Global nonautonomous Schrodinger flows on Hermitian locally symmetric spaces
Institute of Scientific and Technical Information of China (English)
王宏玉; 王友德
2002-01-01
In this paper, we consider the global existence of one-dimensional nonautonomous (inhomogeneous) Schrodinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the nonautonomous (inhomogeneous) Schrodinger flow from S1 into a Hermitian locally symmetric space admits a unique global smooth solution, and then we address the global existence of the Cauchy problem of inhomogeneous Heisenberg spin ferromagnet system.
20th International Workshop on Hermitian Symmetric Spaces and Submanifolds
Ohnita, Yoshihiro; Zhou, Jiazu; Kim, Byung; Lee, Hyunjin
2017-01-01
This book presents the proceedings of the 20th International Workshop on Hermitian Symmetric Spaces and Submanifolds, which was held at the Kyungpook National University from June 21 to 25, 2016. The Workshop was supported by the Research Institute of Real and Complex Manifolds (RIRCM) and the National Research Foundation of Korea (NRF). The Organizing Committee invited 30 active geometers of differential geometry and related fields from all around the globe to discuss new developments for research in the area. These proceedings provide a detailed overview of recent topics in the field of real and complex submanifolds.
Indian Academy of Sciences (India)
XIE QIONGTAO; YAN LINA; WANG LINMAO; FU JUN
2016-05-01
We investigate both the non-Hermitian parity–time-(PT-)symmetric and Hermitianasymmetric volcano potentials, and present the analytical solution in terms of the confluent Heun function. Under certain special conditions, the confluent Heun function can be terminated as a polynomial, thereby leading to certain exact analytical results. It is found that the non-Hermitian PTsymmetric volcano potentials support the normalizable and non-normalizable reflectionless stateswith real energies. The Hermitian asymmetric volcano potentials allow normalizable reflectionless states with complex energies.
Symmetric Functional Model for Extensions of Hermitian
Ryzhov, V
2006-01-01
This paper offers the functional model of a class of non-selfadjoint extensions of a Hermitian operator with equal deficiency indices. The explicit form of dilation of a dissipative extension is offered and the symmetric form of Sz.Nagy-Foia\\c{s} model as developed by B.~Pavlov is constructed. A variant of functional model for a general non-selfadjoint non-dissipative extension is formulated. We illustrate the theory by two examples: singular perturbations of the Laplace operator in~$L_2(\\Real^3)$ by a finite number of point interactions, and the Schr\\"odinger operator on the half axis~$(0, \\infty)$ in the Weyl limit circle case at infinity.
PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics
Bender, C M; Milton, K A; Bender, Carl M.; Chen, Jun-Hua; Milton, Kimball A.
2006-01-01
A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian $ix^3$ quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT-symmetric Hamiltonian. The $\\mathcal{O}(g^4)$ contribution to the ground-state energy of the Hermitian version ...
Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation
Energy Technology Data Exchange (ETDEWEB)
Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar
2016-06-15
We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit. -- Highlights: •Symmetric quadratic operators are useful models for many physical applications. •Any such operator exhibits a pseudo-Hermitian matrix representation. •Its eigenvalues are the natural frequencies of the Hamiltonian operator. •The eigenvalues may be real or complex and describe a phase transition.
A possible method for non-Hermitian and Non-PT-symmetric Hamiltonian systems.
Li, Jun-Qing; Miao, Yan-Gang; Xue, Zhao
2014-01-01
A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution) are found not to be altered by the noncommutativity.
A possible method for non-Hermitian and Non-PT-symmetric Hamiltonian systems.
Directory of Open Access Journals (Sweden)
Jun-Qing Li
Full Text Available A possible method to investigate non-Hermitian Hamiltonians is suggested through finding a Hermitian operator η+ and defining the annihilation and creation operators to be η+ -pseudo-Hermitian adjoint to each other. The operator η+ represents the η+ -pseudo-Hermiticity of Hamiltonians. As an example, a non-Hermitian and non-PT-symmetric Hamiltonian with imaginary linear coordinate and linear momentum terms is constructed and analyzed in detail. The operator η+ is found, based on which, a real spectrum and a positive-definite inner product, together with the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution, are obtained for the non-Hermitian and non-PT-symmetric Hamiltonian. Moreover, this Hamiltonian turns out to be coupled when it is extended to the canonical noncommutative space with noncommutative spatial coordinate operators and noncommutative momentum operators as well. Our method is applicable to the coupled Hamiltonian. Then the first and second order noncommutative corrections of energy levels are calculated, and in particular the reality of energy spectra, the positive-definiteness of inner products, and the related properties (the probability explanation of wave functions, the orthogonality of eigenstates, and the unitarity of time evolution are found not to be altered by the noncommutativity.
Non Hermitian quantum mechanics in non commutative space
Giri, Pulak Ranjan
2008-01-01
We study non Hermitian quantum systems in noncommutative space as well as a \\cal{PT} symmetric deformation of this space. Specifically, a \\mathcal{PT}-symmetric harmonic oscillator together with iC(x_1+x_2) interaction is discussed in this space and solutions are obtained. It is shown that in the \\cal{PT} deformed noncommutative space the Hamiltonian may or may not possess real eigenvalues depending on the choice of the noncommutative parameters. However, it is shown that in standard noncommutative space, the iC(x_1+x_2) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not \\mathcal{PT}-symmetric. A complex interacting anisotropic oscillator system has also been discussed.
Hermitian realizations of κ-Minkowski space-time
Kovačević, Domagoj; Meljanac, Stjepan; Samsarov, Andjelo; Škoda, Zoran
2015-01-01
General realizations, star products and plane waves for κ-Minkowski space-time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.
Non-Hermitian supersymmetry and singular PT symmetrized oscillators
Znojil, M
2002-01-01
SUSY partnership between singular potentials often breaks down. Via regularization it can be restored on certain ad hoc subspaces of Hilbert space [Das and Pernice, Nucl. Phys. B 561 (1999) 357]. Within the naturally complexified (so called PT symmetric) quantum mechanics we show how SUSY between strongly singular harmonic oscillators can completely be re-established. Our recipe leads to a new form of the bosonic creation and annihilation operators and proves continuous near the usual regular (i.e., linear harmonic) limit.
Momentum-independent reflectionless transmission in the non-Hermitian time-reversal symmetric system
Energy Technology Data Exchange (ETDEWEB)
Zhang, X.Z.; Song, Z., E-mail: nkquantum@gmail.com
2013-12-15
We theoretically study the non-Hermitian systems, the non-Hermiticity of which arises from the unequal hopping amplitude (UHA) dimers. The distinguishing features of these models are that they have full real spectra if all of the eigenvectors are time-reversal (T) symmetric rather than parity-time-reversal (PT) symmetric, and that their Hermitian counterparts are shown to be an experimentally accessible system, which have the same topological structures as that of the original ones but modulated hopping amplitudes within the unbroken region. Under the reflectionless transmission condition, the scattering behavior of momentum-independent reflectionless transmission (RT) can be achieved in the concerned non-Hermitian system. This peculiar feature indicates that, for a certain class of non-Hermitian systems with a balanced combination of the RT dimers, the defects can appear fully invisible to an outside observer. -- Highlights: •We investigate the non-Hermitian system with time reversal symmetry. •The Hermitian counterpart is experimentally accessible system. •The behavior of momentum-independent reflectionless transmission can be achieved. •A balanced combination of reflectionless transmission dimers leads to invisibility. •It paves an alternative way for the design of invisible cloaking devices.
Pseudo-Hermitian Systems with PT-Symmetry: Degeneracy and Krein Space
Choutri, B.; Cherbal, O.; Ighezou, F. Z.; Drir, M.
2017-02-01
We show in the present paper that pseudo-Hermitian Hamiltonian systems with even PT-symmetry (P2=1,T2=1) admit a degeneracy structure. This kind of degeneracy is expected traditionally in the odd PT-symmetric systems (P2=1,T2=-1) which is appropriate to the fermions (Scolarici and Solombrino, Phys. Lett. A 303, 239 2002; Jones-Smith and Mathur, Phys. Rev. A 82, 042101 2010). We establish that the pseudo-Hermitian Hamiltonians with even PT-symmetry admit a degeneracy structure if the operator PT anticommutes with the metric operator η σ which is necessarily indefinite. We also show that the Krein space formulation of the Hilbert space is the convenient framework for the implementation of unbroken PT-symmetry. These general results are illustrated with great details for four-level pseudo-Hermitian Hamiltonian with even PT -symmetry.
Green's Function of a General PT-Symmetric Non-Hermitian Non-central Potential
Mourya, Brijesh Kumar
2016-01-01
We study the path integral solution of a system of particle moving in certain class of PT symmetric non-Hermitian and non-central potential. The Hamil- tonian of the system is converted to a separable Hamiltonian of Liouville type in parabolic coordinates and is further mapped into a Hamiltonian corresponding to two 2-dimensional simple harmonic oscillators (SHOs). Thus the explicit Green's functions for a general non-central PT symmetric non hermitian potential are cal- culated in terms of that of 2d SHOs. The entire spectrum for this three dimensional system is shown to be always real leading to the fact that the system remains in unbroken PT phase all the time.
Non-Hermitian Neutrino Oscillations in Matter with PT Symmetric Hamiltonians
Ohlsson, Tommy
2015-01-01
We introduce and develop a novel approach to extend the ordinary two-flavor neutrino oscillation formalism in matter using a non-Hermitian PT symmetric effective Hamiltonian. The condition of PT symmetry is weaker and less mathematical than that of Hermicity, but more physical, and such an extension of the formalism can give rise to sub-leading effects in neutrino flavor transitions similar to the effects by so-called non-standard neutrino interactions. We derive the necessary conditions for the spectrum of the effective Hamiltonian to be real as well as the mappings between the fundamental and effective parameters including the corresponding two-flavor neutrino transition probability. We find that the effective leptonic mixing must always be maximal and that the real spectrum of the effective Hamiltonian will depend on all new fundamental parameters introduced in the non-Hermitian PT symmetric extension of the usual neutrino oscillation formalism.
