and the equations expressing the conservation of mass, momentum, and energy for the vapor, liquid, and mixture using the control volume approach are derived. The phenomenological laws describing the exchange of mass, momentum, and energy between phases are also reviewed. The results have application to development of water-dominated geothermal resources
On the Saha Ionization Equation
the equation in terms of rate theory. ... that the said theory is said to be the harbinger of modern astro- ... Parichay (An Introduction to the Universe). Tagore ..... where |e| is the magnitude of the electron's charge and E is the electric field intensity ...
Saha equation in Rindler space
Sanchari De
2017-05-31
May 31, 2017 ... scenario, the flat local geometry is called the Rindler space. For an illustration, let us consider two reference ... the local acceleration of the frame. To investigate Saha equation in a uniformly acceler- ... the best of our knowledge, the study of Saha equa- tion in Rindler space has not been reported earlier.
Slave equations for spin models
Catterall, S.M.; Drummond, I.T.; Horgan, R.R.
1992-01-01
We apply an accelerated Langevin algorithm to the simulation of continuous spin models on the lattice. In conjunction with the evolution equation for the spins we use slave equations to compute estimators for the connected correlation functions of the model. In situations for which the symmetry of the model is sufficiently strongly broken by an external field these estimators work well and yield a signal-to-noise ratio for the Green function at large time separations more favourable than that resulting from the standard method. With the restoration of symmetry, however, the slave equation estimators exhibit an intrinsic instability associated with the growth of a power law tail in the probability distributions for the measured quantities. Once this tail has grown sufficiently strong it results in a divergence of the variance of the estimator which then ceases to be useful for measurement purposes. The instability of the slave equation method in circumstances of weak symmetry breaking precludes its use in determining the mass gap in non-linear sigma models. (orig.)
Pendulum Motion and Differential Equations
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
Elizarova, Tatiana G
2009-01-01
This book presents two interconnected mathematical models generalizing the Navier-Stokes system. The models, called the quasi-gas-dynamic and quasi-hydrodynamic equations, are then used as the basis of numerical methods solving gas- and fluid-dynamic problems.
Stability of Functional Differential Equations
Lemm, Jeffrey M
1986-01-01
This book provides an introduction to the structure and stability properties of solutions of functional differential equations. Numerous examples of applications (such as feedback systrems with aftereffect, two-reflector antennae, nuclear reactors, mathematical models in immunology, viscoelastic bodies, aeroautoelastic phenomena and so on) are considered in detail. The development is illustrated by numerous figures and tables.
Quantum adiabatic Markovian master equations
Albash, Tameem; Zanardi, Paolo; Boixo, Sergio; Lidar, Daniel A
2012-01-01
We develop from first principles Markovian master equations suited for studying the time evolution of a system evolving adiabatically while coupled weakly to a thermal bath. We derive two sets of equations in the adiabatic limit, one using the rotating wave (secular) approximation that results in a master equation in Lindblad form, the other without the rotating wave approximation but not in Lindblad form. The two equations make markedly different predictions depending on whether or not the Lamb shift is included. Our analysis keeps track of the various time and energy scales associated with the various approximations we make, and thus allows for a systematic inclusion of higher order corrections, in particular beyond the adiabatic limit. We use our formalism to study the evolution of an Ising spin chain in a transverse field and coupled to a thermal bosonic bath, for which we identify four distinct evolution phases. While we do not expect this to be a generic feature, in one of these phases dissipation acts to increase the fidelity of the system state relative to the adiabatic ground state. (paper)
Weak solutions of magma equations
Krishnan, E.V.
1999-01-01
Periodic solutions in terms of Jacobian cosine elliptic functions have been obtained for a set of values of two physical parameters for the magma equation which do not reduce to solitary-wave solutions. It was also obtained solitary-wave solutions for another set of these parameters as an infinite period limit of periodic solutions in terms of Weierstrass and Jacobian elliptic functions
Wave-equation dispersion inversion
Li, Jing
2016-12-08
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-equation traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-equation dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.