Various scattering properties of a new PT-symmetric non-Hermitian potential
Energy Technology Data Exchange (ETDEWEB)
Ghatak, Ananya, E-mail: gananya04@gmail.com [Department of Physics, Banaras Hindu University, Varanasi-221005 (India); Mandal, Raka Dona Ray, E-mail: rakad.ray@gmail.com [Department of Physics, Rajghat Besant School, Varanasi-221001 (India); Mandal, Bhabani Prasad, E-mail: bhabani.mandal@gmail.com [Department of Physics, Banaras Hindu University, Varanasi-221005 (India)
2013-09-15
We complexify a 1-d potential V(x)=V{sub 0}cosh{sup 2}μ(tanh[(x−μd)/d]+tanh(μ)){sup 2} which exhibits bound, reflecting and free states to study various properties of a non-Hermitian system. This potential turns out a PT-symmetric non-Hermitian potential when one of the parameters (μ,d) becomes imaginary. For the case of μ→iμ, we have an entire real bound state spectrum. Explicit scattering states are constructed to show reciprocity at certain discrete values of energy even though the potential is not parity symmetric. Coexistence of deep energy minima of transmissivity with the multiple spectral singularities (MSS) is observed. We further show that this potential becomes invisible from the left (or right) at certain discrete energies. The penetrating states in the other case (d→id) are always reciprocal even though it is PT-invariant and no spectral singularity (SS) is present in this case. The presence of MSS and reflectionlessness is also discussed for the free states in the later case. -- Highlights: •Existence of multiple spectral singularities (MSS) in PT-symmetric non-Hermitian system is shown. •Reciprocity is restored at discrete positive energies even for parity non-invariant complex system. •Co-existence of MSS with deep energy minima of transitivity is obtained. •Possibilities of both unidirectional and bidirectional invisibility are explored for a non-Hermitian system. •Penetrating states are shown to be reciprocal for all energies for PT-symmetric system.
Robust PT-symmetric chain and properties of its Hermitian counterpart
Joglekar, Yogesh N.; Saxena, Avadh
2011-05-01
We study the properties of a parity- and time-reversal- (PT) symmetric tight-binding chain of size N with position-dependent hopping amplitude. In contrast to the fragile PT-symmetric phase of a chain with constant hopping and imaginary impurity potentials, we show that, under very general conditions, our model is always in the PT-symmetric phase. We numerically obtain the energy spectrum and the density of states of such a chain, and show that they are widely tunable. By studying the size dependence of inverse participation ratios, we show that although the chain is not translationally invariant, most of its eigenstates are extended. Our results indicate that tight-binding models with non-Hermitian, PT-symmetric hopping have a robust PT-symmetric phase and rich dynamics which may be explored in coupled waveguides.
Symmetric Spaces in Supergravity
Ferrara, Sergio
2008-01-01
We exploit the relation among irreducible Riemannian globally symmetric spaces (IRGS) and supergravity theories in 3, 4 and 5 space-time dimensions. IRGS appear as scalar manifolds of the theories, as well as moduli spaces of the various classes of solutions to the classical extremal black hole Attractor Equations. Relations with Jordan algebras of degree three and four are also outlined.
Non-Hermitian neutrino oscillations in matter with PT symmetric Hamiltonians
Ohlsson, Tommy
2016-03-01
We introduce and develop a novel approach to extend the ordinary two-flavor neutrino oscillation formalism in matter using a non-Hermitian PT symmetric effective Hamiltonian. The condition of PT symmetry is weaker and less mathematical than that of hermicity, but more physical, and such an extension of the formalism can give rise to sub-leading effects in neutrino flavor transitions similar to the effects by so-called non-standard neutrino interactions. We derive the necessary conditions for the spectrum of the effective Hamiltonian to be real as well as the mappings between the fundamental and effective parameters. We find that the real spectrum of the effective Hamiltonian will depend on all new fundamental parameters introduced in the non-Hermitian PT symmetric extension of the usual neutrino oscillation formalism and that either i) the spectrum is exact and the effective leptonic mixing must always be maximal or ii) the spectrum is approximate and all new fundamental parameters must be small.
Non-Hermitian ${\\cal PT}$-symmetric relativistic quantum theory in an intensive magnetic field
Rodionov, V N
2016-01-01
We develop relativistic non-Hermitian quantum theory and its application to neutrino physics in a strong magnetic field. It is well known, that one of the fundamental postulates of quantum theory is the requirement of Hermiticity of physical parameters. This condition not only guarantees the reality of the eigenvalues of Hamiltonian operators, but also implies the preservation of the probabilities of the considered quantum processes. However as it was shown relatively recently (Bender, Boettcher 1998), Hermiticity is a sufficient but it is not a necessary condition. It turned out that among non-Hermitian Hamiltonians it is possible to allocate a number of such which have real energy spectra and can ensure the development of systems over time with preserving unitarity. This type of Hamiltonians includes so-called parity-time (${\\cal PT}$) symmetric models which is already used in various fields of modern physics. The most developed in this respect are models, which used in the field of ${\\cal PT}$-symmetric op...
Bender, C M; Chen, J H; Jones, H F; Milton, K A; Ogilvie, M C; Bender, Carl M.; Brody, Dorje C.; Chen, Jun-Hua; Jones, Hugh F.; Milton, Kimball A.; Ogilvie, Michael C.
2006-01-01
In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\\cP\\cT$-symmetric wrong-sign quartic Hamiltonian $H=\\half p^2-gx^4$ has the same spectrum as the conventional Hermitian Hamiltonian $\\tilde H=\\half p^2+4g x^4-\\sqrt{2g} x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian $\\cP\\cT$-symmetric Hamiltonian. This anomaly in the Hermitian form of a $\\cP\\cT$-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into $H$. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to...
Problem of the coexistence of several non-Hermitian observables in PT -symmetric quantum mechanics
Znojil, Miloslav; Semorádová, Iveta; RÅ¯žička, František; Moulla, Hafida; Leghrib, Ilhem
2017-04-01
During recent developments in quantum theory it has been clarified that observable quantities (such as energy and position) may be represented by operators Λ (with real spectra) that are manifestly non-Hermitian in a preselected friendly Hilbert space H(F ). The consistency of these models is known to require an upgrade of the inner product, i.e., mathematically speaking, a transition H(F )→H(S ) to another, standard, Hilbert space. We prove that whenever we are given more than one candidate for an observable (i.e., say, two operators Λ0 and Λ1) in advance, such an upgrade need not exist in general.
EQUIFOCAL HYPERSURFACES IN SYMMETRIC SPACES
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
This note investigates the multiplicity problem of principal curvatures of equifocal hyper surfaces in simply connected rank 1 symmetric spaces. Using Clifford representation theory, and the author also constructs infinitely many equifocal hypersurfaces in the symmetric spaces.
Harmonic analysis on symmetric spaces
Terras, Audrey
This text explores the geometry and analysis of higher rank analogues of the symmetric spaces introduced in volume one. To illuminate both the parallels and differences of the higher rank theory, the space of positive matrices is treated in a manner mirroring that of the upper-half space in volume one. This concrete example furnishes motivation for the general theory of noncompact symmetric spaces, which is outlined in the final chapter. The book emphasizes motivation and comprehensibility, concrete examples and explicit computations (by pen and paper, and by computer), history, and, above all, applications in mathematics, statistics, physics, and engineering. The second edition includes new sections on Donald St. P. Richards’s central limit theorem for O(n)-invariant random variables on the symmetric space of GL(n, R), on random matrix theory, and on advances in the theory of automorphic forms on arithmetic groups.
Vassiliev Invariants from Symmetric Spaces
DEFF Research Database (Denmark)
Biswas, Indranil; Gammelgaard, Niels Leth
We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are ......, they are exactly characterized by having the symmetries of the Riemann curvature tensor....
The antipodal sets of compact symmetric spaces
National Research Council Canada - National Science Library
Liu, Xingda; Deng, Shaoqiang
2014-01-01
We study the antipodal set of a point in a compact Riemannian symmetric space. It turns out that we can give an explicit description of the antipodal set of a point in any connected simply connected compact Riemannian symmetric space...
A K-theoretic approach to the classification of symmetric spaces
Bohle, Dennis
2011-01-01
We show that the classification of the symmetric spaces can be achieved by K-theoretical methods. We focus on Hermitian symmetric spaces of non-compact type, and define K-theory for JB*-triples along the lines of C*-theory. K-groups have to be provided with further invariants in order to classify. Among these are the cycles obtained from so called grids, intimately connected to the root systems of an underlying Lie-algebra and thus reminiscent of the classical classification scheme.
Entanglement in non-Hermitian quantum theory
Indian Academy of Sciences (India)
Arun K Pati
2009-09-01
Entanglement is one of the key features of quantum world that has no classical counterpart. This arises due to the linear superposition principle and the tensor product structure of the Hilbert space when we deal with multiparticle systems. In this paper, we will introduce the notion of entanglement for quantum systems that are governed by non-Hermitian yet $\\mathcal{PT}$ -symmetric Hamiltonians. We will show that maximally entangled states in usual quantum theory behave like non-maximally entangled states in $\\mathcal{PT}$ -symmetric quantum theory. Furthermore, we will show how to create entanglement between two $\\mathcal{PT}$ qubits using non-Hermitian Hamiltonians and discuss the entangling capability of such interaction Hamiltonians that are non-Hermitian in nature.
On Finite $J$-Hermitian Quantum Mechanics
Lee, Sungwook
2014-01-01
In his recent paper arXiv:1312.7738, the author discussed $J$-Hermitian quantum mechanics and showed that $PT$-symmetric quantum mechanics is essentially $J$-Hermitian quantum mechanics. In this paper, the author discusses finite $J$-Hermitian quantum mechanics which is derived naturally from its continuum one and its relationship with finite $PT$-symmetric quantum mechanics.
Advanced Topic: Quasi-Hermitian Quantum Systems
Curtright, Thomas L.; Fairlie, David B.; Zachos, Cosmas K.
2014-11-01
So far, the discussion has limited itself to hermitian operators and systems. However, superficially non-hermitian Hamiltonian quantum systems are also of considerable current interest, especially in the context of PT symmetric models [Ben07, Mos05], although many of the main ideas appeared earlier [SGH92, XA96]. For such systems, the Hilbert space structure is at first sight very different from that for hermitian Hamiltonian systems, inasmuch as the dual wavefunctions are not just the complex conjugates of the wavefunctions, or, equivalently, the Hilbert space metric is not the usual one. While it is possible to keep most of the compact Dirac notation in analyzing such systems, here we work with explicit functions and avoid abstract notation, in the hope to fully expose all the structure, rather than to hide it...
Random matrix theory and symmetric spaces
Energy Technology Data Exchange (ETDEWEB)
Caselle, M.; Magnea, U
2004-05-01
In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important results can be obtained from this identification. In particular the Cartan classification of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classification of random matrix ensembles. The review is organized into two main parts. In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories. In Part II we discuss various applications of symmetric spaces to random matrix theories and in particular the new classification of disordered systems derived from the classification of symmetric spaces. We also review how the mapping from integrable Calogero-Sutherland models to symmetric spaces can be used in the theory of random matrices, with particular consequences for quantum transport problems. We conclude indicating some interesting new directions of research based on these identifications.
Martingale Rosenthal inequalities in symmetric spaces
Energy Technology Data Exchange (ETDEWEB)
Astashkin, S V [Samara State University, Samara (Russian Federation)
2014-12-31
We establish inequalities similar to the classical Rosenthal inequalities for sequences of martingale differences in general symmetric spaces; a central role is played here by the predictable quadratic characteristic of a martingale. Bibliography: 26 titles.
15th International Conference on Non-Hermitian Hamiltonians in Quantum Physics
Passante, Roberto; Trapani, Camillo
2016-01-01
This book presents the Proceedings of the 15th International Conference on Non-Hermitian Hamiltonians in Quantum Physics, held in Palermo, Italy, from 18 to 23 May 2015. Non-Hermitian operators, and non-Hermitian Hamiltonians in particular, have recently received considerable attention from both the mathematics and physics communities. There has been a growing interest in non-Hermitian Hamiltonians in quantum physics since the discovery that PT-symmetric Hamiltonians can have a real spectrum and thus a physical relevance. The main subjects considered in this book include: PT-symmetry in quantum physics, PT-optics, Spectral singularities and spectral techniques, Indefinite-metric theories, Open quantum systems, Krein space methods, and Biorthogonal systems and applications. The book also provides a summary of recent advances in pseudo-Hermitian Hamiltonians and PT-symmetric Hamiltonians, as well as their applications in quantum physics and in the theory of open quantum systems.
Qp-spaces on bounded symmetric domains
Directory of Open Access Journals (Sweden)
Jonathan Arazy
2008-01-01
Full Text Available We generalize the theory of Qp spaces, introduced on the unit disc in 1995 by Aulaskari, Xiao and Zhao, to bounded symmetric domains in Cd, as well as to analogous Moebius-invariant function spaces and Bloch spaces defined using higher order derivatives; the latter generalization contains new results even in the original context of the unit disc.
On noncommutative spherically symmetric spaces
Energy Technology Data Exchange (ETDEWEB)
Buric, Maja [University of Belgrade, Faculty of Physics, P.O. Box 44, Belgrade (Serbia); Madore, John [Laboratoire de Physique Theorique, Orsay (France)
2014-03-15
Two families of noncommutative extensions are given of a general space-time metric with spherical symmetry, both based on the matrix truncation of the functions on the sphere of symmetry. The first family uses the truncation to foliate space as an infinite set of spheres, and it is of dimension four and necessarily time-dependent; the second can be time-dependent or static, is of dimension five, and uses the truncation to foliate the internal space. (orig.)
On noncommutative spherically symmetric spaces
Buric, Maja
2014-01-01
Two families of noncommutative extensions are given of a general space-time metric with spherical symmetry, both based on the matrix truncation of the functions on the sphere of symmetry. The first family uses the truncation to foliate space as an infinite set of spheres, is of dimension four and necessarily time-dependent; the second can be time-dependent or static, is of dimension five and uses the truncation to foliate the internal space.
Fourier inversion on a reductive symmetric space
Ban, E.P. van den
2001-01-01
Let X be a semisimple symmetric space. In previous papers, [8] and [9], we have dened an explicit Fourier transform for X and shown that this transform is injective on the space C 1 c (X) ofcompactly supported smooth functions on X. In the present paper, which is a continuation of these papers, we e
Faster than Hermitian Time Evolution
Directory of Open Access Journals (Sweden)
Carl M. Bender
2007-12-01
Full Text Available For any pair of quantum states, an initial state $|I angle$ and afinal quantum state $|F angle$, in a Hilbert space, there are many Hamiltonians $H$ under which $|I angle$ evolves into $|F angle$. Let us impose the constraint that the difference between the largest and smallest eigenvalues of $H$, $E_{max}$ and $E_{min}$, is held fixed. We can then determine the Hamiltonian $H$ that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time $au$. For Hermitian Hamiltonians, $au$ has a nonzero lower bound. However, amongnon-Hermitian ${cal PT}$-symmetric Hamiltonians satisfying the same energy constraint, $au$ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of $au$ can be made arbitrarily small because for ${cal PT}$-symmetric Hamiltonians the path from the vector $|I angle$ to the vector $|F angle$, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here is similar to that in general relativity in whichthe distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.
Bloch spaces on bounded symmetric domains in complex Banach spaces
Institute of Scientific and Technical Information of China (English)
DENG; Fangwen
2006-01-01
We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied.
On integrability of strings on symmetric spaces
Energy Technology Data Exchange (ETDEWEB)
Wulff, Linus [Blackett Laboratory, Imperial College,London SW7 2AZ (United Kingdom)
2015-09-17
In the absence of NSNS three-form flux the bosonic string on a symmetric space is described by a symmetric space coset sigma-model. Such models are known to be classically integrable. We show that the integrability extends also to cases with non-zero NSNS flux (respecting the isometries) provided that the flux satisfies a condition of the form H{sub abc}H{sup cde}∼R{sub ab}{sup de}. We then turn our attention to the type II Green-Schwarz superstring on a symmetric space. We prove that if the space preserves some supersymmetry there exists a truncation of the full superspace to a supercoset space and derive the general form of the superisometry algebra. In the case of vanishing NSNS flux the corresponding supercoset sigma-model for the string is known to be integrable. We prove that the integrability extends to the full string by augmenting the supercoset Lax connection with terms involving the fermions which are not captured by the supercoset model. The construction is carried out to quadratic order in these fermions. This proves the integrability of strings on symmetric spaces supported by RR flux which preserve any non-zero amount of supersymmetry. Finally we also construct Lax connections for some supercoset models with non-zero NSNS flux describing strings in AdS{sub 2,3}×S{sup 2,3}×S{sup 2,3}×T{sup 2,3,4} backgrounds preserving eight supersymmetries.
Adjacency Preserving Bijection Maps of Hermitian Matrices over any Division Ring with an Involution
Institute of Scientific and Technical Information of China (English)
Li Ping HUANG
2007-01-01
Let D be any division ring with an involution, H(D) be the space of all n × n hermitian matrices over D. Two hermitian matrices A and B are said to be adjacent if rank(A - B) =1. It is proved that if ψ is a bijective map from H(D)(n ≥ 2) to itself such that ψ preserves the adjacency,then ψ-1 also preserves the adjacency. Moreover, if H(D)≠F3(F2), then ψ preserves the arithmetic distance. Thus, an open problem posed by Wan Zhe-Xian is answered for geometry of symmetric and hermitian matrices.
Automorphism groups of causal symmetric spaces of Cayley type and bounded symmetric domains
Institute of Scientific and Technical Information of China (English)
Soji; Kaneyuki
2005-01-01
Symmetric spaces of Cayley type are a higher dimensional analogue of a onesheeted hyperboloid in R3. They form an important class of causal symmetric spaces. To a symmetric space of Cayley type M, one can associate a bounded symmetric domain of tube type D. We determine the full causal automorphism group of M. This clarifies the relation between the causal automorphism group and the holomorphic automorphism group of D.
Pseudo-Z symmetric space-times
Energy Technology Data Exchange (ETDEWEB)
Mantica, Carlo Alberto, E-mail: carloalberto.mantica@libero.it [Physics Department, Università degli Studi di Milano, Via Celoria 16, 20133 Milano (Italy); Suh, Young Jin, E-mail: yjsuh@knu.ac.kr [Department of Mathematics, Kyungpook National University, Taegu 702-701 (Korea, Republic of)
2014-04-15
In this paper, we investigate Pseudo-Z symmetric space-time manifolds. First, we deal with elementary properties showing that the associated form A{sub k} is closed: in the case the Ricci tensor results to be Weyl compatible. This notion was recently introduced by one of the present authors. The consequences of the Weyl compatibility on the magnetic part of the Weyl tensor are pointed out. This determines the Petrov types of such space times. Finally, we investigate some interesting properties of (PZS){sub 4} space-time; in particular, we take into consideration perfect fluid and scalar field space-time, and interesting properties are pointed out, including the Petrov classification. In the case of scalar field space-time, it is shown that the scalar field satisfies a generalized eikonal equation. Further, it is shown that the integral curves of the gradient field are geodesics. A classical method to find a general integral is presented.
Kashiwara-Vergne-Rouviere methods for symmetric spaces
Torossian, Charles
2002-01-01
This article follows our previous work on Campbell-Hausdorff formula. We study the case of symmetric spaces. We recover, by using a Kontsevich's deformation of the Baker-Campbell-Hausdorff formula, Rouviere's results on the convolution of invariant distributions, for solvable symmetric spaces and "very symmetric spaces".
Kashiwara-Vergne-Rouviere methods for symmetric spaces
Torossian, Charles
2002-01-01
This article follows our previous work on Campbell-Hausdorff formula. We study the case of symmetric spaces. We recover, by using a Kontsevich's deformation of the Baker-Campbell-Hausdorff formula, Rouviere's results on the convolution of invariant distributions, for solvable symmetric spaces and "very symmetric spaces".