Solutions of Einstein's field equations
Tomonaga, Y [Utsunomiya Univ. (Japan). Faculty of Education
1978-12-01
In this paper the author investigates the Einstein's field equations of the non-vacuum case and generalizes the solution of Robertson-Walker by the three dimensional Einstein spaces. In Section 2 the author shortly generalizes the dynamic space-time of G. Lemetre and A. Friedmann by a simple transformation.
Equations for formally real meadows
Bergstra, J.A.; Bethke, I.; Ponse, A.
2015-01-01
We consider the signatures Σm = (0,1,−,+,⋅,−1) of meadows and (Σm,s) of signed meadows. We give two complete axiomatizations of the equational theories of the real numbers with respect to these signatures. In the first case, we extend the axiomatization of zero-totalized fields by a single axiom
Wave equation of hydrogen atom
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)
Transport equation and shock waves
Besnard, D.
1981-04-01
A multi-group method is derived from a one dimensional transport equation for the slowing down and spatial transport of energetic positive ions in a plasma. This method is used to calculate the behaviour of energetic charged particles in non homogeneous and non stationary plasma, and the effect of energy deposition of the particles on the heating of the plasma. In that purpose, an equation for the density of fast ions is obtained from the Fokker-Planck equation, and a closure condition for the second moment of this equation is deduced from phenomenological considerations. This method leads to a numerical method, simple and very efficient, which doesn't require much computer storage. Two types of numerical results are obtained. First, results on the slowing down of 3.5 MeV alpha particles in a 50 keV plasma plublished by Corman and al and Moses are compared with the results obtained with both our method and a Monte Carlo type method. Good agreement was obtained, even for energy deposition on the ions of the plasma. Secondly, we have calculated propagation of alpha particles heating a cold plasma. These results are in very good agreement with those given by an accurate Monte Carlo method, for both the thermal velocity, and the energy deposition in the plasma
Structural equations in language learning
Moortgat, M.J.
In categorial systems with a fixed structural component, the learning problem comes down to finding the solution for a set of typeassignment equations. A hard-wired structural component is problematic if one want to address issues of structural variation. Our starting point is a type-logical
Fractional Diffusion Equations and Anomalous Diffusion
Evangelista, Luiz Roberto; Kaminski Lenzi, Ervin
2018-01-01
Preface; 1. Mathematical preliminaries; 2. A survey of the fractional calculus; 3. From normal to anomalous diffusion; 4. Fractional diffusion equations: elementary applications; 5. Fractional diffusion equations: surface effects; 6. Fractional nonlinear diffusion equation; 7. Anomalous diffusion: anisotropic case; 8. Fractional Schrödinger equations; 9. Anomalous diffusion and impedance spectroscopy; 10. The Poisson–Nernst–Planck anomalous (PNPA) models; References; Index.
Painleve test and discrete Boltzmann equations
Euler, N.; Steeb, W.H.
1989-01-01
The Painleve test for various discrete Boltzmann equations is performed. The connection with integrability is discussed. Furthermore the Lie symmetry vector fields are derived and group-theoretical reduction of the discrete Boltzmann equations to ordinary differentiable equations is performed. Lie Backlund transformations are gained by performing the Painleve analysis for the ordinary differential equations. 16 refs
Plas, R.
1962-07-01
The author reports a study on kinetics equations for a reactor. He uses the conventional form of these equations but by using a dynamic multiplication factor. Thus, constants related to delayed neutrons are not modified by efficiency factors. The author first describes the theoretic kinetic operation of a reactor and develops the associated equations. He reports the development of equations for multiplication factors.
Algebraic entropy for differential-delay equations
Viallet, Claude M.
2014-01-01
We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.
Invariant imbedding equations for linear scattering problems
Apresyan, L.
1988-01-01
A general form of the invariant imbedding equations is investigated for the linear problem of scattering by a bounded scattering volume. The conditions for the derivability of such equations are described. It is noted that the possibility of the explicit representation of these equations for a sphere and for a layer involves the separation of variables in the unperturbed wave equation
Confinement contains condensates
Brodsky, S. J.; Roberts, C. D.; Shrock, R.