CANONICAL EXTENSIONS OF SYMMETRIC LINEAR RELATIONS
Sandovici, Adrian; Davidson, KR; Gaspar, D; Stratila, S; Timotin, D; Vasilescu, FH
2006-01-01
The concept of canonical extension of Hermitian operators has been recently introduced by A. Kuzhel. This paper deals with a generalization of this notion to the case of symmetric linear relations. Namely, canonical regular extensions of symmetric linear relations in Hilbert spaces are studied. The
Quasiclassical analysis of Bloch oscillations in non-Hermitian tight-binding lattices
Graefe, E M; Rush, A
2016-01-01
Many features of Bloch oscillations in one-dimensional quantum lattices with a static force can be described by quasiclassical considerations for example by means of the acceleration theorem, at least for Hermitian systems. Here the quasiclassical approach is extended to non-Hermitian lattices, which are of increasing interest. The analysis is based on a generalised non-Hermitian phase space dynamics developed recently. Applications to a single-band tight-binding system demonstrate that many features of the quantum dynamics can be understood from this classical description qualitatively and even quantitatively. Two non-Hermitian and $PT$-symmetric examples are studied, a Hatano-Nelson lattice with real coupling constants and a system with purely imaginary couplings, both for initially localised states in space or in momentum. It is shown that the time-evolution of the norm of the wave packet and the expectation values of position and momentum can be described in a classical picture.
Rodionov, V N
2013-01-01
The modified Dirac equations for the massive particles with the replacement of the physical mass $m$ with the help of the relation $m\\rightarrow m_1+ \\gamma_5 m_2$ are investigated. It is shown that for a fermion theory with a $\\gamma_5$-mass term, the limiting of the mass specter by the value $ m_{max}= {m_1}^2/2m_2$ takes place. In this case the different regions of the unbroken $\\cal PT$ symmetry may be expressed by means of the restriction of the physical mass $m\\leq m_{max}$. It should be noted that in the approach which was developed by C.Bender et al. for the $\\cal PT$-symmetric version of the massive Thirring model with $\\gamma_5$-mass term, the region of the unbroken $\\cal PT$-symmetry was found in the form $m_1\\geq m_2$ \\cite{ft12}. However on the basis of the mass limitation $m\\leq m_{max}$ we obtain that the domain $m_1\\geq m_2$ consists of two different parametric sectors: i) $0\\leq m_2 \\leq m_1/\\sqrt{2}$ -this values of mass parameters $m_1,m_2$ correspond to the traditional particles for which ...
Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces
Directory of Open Access Journals (Sweden)
Doug Pickrell
2008-10-01
Full Text Available This paper is a sequel to [Caine A., Pickrell D., Int. Math. Res. Not., to appear, arXiv:0710.4484], where we studied the Hamiltonian systems which arise from the Evens-Lu construction of homogeneous Poisson structures on both compact and noncompact type symmetric spaces. In this paper we consider loop space analogues. Many of the results extend in a relatively routine way to the loop space setting, but new issues emerge. The main point of this paper is to spell out the meaning of the results, especially in the SU(2 case. Applications include integral formulas and factorizations for Toeplitz determinants.
Integrable Deformations of Strings on Symmetric Spaces
Hollowood, Timothy J; Schmidtt, David M
2014-01-01
A general class of deformations of integrable sigma-models with symmetric space F/G target-spaces are found. These deformations involve defining the non-abelian T dual of the sigma-model and then replacing the coupling of the Lagrange multiplier imposing flatness with a gauged F/F WZW model. The original sigma-model is obtained in the limit of large level. The resulting deformed theories are shown to preserve both integrability and the equations-of-motion, but involve a deformation of the symplectic structure. It is shown that this deformed symplectic structure involves a linear combination of the original Poisson bracket and a generalization of the Faddeev-Reshetikhin Poisson bracket which we show can be re-expressed as two decoupled F current algebras. It is then shown that the deformation can be incorporated into the classical model of strings on R x F/G via a generalization of the Pohlmeyer reduction. In this case, in the limit of large sigma-model coupling it is shown that the theory becomes the relativi...
Wrapping Brownian motion and heat kernels II: symmetric spaces
Maher, David G
2010-01-01
In this paper we extend our previous results on wrapping Brownian motion and heat kernels onto compact Lie groups to various symmetric spaces, where a global generalisation of Rouvi\\`ere's formula and the $e$-function are considered. Additionally, we extend some of our results to complex Lie groups, and certain non-compact symmetric spaces.
Fourier transforms of spherical distributions on compact symmetric spaces
Olafsson, Gestur; Schlichtkrull, Henrik
2008-01-01
In our previous articles "A local Paley-Wiener theorem for compact symmetric spaces", Adv. Math. 218 (2008), 202--215, and "Fourier series on compact symmetric spaces" (submitted) we studied Fourier series on a compact symmetric space M=U/K. In particular, we proved a Paley-Wiener type theorem for the smooth functions on M, which have sufficiently small support and are K-invariant, respectively K-finite. In this article we extend those results to K-invariant distributions on M. We show that t...
Saraceno, Marcos; Ermann, Leonardo; Cormick, Cecilia
2017-03-01
The problem of finding symmetric informationally complete positive-operator-valued-measures (SIC-POVMs) has been solved numerically for all dimensions d up to 67 [A. J. Scott and M. Grassl, J. Math. Phys. 51, 042203 (2010), 10.1063/1.3374022], but a general proof of existence is still lacking. For each dimension, it was shown that it is possible to find a SIC-POVM that is generated from a fiducial state upon application of the operators of the Heisenberg-Weyl group. We draw on the numerically determined fiducial states to study their phase-space features, as displayed by the characteristic function and the Wigner, Bargmann, and Husimi representations, adapted to a Hilbert space of finite dimension. We analyze the phase-space localization of fiducial states, and observe that the SIC-POVM condition is equivalent to a maximal delocalization property. Finally, we explore the consequences in phase space of the conjectured Zauner symmetry. In particular, we construct a Hermitian operator commuting with this symmetry that leads to a representation of fiducial states in terms of eigenfunctions with definite semiclassical features.
Some questions on spectrum and arithmetic of locally symmetric spaces
Rajan, C S
2010-01-01
We consider the question that the spectrum and arithmetic of locally symmetric spaces defined by congruent arithmetical lattices should mutually determine each other. We frame these questions in the context of automorphic representations.
The Phase Space Formulation of Time-Symmetric Quantum Mechanics
National Research Council Canada - National Science Library
Charlyne de Gosson; Maurice A. de Gosson
2015-01-01
Time-symmetric quantum mechanics can be described in the Weyl–Wigner–Moyal phase space formalism by using the properties of the cross-terms appearing in the Wigner distribution of a sum of states...
Polynomial Estimates for c-functions on Reductive Symmetric Spaces
DEFF Research Database (Denmark)
van den Ban, Erik; Schlichtkrull, Henrik
2012-01-01
The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions.......The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions....
Uniqueness of symmetric basis in quasi-Banach spaces
Albiac, F.; Leránoz, C.
2008-12-01
We show that if X is a nonlocally convex natural quasi-Banach space with symmetric basis whose Banach envelope is isomorphic to l1, then all symmetric bases of X are equivalent. The scope of this result is quite ample since the Banach envelopes of natural quasi-Banach spaces with basis always exhibit an l1-like behavior, in the sense that they contain copies of 's uniformly complemented.
(M-theory-)Killing spinors on symmetric spaces
Energy Technology Data Exchange (ETDEWEB)
Hustler, Noel, E-mail: n.hustler@ed.ac.uk [Maxwell and Tait Institutes, School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JZ, Scotland (United Kingdom); Lischewski, Andree, E-mail: lischews@mathematik.hu-berlin.de [Humboldt-Universität zu Berlin, Institut für Mathematik, Rudower Chaussee 25, Room 1.310, D12489 Berlin (Germany)
2015-08-15
We show how the theory of invariant principal bundle connections for reductive homogeneous spaces can be applied to determine the holonomy of generalised Killing spinor covariant derivatives of the form D = ∇ + Ω in a purely algebraic and algorithmic way, where Ω : TM → Λ{sup ∗}(TM) is a left-invariant homomorphism. Specialising this to the case of symmetric M-theory backgrounds (i.e., (M, g, F) with (M, g) an eleven-dimensional Lorentzian (locally) symmetric space and F an invariant closed 4-form), we derive several criteria for such a background to preserve some supersymmetry and consequently find all supersymmetric symmetric M-theory backgrounds.
Symmetric spaces and the Kashiwara-Vergne method
Rouvière, François
2014-01-01
Gathering and updating results scattered in journal articles over thirty years, this self-contained monograph gives a comprehensive introduction to the subject. Its goal is to: - motivate and explain the method for general Lie groups, reducing the proof of deep results in invariant analysis to the verification of two formal Lie bracket identities related to the Campbell-Hausdorff formula (the "Kashiwara-Vergne conjecture"); - give a detailed proof of the conjecture for quadratic and solvable Lie algebras, which is relatively elementary; - extend the method to symmetric spaces; here an obstruction appears, embodied in a single remarkable object called an "e-function"; - explain the role of this function in invariant analysis on symmetric spaces, its relation to invariant differential operators, mean value operators and spherical functions; - give an explicit e-function for rank one spaces (the hyperbolic spaces); - construct an e-function for general symmetric spaces, in the spirit of Kashiwara and Vergne's or...
Lagrangian formulation of symmetric space sine-Gordon models
Bakas, Ioannis; Shin, H J; Park, Q Han
1996-01-01
The symmetric space sine-Gordon models arise by conformal reduction of ordinary 2-dim \\sigma-models, and they are integrable exhibiting a black-hole type metric in target space. We provide a Lagrangian formulation of these systems by considering a triplet of Lie groups F \\supset G \\supset H. We show that for every symmetric space F/G, the generalized sine-Gordon models can be derived from the G/H WZW action, plus a potential term that is algebraically specified. Thus, the symmetric space sine-Gordon models describe certain integrable perturbations of coset conformal field theories at the classical level. We also briefly discuss their vacuum structure, Backlund transformations, and soliton solutions.
Convergence of Symmetric Diffusions on Wiener Spaces
Institute of Scientific and Technical Information of China (English)
A.Posilicano; T.S.Zhang
2004-01-01
In this paper,we study the distorted Ornstein-Uhlenbeck processes associated with given densities on an abstract Wiener space.We prove that the laws of distorted Ornstein-Uhlenbeck processes converge in total variation norm if the densities converge in Sobolev space D 1/2.
BiHermitian supersymmetric quantum mechanics
Energy Technology Data Exchange (ETDEWEB)
Zucchini, Roberto [Dipartimento di Fisica, Universita degli Studi di Bologna, V Irnerio 46, I-40126 Bologna (Italy)
2007-04-21
BiHermitian geometry, discovered long ago by Gates, Hull and Rocek, is the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. In this paper, we work out supersymmetric quantum mechanics for a biHermitian target space. We display the full supersymmetry of the model and illustrate in detail its quantization procedure. Finally, we show that the quantized model reproduces the Hodge theory for compact twisted generalized Kaehler manifolds recently developed by Gualtieri. This allows us to recover and put in a broader context the results on the biHermitian topological sigma models obtained by Kapustin and Li.
BiHermitian Supersymmetric Quantum Mechanics
Zucchini, R
2006-01-01
BiHermitian geometry, discovered long ago by Gates, Hull and Rocek, is the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. In this paper, we work out supersymmetric quantum mechanics for a biHermitian target space. We display the full supersymmetry of the model and illustrate in detail its quantization procedure. Finally, we show that the quantized model reproduces the Hodge theory for compact twisted generalized Kaehler manifolds recently developed by Gualtieri. This allows us to recover and put in a broader context the results on the biHermitian topological sigma models obtained by Kapustin and Li.
BiHermitian supersymmetric quantum mechanics
Zucchini, Roberto
2007-04-01
BiHermitian geometry, discovered long ago by Gates, Hull and Rocek, is the most general sigma model target space geometry allowing for (2, 2) world sheet supersymmetry. In this paper, we work out supersymmetric quantum mechanics for a biHermitian target space. We display the full supersymmetry of the model and illustrate in detail its quantization procedure. Finally, we show that the quantized model reproduces the Hodge theory for compact twisted generalized Kähler manifolds recently developed by Gualtieri in [33]. This allows us to recover and put in a broader context the results on the biHermitian topological sigma models obtained by Kapustin and Li in [9].
Integrability of Invariant Geodesic Flows on n-Symmetric Spaces
Jovanovic, Bozidar
2010-01-01
In this paper, by modifying the argument shift method,we prove Liouville integrability of geodesic flows of normal metrics (invariant Einstein metrics) on the Ledger-Obata $n$-symmetric spaces $K^n/\\diag(K)$, where $K$ is a semisimple (respectively, simple) compact Lie group.
Injectivity Radius and Cartan Polyhedron for Simply Connected Symmetric Spaces
Institute of Scientific and Technical Information of China (English)
Ling YANG
2007-01-01
The author explores the relationship between the cut locus of. an arbitrary simply connected and compact Riemannian symmetric space and the Cartan polyhedron of corresponding restricted root system, and computes the injectivity radius and diameter for every type of irreducible ones.
Some remarks on quasi-Hermitian operators
Energy Technology Data Exchange (ETDEWEB)
Antoine, Jean-Pierre, E-mail: jean-pierre.antoine@uclouvain.be [Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, B-1348 Louvain-la-Neuve (Belgium); Trapani, Camillo, E-mail: camillo.trapani@unipa.it [Dipartimento di Matematica e Informatica, Università di Palermo, I-90123, Palermo (Italy)
2014-01-15
A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.
Hermitian K-theory of the integers
Berrick, A. J.; Karoubi, M.
2005-01-01
The 2-primary torsion of the higher algebraic K-theory of the integers has been computed by Rognes and Weibel. In this paper we prove analogous results for the Hermitian K-theory of the integers with 2 inverted (denoted by Z'). We also prove in this case the analog of the Lichtenbaum conjecture for the hermitian K-theory of Z' : the homotopy fixed point set of a suitable Z/2 action on the classifying space of the algebraic K-theory of Z' is the hermitian K-theory of Z' after 2-adic completion.
Bender, Carl M.
2015-07-01
The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p2 + ix3, which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H = p2 + ix3 is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been verified in laboratory experiments. A particularly interesting PT-symmetric Hamiltonian is H = p2 - x4, which contains an upside-down potential. This potential is discussed in detail, and it is explained in intuitive as well as in rigorous terms why the energy levels of this potential are real, positive, and discrete. Applications of PT-symmetry in quantum field theory are also discussed.
Constraining Torsion in Maximally symmetric (sub)spaces
Sur, Sourav
2013-01-01
We look into the general aspects of space-time symmetries in presence of torsion, and how the latter is affected by such symmetries. Focusing in particular to space-times which either exhibit maximal symmetry on their own, or could be decomposed to maximally symmetric subspaces, we work out the constraints on torsion in two different theoretical schemes. We show that at least for a completely antisymmetric torsion tensor (for e.g. the one motivated from string theory), an equivalence is set between these two schemes, as the non-vanishing independent torsion tensor components turn out to be the same.
A Haar component for quantum limits on locally symmetric spaces
Anantharaman, Nalini
2010-01-01
We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite $n\\times n$ matrices (any quotient for $n=3$, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Lebesgue component. This is consistent with the conjecture that the limit measures are absolutely continuous.
Minimal surfaces in symmetric spaces with parallel second fundamental form
Indian Academy of Sciences (India)
XIAOXIANG JIAO; MINGYAN LI
2017-09-01
In this paper, we study geometry of isometric minimal immersions of Riemannian surfaces in a symmetric space by moving frames and prove that the Gaussian curvature must be constant if the immersion is of parallel second fundamental form. In particular, when the surface is $S^2$, we discuss the special case and obtain a necessary and sufficient condition such that its second fundamental form is parallel. We alsoconsider isometric minimal two-spheres immersed in complex two-dimensional Kählersymmetric spaces with parallel second fundamental form, and prove that the immersionis totally geodesic with constant Kähler angle if it is neither holomorphic nor antiholomorphicwith Kähler angle $\\alpha\
Spherically symmetric solution in a space-time with torsion
Farfan, Filemon; Loaiza-Brito, Oscar; Moreno, Claudia; Yakhno, Alexander
2011-01-01
By using the analysis group method, we obtain a new exact evolving and spherically symmetric solution of the Einstein-Cartan equations of motion, corresponding to a space-time threaded with a three-form Kalb-Ramond field strength. The solution describes in its more generic form, a space-time which scalar curvature vanishes for large distances and for large time. In static conditions, it reduces to a classical wormhole solution already reported in literature. In the process we have found evidence towards the construction of more new solutions.
On the generation of non-Hermitian Hamiltonians with real spectra by a modified Darboux transform
Energy Technology Data Exchange (ETDEWEB)
Munoz, R. [Universidad Autonoma de la Ciudad de Mexico, Fray Servando Teresa de Mier 99, 4to piso, Col. Obrera and Departamento de Fisica, CINVESTAV-IPN, A.P. 14-740, 7000 Mexico, DF (Mexico)]. E-mail: rodrigom@fis.cinvestav.mx
2005-10-03
Non-PT-symmetric, as well as PT-symmetric, non-Hermitian Hamiltonians with real spectra are generated by a modified Darboux transform. Isospectral partners of the harmonic oscillator are discussed as an example.
Pseudo-Hermitian quantum mechanics with unbounded metric operators.
Mostafazadeh, Ali
2013-04-28
I extend the formulation of pseudo-Hermitian quantum mechanics to η(+)-pseudo-Hermitian Hamiltonian operators H with an unbounded metric operator η(+). In particular, I give the details of the construction of the physical Hilbert space, observables and equivalent Hermitian Hamiltonian for the case that H has a real and discrete spectrum and its eigenvectors belong to the domain of η(+) and consequently √η(+).
The Phase Space Formulation of Time-Symmetric Quantum Mechanics
Directory of Open Access Journals (Sweden)
Charlyne de Gosson
2015-11-01
Full Text Available Time-symmetric quantum mechanics can be described in the Weyl–Wigner–Moyal phase space formalism by using the properties of the cross-terms appearing in the Wigner distribution of a sum of states. These properties show the appearance of a strongly oscillating interference between the pre-selected and post-selected states. It is interesting to note that the knowledge of this interference term is sufficient to reconstruct both states.Quanta 2015; 4: 27–34.
Quantum unique ergodicity on locally symmetric spaces: the degenerate lift
Silberman, Lior
2011-01-01
Given a measure $\\bar\\mu$ on a locally symmetric space $Y=\\Gamma\\backslash G/K$, obtained as a weak-{*} limit of probability measures associated to eigenfunctions of the ring of invariant differential operators, we construct a measure $\\mu$ on the homogeneous space $X=\\Gamma\\backslash G$ which lifts $\\bar\\mu$ and which is invariant by a connected subgroup $A_{1}\\subset A$ of positive dimension, where $G=NAK$ is an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, then $\\mu$ is also the limit of measures associated to Hecke eigenfunctions on $X$. This generalizes previous results of the author and A.\\ Venkatesh to the case of "degenerate" limiting spectral parameters.
Finding of the Metric Operator for a Quasi-Hermitian Model
Directory of Open Access Journals (Sweden)
Ebru Ergun
2013-01-01
Full Text Available We consider on an appropriate Sobolev space a non-Hermitian Hamiltonian depending on the two complex parameters α and β and having real spectrum. We derive a closed formula for a family of the metric operators, which render the Hamiltonian Hermitian. In some particular cases, we calculate the Hermitian counterpart of .
Deconstructing non-dissipative non-Dirac-hermitian relativistic quantum systems
2011-01-01
A method to construct non-dissipative non-Dirac-hermitian relativistic quantum system that is isospectral with a Dirac-hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-hermitian operators, which are hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvabl...