2012-01-01
been viewed as constant empirical mass scales that fill all space-time, are instead wholly contained within hadrons; i.e., they are a property of hadrons themselves and expressed, e.g., in their Bethe-Salpeter or light-front wave functions. We explain that this paradigm is consistent with empirical...
Superconductivity in a Repulsive Model
Feldman, Joel; Knoerrer, Horst; Sinclair, Robert
1997-01-01
A two-dimensional system of Fermions with classical dispersion relationand a purely repulsive delta function pair potential generates the dominant attractive coupling in the third order Bethe-Salpeter approximation for the Cooper channel. This suggests that the ground state is an l=1 superconductor....
Heavy baryon transitions and the heavy quark effective theory
Hussain, F.
1992-01-01
Heavy baryon decays are studied in the context of the Bethe-Salpeter approach to the heavy quark effective theory. A drastic reduction, in the number of independent form factors, is found. Results are presented both for heavy to heavy and heavy to light baryon decays. (orig.)
On single-time reduction in quantum field theory
Arkhipov, A.A.
1984-01-01
It is shown, how the causality and spectrality properties in qUantum field theory may help one to carry out a single-time reduction of the Bethe-Salpeter wave fUnction. The single-time reduction technique is not based on any concrete model of the quantum field theory. Axiomatic formulations underline the quantum field theory
The AGL equation from the dipole picture
Gay Ducati, M.B.; Goncalves, V.P.
1999-01-01
The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to a unitarized gluon distribution in the small x regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this paper that the AGL equation and, consequently, the GLR equation, can also be obtained from the dipole picture in the double logarithmic limit, using an evolution equation, recently proposed, which includes all multiple pomeron exchanges in the leading logarithmic approximation. Our conclusion is that the AGL equation is a good candidate for a unitarized evolution equation at small x in the DLA limit
Thermoviscous Model Equations in Nonlinear Acoustics
Rasmussen, Anders Rønne
Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....
Manhattan equation for the operational amplifier
Mishonov, Todor M.; Danchev, Victor I.; Petkov, Emil G.; Gourev, Vassil N.; Dimitrova, Iglika M.; Varonov, Albert M.
2018-01-01
A differential equation relating the voltage at the output of an operational amplifier $U_0$ and the difference between the input voltages ($U_{+}$ and $U_{-}$) has been derived. The crossover frequency $f_0$ is a parameter in this operational amplifier master equation. The formulas derived as a consequence of this equation find applications in thousands of specifications for electronic devices but as far as we know, the equation has never been published. Actually, the master equation of oper...
Reduced kinetic equations: An influence functional approach
Wio, H.S.
1985-01-01
The author discusses a scheme for obtaining reduced descriptions of multivariate kinetic equations based on the 'influence functional' method of Feynmann. It is applied to the case of Fokker-Planck equations showing the form that results for the reduced equation. The possibility of Markovian or non-Markovian reduced description is discussed. As a particular example, the reduction of the Kramers equation to the Smoluchwski equation in the limit of high friction is also discussed
Dynamical equations for the optical potential
Kowalski, K.L.
1981-01-01
Dynamical equations for the optical potential are obtained starting from a wide class of N-particle equations. This is done with arbitrary multiparticle interactions to allow adaptation to few-body models of nuclear reactions and including all effects of nucleon identity. Earlier forms of the optical potential equations are obtained as special cases. Particular emphasis is placed upon obtaining dynamical equations for the optical potential from the equations of Kouri, Levin, and Tobocman including all effects of particle identity
Group foliation of finite difference equations
Thompson, Robert; Valiquette, Francis
2018-06-01
Using the theory of equivariant moving frames, a group foliation method for invariant finite difference equations is developed. This method is analogous to the group foliation of differential equations and uses the symmetry group of the equation to decompose the solution process into two steps, called resolving and reconstruction. Our constructions are performed algorithmically and symbolically by making use of discrete recurrence relations among joint invariants. Applications to invariant finite difference equations that approximate differential equations are given.