Foundations of symmetric spaces of measurable functions Lorentz, Marcinkiewicz and Orlicz spaces
Rubshtein, Ben-Zion A; Muratov, Mustafa A; Pashkova, Yulia S
2016-01-01
Key definitions and results in symmetric spaces, particularly Lp, Lorentz, Marcinkiewicz and Orlicz spaces are emphasized in this textbook. A comprehensive overview of the Lorentz, Marcinkiewicz and Orlicz spaces is presented based on concepts and results of symmetric spaces. Scientists and researchers will find the application of linear operators, ergodic theory, harmonic analysis and mathematical physics noteworthy and useful. This book is intended for graduate students and researchers in mathematics and may be used as a general reference for the theory of functions, measure theory, and functional analysis. This self-contained text is presented in four parts totaling seventeen chapters to correspond with a one-semester lecture course. Each of the four parts begins with an overview and is subsequently divided into chapters, each of which concludes with exercises and notes. A chapter called “Complements” is included at the end of the text as supplementary material to assist students with independent work.
On the Schwartz Space Isomorphism Theorem for Rank One Symmetric Space
Indian Academy of Sciences (India)
Joydip Jana; Rudra P Sarkar
2007-08-01
In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism (0 < ≤ 2) under the Fourier transform for the class of functions of left -type on a Riemannian symmetric space of rank one. Our treatment rests on Anker’s [2] proof of the corresponding result in the case of left -invariant functions on . Thus we give a proof which relies only on the Paley–Wiener theorem.
On integrable wave interactions and Lax pairs on symmetric spaces
Gerdjikov, Vladimir S; Ivanov, Rossen I
2016-01-01
Multi-component generalizations of derivative nonlinear Schrodinger (DNLS) type of equations having quadratic bundle Lax pairs related to Z_2-graded Lie algebras and A.III symmetric spaces are studied. The Jost solutions and the minimal set of scattering data for the case of local and nonlocal reductions are constructed. The latter lead to multi-component integrable equations with CPT-symmetry. Furthermore, the fundamental analytic solutions (FAS) are constructed and the spectral properties of the associated Lax operators are briefly discussed. The Riemann-Hilbert problem (RHP) for the multi-component generalizations of DNLS equation of Kaup-Newell (KN) and Gerdjikov-Ivanov (GI) types are derived. A modification of the dressing method is presented allowing the explicit derivation of the soliton solutions for the multi-component GI equation with both local and nonlocal reductions. It is shown that for specific choices of the reduction these solutions can have regular behavior for all finite x and t. The fundam...
Deconstructing non-dissipative non-Dirac-hermitian relativistic quantum systems
Ghosh, Pijush K
2011-01-01
A method to construct non-dissipative non-Dirac-hermitian relativistic quantum system that is isospectral with a Dirac-hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-hermitian operators, which are hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-hermitian relativistic quantum systems are presented by establishing/employing a connection between Dirac equation and supersymmetry
Deconstructing non-dissipative non-Dirac-Hermitian relativistic quantum systems
Ghosh, Pijush K.
2011-08-01
A method to construct non-dissipative non-Dirac-Hermitian relativistic quantum system that is isospectral with a Dirac-Hermitian Hamiltonian is presented. The general technique involves a realization of the basic canonical (anti-)commutation relations involving the Dirac matrices and the bosonic degrees of freedom in terms of non-Dirac-Hermitian operators, which are Hermitian in a Hilbert space that is endowed with a pre-determined positive-definite metric. Several examples of exactly solvable non-dissipative non-Dirac-Hermitian relativistic quantum systems are presented by establishing/employing a connection between Dirac equation and supersymmetry.
Spherically Symmetric Space Time with Regular de Sitter Center
Dymnikova, Irina
We formulate the requirements which lead to the existence of a class of globally regular solutions of the minimally coupled GR equations asymptotically de Sitter at the center.REFID="S021827180300358XFN001"> The source term for this class, invariant under boosts in the radial direction, is classified as spherically symmetric vacuum with variable density and pressure Tμ ν vac associated with an r-dependent cosmological term Λ μ ν = 8π GTμ ν vac, whose asymptotic at the origin, dictated by the weak energy condition, is the Einstein cosmological term Λgμν, while asymptotic at infinity is de Sitter vacuum with λ < Λ or Minkowski vacuum. For this class of metrics the mass m defined by the standard ADM formula is related to both the de Sitter vacuum trapped at the origin and the breaking of space time symmetry. In the case of the flat asymptotic, space time symmetry changes smoothly from the de Sitter group at the center to the Lorentz group at infinity through radial boosts in between. Geometry is asymptotically de Sitter as r → 0 and asymptotically Schwarzschild at large r. In the range of masses m ≥ mcrit, the de Sitter Schwarzschild geometry describes a vacuum nonsingular black hole (ΛBH), and for m < mcrit it describes G-lump — a vacuum selfgravitating particle-like structure without horizons. In the case of de Sitter asymptotic at infinity, geometry is asymptotically de Sitter as r → 0 and asymptotically Schwarzschild de Sitter at large r. Λμν geometry describes, dependently on parameters m and q = √ {Λ /λ } and choice of coordinates, a vacuum nonsingular cosmological black hole, self-gravitating particle-like structure at the de Sitter background λgμν, and regular cosmological models with cosmological constant evolving smoothly from Λ to λ.
Weighted Composition Operators between Different Bergman Spaces of Bounded Symmetric Domains
Indian Academy of Sciences (India)
Xiaofen Lv; Xiaomin Tang
2009-09-01
In this paper, we consider the boundedness and compactness of the weighted composition operators between different Bergman spaces of bounded symmetric domains in terms of the Carleson measure. As an application, we study the multipliers between different Bergman spaces.
Directory of Open Access Journals (Sweden)
Cheng-yi Zhang
2016-06-01
Full Text Available Abstract Some convergence conditions on successive over-relaxed (SOR iterative method and symmetric SOR (SSOR iterative method are proposed for non-Hermitian positive definite linear systems. Some examples are given to demonstrate the results obtained.
Level statistics of a pseudo-Hermitian Dicke model.
Deguchi, Tetsuo; Ghosh, Pijush K; Kudo, Kazue
2009-08-01
A non-Hermitian operator that is related to its adjoint through a similarity transformation is defined as a pseudo-Hermitian operator. We study the level statistics of a pseudo-Hermitian Dicke Hamiltonian that undergoes quantum phase transition (QPT). We find that the level-spacing distribution of this Hamiltonian near the integrable limit is close to Poisson distribution, while it is Wigner distribution for the ranges of the parameters for which the Hamiltonian is nonintegrable. We show that the assertion in the context of the standard Dicke model that QPT is a precursor to a change in the level statistics is not valid in general.
The bi Hermitian topological sigma model
Zucchini, R
2006-01-01
BiHermitian geometry, discovered long ago by Gates, Hull and Roceck, is the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target space, we write down the explicit expression of the sigma model's action and BRST transformations and present a computation of the topological gauge fermion.
Ricci Collineations of Static Space Times with Maximal Symmetric Transverse Spaces
Institute of Scientific and Technical Information of China (English)
M. Akbar; CAI Rong-Gen
2006-01-01
A complete classification of static space times with maximal symmetric transverse spaces is provided,according to their Ricci collineations. The classification is made when one component of Ricci collineation vector field V is non-zero (cases 1～4), two components of V are non-zero (cases 5～10), and three components of V are non-zero (cases 11～14), respectivily. Both non-degenerate (det Rab ≠ 0) as well as the degenerate (det Rab = 0) cases are discussed and some new metrics are found.
On the Moduli Space of non-BPS Attractors for N=2 Symmetric Manifolds
Ferrara, Sergio
2007-01-01
We study the ``flat'' directions of non-BPS extremal black hole attractors for N=2, d=4 supergravities whose vector multiplets' scalar manifold is endowed with homogeneous symmetric special Kahler geometry. The non-BPS attractors with non-vanishing central charge have a moduli space described by real special geometry (and thus related to the d=5 parent theory), whereas the moduli spaces of non-BPS attractors with vanishing central charge are certain Kahler homogeneous symmetric manifolds. The moduli spaces of the non-BPS attractors of the corresponding N=2, d=5 theories are also indicated, and shown to be rank-1 homogeneous symmetric manifolds.
Exceptional (Z/2Z) x (Z/2Z)-symmetric spaces
2008-01-01
The notion of (Z/2Z) x (Z/2Z)-symmetric spaces is a generalization of classical symmetric spaces, where the group Z/2Z is replaced by (Z/2Z) x (Z/2Z). In this article, a classification is given of the (Z/2Z) x (Z/2Z)-symmetric spaces G/K where G is an exceptional compact Lie group or Spin(8), complementing recent results of Bahturin and Goze. Our results are equivalent to a classification of (Z/2Z) x (Z/2Z)-gradings on the exceptional simple Lie algebras e6, e7, e8, f4, g2 and so(8).
Institute of Scientific and Technical Information of China (English)
Mohammed Ashraful Islam
2000-01-01
The analytic cosmological solutions of Einstein's field equations for a type of static metric representing plane, spherical and hyperbolic symmetric spaces are presented and their properties are discussed separately. A general type of solution is obtained which represents the plane, spherical and hyperbolic symmetric cosmological models. Its physical properties are also discussed in details.
Indian Academy of Sciences (India)
K D Patil; S H Ghate; R V Saraykar
2001-04-01
We consider a collapsing spherically symmetric inhomogeneous dust cloud in higher dimensional space-time. We show that the central singularity of collapse can be a strong curvature or a weak curvature naked singularity depending on the initial density distribution.
Condensed State Spaces for Symmetrical Coloured Petri Nets
DEFF Research Database (Denmark)
Jensen, Kurt
1996-01-01
This paper deals with state spaces. A state space is a directed graph with a node for each reachable state and an arc for each possible state change. We describe how symmetries of the modelled system can be exploited to obtain much more succinct state space analysis. The symmetries induce equival...
Orthogonal Polynomials from Hermitian Matrices II
Odake, Satoru
2016-01-01
This is the second part of the project `unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.' In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big $q$-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended $\\ell^2$ Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schr\\"odinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out...
Trajectories in a space with a spherically symmetric dislocation
Andrade, Alcides F
2012-01-01
We consider a new type of defect in the scope of linear elasticity theory, using geometrical methods. This defect is produced by a spherically symmetric dislocation, or ball dislocation. We derive the induced metric as well as the affine connections and curvature tensors. Since the induced metric is discontinuous, one can expect ambiguity coming from these quantities, due to products between delta functions or its derivatives, plaguing a description of ball dislocations based on the Geometric Theory of Defects. However, exactly as in the previous case of cylindric defect, one can obtain some well-defined physical predictions of the induced geometry. In particular, we explore some properties of test particle trajectories around the defect and show that these trajectories are curved but can not be circular orbits.
The space-time outside a source of gravitational radiation: the axially symmetric null fluid
Energy Technology Data Exchange (ETDEWEB)
Herrera, L. [Universidad Central de Venezuela, Escuela de Fisica, Facultad de Ciencias, Caracas (Venezuela, Bolivarian Republic of); Universidad de Salamanca, Instituto Universitario de Fisica Fundamental y Matematicas, Salamanca (Spain); Di Prisco, A. [Universidad Central de Venezuela, Escuela de Fisica, Facultad de Ciencias, Caracas (Venezuela, Bolivarian Republic of); Ospino, J. [Universidad de Salamanca, Departamento de Matematica Aplicada and Instituto Universitario de Fisica Fundamental y Matematicas, Salamanca (Spain)
2016-11-15
We carry out a study of the exterior of an axially and reflection symmetric source of gravitational radiation. The exterior of such a source is filled with a null fluid produced by the dissipative processes inherent to the emission of gravitational radiation, thereby representing a generalization of the Vaidya metric for axially and reflection symmetric space-times. The role of the vorticity, and its relationship with the presence of gravitational radiation is put in evidence. The spherically symmetric case (Vaidya) is, asymptotically, recovered within the context of the 1 + 3 formalism. (orig.)
Compact Hermitian Young Projection Operators
Alcock-Zeilinger, Judith
2016-01-01
In this paper, we describe a compact and practical algorithm to construct Hermitian Young projection operators for irreducible representations of the special unitary group SU(N), and discuss why ordinary Young projection operators are unsuitable for physics applications. The proof of this construction algorithm uses the iterative method described by Keppeler and Sj\\"odahl. We further show that Hermitian Young projection operators share desirable properties with Young tableaux, namely a nested hierarchy when "adding a particle". We end by exhibiting the enormous advantage of the Hermitian Young projection operators constructed in this paper over those given by Keppeler and Sj\\"odahl.
Energy Technology Data Exchange (ETDEWEB)
Castrillon Lopez, M. [Facultad de Matematicas, Departamento de Geometria y Topologia (Spain)], E-mail: mcastri@mat.ucm.es; Gadea, P. M. [CSIC, Institute of Fundamental Physics (Spain)], E-mail: pmgadea@iec.csic.es; Oubina, J. A. [Universidade de Santiago de Compostela, Departamento de Xeometria e Topoloxia, Facultade de Matematicas (Spain)], E-mail: jaoubina@usc.es
2009-02-15
For each non-compact quaternion-Kaehler symmetric space of dimension eight, all of its descriptions as a homogeneous Riemannian space, and the associated homogeneous quaternionic Kaehler structures obtained through the Witte's refined Langlands decomposition, are studied.
Quasi-linear symmetric hyperbolic Fuchsian systems in several space dimensions
Ames, Ellery; Isenberg, James; LeFloch, Philippe G
2012-01-01
We establish well-posedness results for the singular initial value problem associated with a class of quasi-linear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an extension of earlier work by the authors for the same problem in one space dimension.
Dirichlet Problem for Hermitian-Einstein Equation over Almost Hermitian Manifold
Institute of Scientific and Technical Information of China (English)
Yue WANG; Xi ZHANG
2012-01-01
In this paper,we investigate the Dirichlet problem for Hermitian-Einstein equation on complex vector bundle over almost Hermitian manifold,and we obtain the unique solution of the Dirichlet problem for Hermitian-Einstein equation.
Contractively complemented subspaces of pre-symmetric spaces
Neal, Matthew
2008-01-01
In 1965, Ron Douglas proved that if $X$ is a closed subspace of an $L^1$-space and $X$ is isometric to another $L^1$-space, then $X$ is the range of a contractive projection on the containing $L^1$-space. In 1977 Arazy-Friedman showed that if a subspace $X$ of $C_1$ is isometric to another $C_1$-space (possibly finite dimensional), then there is a contractive projection of $C_1$ onto $X$. In 1993 Kirchberg proved that if a subspace $X$ of the predual of a von Neumann algebra $M$ is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of $M$ onto $X$. We widen significantly the scope of these results by showing that if a subspace $X$ of the predual of a $JBW^*$-triple $A$ is isometric to the predual of another $JBW^*$-triple $B$, then there is a contractive projection on the predual of $A$ with range $X$, as long as $B$ does not have a direct summand which is isometric to a space of the form $L^\\infty(\\Omega,H)$, where $H$ is a Hilbert space of dimensio...
The biHermitian topological sigma model
Energy Technology Data Exchange (ETDEWEB)
Zucchini, Roberto [Dipartimento di Fisica, Universita degli Studi di Bologna, V. Irnerio 46, I-40126 Bologna (Italy); I.N.F.N., sezione di Bologna (Italy)
2006-12-15
BiHermitian geometry, discovered long ago by Gates, Hull and Rocek, is the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. By using the twisting procedure proposed by Kapustin and Li, we work out the type A and B topological sigma models for a general biHermtian target space, we write down the explicit expression of the sigma model's action and BRST transformations and present a computation of the topological gauge fermion and the topological action.
Sirsi, Swarnamala; Hegde, Subramanya
2011-01-01
Quantum computation on qubits can be carried out by an operation generated by a Hamiltonian such as application of a pulse as in NMR, NQR. Quantum circuits form an integral part of quan- tum computation. We investigate the nonlocal operations generated by a given Hamiltonian. We construct and study the properties of perfect entanglers, that is, the two-qubit operations that can generate maximally entangled states from some suitably chosen initial separable states in terms of their entangling power. Our work addresses the problem of analyzing the quantum evolution in the special case of two qubit symmetric states. Such a symmetric space can be considered to be spanned by the angular momentum states {|j = 1,m>;m = +1, 0,-1}. Our technique relies on the decomposition of a Hamiltonian in terms of newly defined Hermitian operators Mk's (k= 0.....8) which are constructed out of angular momentum operators Jx, Jy, Jz. These operators constitute a linearly independent set of traceless matrices (except for M0). Further...
Harmonic Analysis on semisimple symmetric spaces. A survey of some general results.
Ban, E.P. van den; Flensted-Jensen, M.; Schlichtkrull, H.
2001-01-01
We give a survey of the status of some of the fundamental problems in harmonic analysis on semisimple symmetric spaces, including the description of the discrete series, the denition of the Fourier transform, the inversion formula, the Plancherel formula and the Paley{ Wiener theorem.
Azizov, Tomas; Ćurgus, Branko; Dijksma, Aad
2003-01-01
Certain meromorphic matrix valued functions on C\\R, the so-called boundary coefficients, are characterized in terms of a standard symmetric operator S in a Pontryagin space with finite (not necessarily equal) defect numbers, a meromorphic mapping into the defect subspaces of S, and a boundary mappin
The Plancherel decomposition for a reductive symmetric space II : representation theory
Ban, E.P. van den; Schlichtkrull, H.
2001-01-01
We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I. The formula for Schwartz functions involves Eisenstein integrals obtained by a residual
Weighted Composition Operators on Weighted Bergman Spaces of Bounded Symmetric Domains
Indian Academy of Sciences (India)
Sanjay Kumar; Kanwar Jatinder Singh
2007-05-01
In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator $W_{\\varphi,\\psi}$ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten -class weighted composition operators.
Structure of the degenerate principal series on symmetric R-spaces and small representations
DEFF Research Database (Denmark)
Möllers, Jan; Schwarz, Benjamin
2014-01-01
Let $G$ be a simple real Lie group with maximal parabolic subgroup $P$ whose nilradical is abelian. Then $X=G/P$ is called a symmetric $R$-space. We study the degenerate principal series representations of $G$ on $C^\\infty(X)$ in the case where $P$ is not conjugate to its opposite parabolic. We...
Cowling–Price Theorem and Characterization of Heat Kernel on Symmetric Spaces
Indian Academy of Sciences (India)
Swagato K Ray; Rudra P Sarkar
2004-05-01
We extend the uncertainty principle, the Cowling–Price theorem, on non-compact Riemannian symmetric spaces . We establish a characterization of the heat kernel of the Laplace–Beltrami operator on from integral estimates of the Cowling–Price type.
On the local form of static plane symmetric space-times in the presence of matter
Gomes, Leandro G
2015-01-01
For any configuration of a static plane-symmetric distribution of matter along space-time, there are coordinates where the metric can be put explicitly as a functional of the energy density and pressures. It satisfies Einstein equations as far as we require the conservation of the energy-momentum tensor, which is the single ODE for self-gravitating hydrostatic equilibrium. As a direct application, a general solution is given when the pressures are linearly related to the energy density, recovering, as special cases, most of known solutions of static plane-symmetric Einstein equations.
Geometry of Vlasov kinetic moments: A bosonic Fock space for the symmetric Schouten bracket
Energy Technology Data Exchange (ETDEWEB)
Gibbons, John [Department of Mathematics, Imperial College London, London SW7 2AZ (United Kingdom); Holm, Darryl D. [Department of Mathematics, Imperial College London, London SW7 2AZ (United Kingdom); Computer and Computational Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545 (United States); Tronci, Cesare [Department of Mathematics, Imperial College London, London SW7 2AZ (United Kingdom); TERA Foundation for Oncological Hadrontherapy, 11 V. Puccini, Novara 28100 (Italy)], E-mail: cesare.tronci@imperial.ac.uk
2008-06-02
The dynamics of Vlasov kinetic moments is shown to be Lie-Poisson on the dual Lie algebra of symmetric contravariant tensor fields. The corresponding Lie bracket is identified with the symmetric Schouten bracket and the moment Lie algebra is related with a bundle of bosonic Fock spaces, where creation and annihilation operators are used to construct the cold plasma closure. Kinetic moments are also shown to define a momentum map, which is infinitesimally equivariant. This momentum map is the dual of a Lie algebra homomorphism, defined through the Schouten bracket. Finally the moment Lie-Poisson bracket is extended to anisotropic interactions.