An inverse problem in a parabolic equation
Zhilin Li
1998-11-01
Full Text Available In this paper, an inverse problem in a parabolic equation is studied. An unknown function in the equation is related to two integral equations in terms of heat kernel. One of the integral equations is well-posed while another is ill-posed. A regularization approach for constructing an approximate solution to the ill-posed integral equation is proposed. Theoretical analysis and numerical experiment are provided to support the method.
One-boson exchange model in the Tobocman-Chulick formalism
Chulick, G.S.
1988-01-01
An alternative method to the standard techniques of field theory for the derivation of few-body dynamical equations is presented here. This new formalism gives rise to a set of coupled, three-dimensional, relativistic equations which represent one or more (coupled channel) nuclear interactive processes. The particles represented by these equations are dressed and/or are composite, with mass and vertex renormalization done in a simple, straightforward manner. The n-boson Tamm-Dancoff approximation is then used to restrict to a reasonable amount the number of coupled equations to be solved. In the one-boson Tamm-Dancoff approximation, the formalism gives rise to relativistic One-Boson Exchange time-ordered perturbation theory: i.e., the basic Bonn potential. Moreover, the formalism gives the Bonn potential a firmer theoretical basis, with physical particles, and with mass and vertex renormalization systematically taken into account. The formalism was tested numerically at two levels. First, it was tested for the simple model of elastic scalar NN scattering via the exchange of a single scalar boson. The resultant phase shifts, when compared to those for the Bethe-Salpeter equation and several of its three-dimensional reductions for the same model, were found to be reasonable. Next, the formalism was tested for the same model expanded to include non-elastic NN scattering processes. Even though the resultant scattering cross-sections were not compatible to the empirical scattering cross-sections, it was possible to discern what must be included in the model to obtain qualitative agreement
Quarks and gluons in the phase diagram of quantum chromodynamics
Welzbacher, Christian Andreas
2016-07-14
In this dissertation we study the phase diagram of strongly interacting matter by approaching the theory of quantum chromodynamics in the functional approach of Dyson-Schwinger equations. With these quantum (field) equations of motions we calculate the non-perturbative quark propagator within the Matsubara formalism. We built up on previous works and extend the so-called truncation scheme, which is necessary to render the infinite tower of Dyson-Schwinger equations finite and study phase transitions of chiral symmetry and the confinement/deconfinement transition. In the first part of this thesis we discuss general aspects of quantum chromodynamics and introduce the Dyson-Schwinger equations in general and present the quark Dyson-Schwinger equation together with its counterpart for the gluon. The Bethe-Salpeter equation is introduced which is necessary to perform two-body bound state calculations. A view on the phase diagram of quantum chromodynamics is given, including the discussion of order parameter for chiral symmetry and confinement. Here we also discuss the dependence of the phase structure on the masses of the quarks. In the following we present the truncation and our results for an unquenched N{sub f} = 2+1 calculation and compare it to previous studies. We highlight some complementary details for the quark and gluon propagator and discus the resulting phase diagram, which is in agreement with previous work. Results for an equivalent of the Columbia plot and the critical surface are discussed. A systematically improved truncation, where the charm quark as a dynamical quark flavour is added, will be presented in Ch. 4. An important aspect in this investigation is the proper adjustment of the scales. This is done by matching vacuum properties of the relevant pseudoscalar mesons separately for N{sub f} = 2+1 and N f = 2+1+1 via a solution of the Bethe-Salpeter equation. A comparison of the resulting N{sub f} = 2+1 and N{sub f} = 2+1+1 phase diagram indicates
Systems of Inhomogeneous Linear Equations
Scherer, Philipp O. J.
Many problems in physics and especially computational physics involve systems of linear equations which arise e.g. from linearization of a general nonlinear problem or from discretization of differential equations. If the dimension of the system is not too large standard methods like Gaussian elimination or QR decomposition are sufficient. Systems with a tridiagonal matrix are important for cubic spline interpolation and numerical second derivatives. They can be solved very efficiently with a specialized Gaussian elimination method. Practical applications often involve very large dimensions and require iterative methods. Convergence of Jacobi and Gauss-Seidel methods is slow and can be improved by relaxation or over-relaxation. An alternative for large systems is the method of conjugate gradients.