Relativistic Zitterbewegung in non-Hermitian photonic waveguide systems
Wang, Guanglei; Xu, Hongya; Huang, Liang; Lai, Ying-Cheng
2017-01-01
Zitterbewegung (ZB) is a phenomenon in relativistic quantum systems where the electron wave packet exhibits a trembling or oscillating behavior during its motion, caused by its interaction or coupling with the negative energy state. To directly observe ZB in electronic systems is difficult, due to the challenges associated with the small amplitude of the motion which is of the order of Compton wavelength. Photonic systems offer an alternative paradigm. We exploit the concept of pseudo parity-time (pseudo { P }{ T }) symmetry to study ZB in non-Hermitian quantum systems implemented as an experimentally feasible optical waveguide array. In particular, the non-Hermitian Hamiltonian is realized through evanescent coupling among the waveguides to form a one-dimensional lattice with periodic modulations in gain and loss along the guiding direction. As the modulation frequency is changed, we obtain a number of phenomena including periodically suppressed ZB trembling, spatial energy localization, and Hermitian-like ZB oscillations. We calculate phase diagrams indicating the emergence of different types of dynamical behaviors of the relativistic non-Hermitian quantum system in an experimentally justified parameter space. We provide numerical results and a physical analysis to explain the distinct dynamical behaviors revealed by the phase diagrams. Our findings provide a deeper understanding of both the relativistic ZB phenomenon and non-Hermitian pseudo-{ P }{ T } systems, with potential applications in controlling/harnessing light propagation in waveguide-based optical systems.
Random matrix theory for pseudo-Hermitian systems: Cyclic blocks
Indian Academy of Sciences (India)
Sudhir R Jain; Shashi C L Srivastava
2009-12-01
We discuss the relevance of random matrix theory for pseudo-Hermitian systems, and, for Hamiltonians that break parity and time-reversal invariance . In an attempt to understand the random Ising model, we present the treatment of cyclic asymmetric matrices with blocks and show that the nearest-neighbour spacing distributions have the same form as obtained for the matrices with scalar entries. We also summarize the theory for random cyclic matrices with scalar entries. We have also found that for block matrices made of Hermitian and pseudo-Hermitian sub-blocks of the form appearing in Ising model depart from the known results for scalar entries. However, there is still similarity in trends even in log–log plots.
Non-Hermitian spin chains with inhomogeneous coupling
Energy Technology Data Exchange (ETDEWEB)
Bytsko, Andrei G. [Rossijskaya Akademiya Nauk, St. Petersburg (Russian Federation). Inst. Matematiki; Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany). Gruppe Theorie
2009-11-15
An open U{sub q}(sl{sub 2})-invariant spin chain of spin S and length N with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter {gamma} are determined for which the spectrum of the model is real. For a certain range of {gamma}, a universal metric operator is constructed and thus the quasi-Hermiticity of the model is established. The constructed metric operator is non-dynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous U{sub q}(sl{sub 2})-invariant integrable spin chains with nearest-neighbour interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite dimensional space is discussed. (orig.)
A differential equation for Lerch's transcendent and associated symmetric operators in Hilbert space
Energy Technology Data Exchange (ETDEWEB)
Kaplitskii, V M [Southern Federal University, Rostov-on-Don (Russian Federation)
2014-08-01
The function Ψ(x,y,s)=e{sup iy}Φ(−e{sup iy},s,x), where Φ(z,s,v) is Lerch's transcendent, satisfies the following two-dimensional formally self-adjoint second-order hyperbolic differential equation, where s=1/2+iλ. The corresponding differential expression determines a densely defined symmetric operator (the minimal operator) on the Hilbert space L{sub 2}(Π), where Π=(0,1)×(0,2π). We obtain a description of the domains of definition of some symmetric extensions of the minimal operator. We show that formal solutions of the eigenvalue problem for these symmetric extensions are represented by functional series whose structure resembles that of the Fourier series of Ψ(x,y,s). We discuss sufficient conditions for these formal solutions to be eigenfunctions of the resulting symmetric differential operators. We also demonstrate a close relationship between the spectral properties of these symmetric differential operators and the distribution of the zeros of some special analytic functions analogous to the Riemann zeta function. Bibliography: 15 titles.
Institute of Scientific and Technical Information of China (English)
Junfeng Yin; Quanyu Dou
2012-01-01
In this paper,a generalized preconditioned Hermitian and skew-Hermitian splitting (GPHSS) iteration method for a non-Hermitian positive-definite matrix is studied,which covers standard Hermitian and skew-Hermitian splitting (HSS) iteration and also many existing variants.Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix.From practical point of view,we have analyzed and implemented inexact generalized preconditioned Hermitian and skew-Hermitian splitting (IGPHSS) iteration,which employs Krylov subspace methods as its inner processes.Numerical experiments from three-dimensional convection-diffusion equation show that the GPHSS and IGPHSS iterations are efficient and competitive with standard HSS iteration and AHSS iteration.
Sums of hermitian squares and the BMV conjecture
Klep, Igor
2007-01-01
Recently Lieb and Seiringer showed that the Bessis-Moussa-Villani conjecture from quantum physics can be restated in the following purely algebraic way: The sum of all words in two positive semidefinite matrices where the number of each of the two letters is fixed is always a matrix with nonnegative trace. We show that this statement holds if the words are of length at most 13. This has previously been known only up to length 7. In our proof, we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming. As a by-product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size, yet not being a sum of hermitian squares and commutators.
Directory of Open Access Journals (Sweden)
Vladimir S. Gerdjikov
2011-10-01
Full Text Available A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM. Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF type integrable models on A.III-type symmetric spaces are briefly analyzed.
Hybrid Fixed Point Theorems in Symmetric Spaces via Common Limit Range Property
Directory of Open Access Journals (Sweden)
Imdad Mohammad
2014-12-01
Full Text Available In this paper, we point out that some recent results of Vijaywar et al. (Coincidence and common fixed point theorems for hybrid contractions in symmetric spaces, Demonstratio Math. 45 (2012, 611-620 are not true in their present form. With a view to prove corrected and improved versions of such results, we introduce the notion of common limit range property for a hybrid pair of mappings and utilize the same to obtain some coincidence and fixed point results for mappings defined on an arbitrary set with values in symmetric (semi-metric spaces. Our results improve, generalize and extend some results of the existing literature especially due to Imdad et al., Javid and Imdad, Vijaywar et al. and some others. Some illustrative examples to highlight the realized improvements are also furnished.
Lie Symmetrical Non-Noether Conserved Quantity of Mechanical System in Phase Space
Institute of Scientific and Technical Information of China (English)
FANG Jian-Hui; PENG Yong; LIAO Yong-Pan; LI Hong
2004-01-01
In this paper, we study the Lie symmetrical non-Noether conserved quantity of the differential equations ofmotion of mechanical system in phase space under the general infinitesimal transformations of groups. Firstly, we givethe determining equations of the Lie symmetry of the system. Secondly, the non-Noether conserved quantity of the Liesymmetry is derived. Finally, an example is given to illustrate the application of the result.
Lie Symmetrical Non-Noether Conserved Quantity of Mechanical System in Phase Space
Institute of Scientific and Technical Information of China (English)
FANGJian-Hui; PENGYong; LIAOYong-Pan; LIHong
2004-01-01
In this paper, we study the Lie symmetrical non-Noether conserved quantity of the differential equations of motion of mechanical system in phase space under the general infinitesimal transformations of groups. Firstly. we give the determining equations of the Lie symmetry of the system. Secondly, the non-Noether conserved quantity of the Lie symmetry is derived. Finally, an example is given to illustrate the application of the result.
On the gradient of Schwarz symmetrization of functions in Sobolev spaces
Bramanti, Marco
2010-01-01
Let S be a Sobolev or Orlicz-Sobolev space of functions not necessarily vanishing at the boundary of the domain. We give sufficient conditions on a nonnegative function in S in order that its spherical rearrangement ("Schwartz symmetrization") still belongs to S. These results are obtained via relative isoperimetric inequalities and somewhat generalize a well-known Polya-Szego's theorem. We also prove that the rearrangement of any function in S is locally in S.
Maj, Omar
2008-01-01
This is the second part of a work aimed to study complex-phase oscillatory solutions of nonlinear symmetric hyperbolic systems. We consider, in particular, the case of one space dimension. That is a remarkable case, since one can always satisfy the \\emph{naive} coherence condition on the complex phases, which is required in the construction of the approximate solution. Formally the theory applies also in several space dimensions, but the \\emph{naive} coherence condition appears to be too restrictive; the identification of the optimal coherence condition is still an open problem.
Non-Hermitian heat engine with all-quantum-adiabatic-process cycle
Lin, S.; Song, Z.
2016-11-01
As a quantum device, a quantum heat engine (QHE) is described by a Hermitian Hamiltonian. However, since it is an open system, reservoirs must be imposed phenomenologically without any description in the context of quantum mechanics. A non-Hermitian system is expected to describe an open system that exchanges energy and particles with external reservoirs. Correspondingly, such an exchange can be adiabatic in the context of quantum mechanics. We first propose a non-Hermitian QHE by a concrete simple two-level system, which is an S=1/2 spin in a complex external magnetic field. The non-Hermitian { P }{ T }-symmetric Hamiltonian, as a self-contained one, describes both the working medium and reservoirs. A heat engine cycle is composed of completely quantum adiabatic processes. Surprisingly, the heat efficiency is obtained to be the same as that of the Hermitian quantum Otto cycle. A classical analog of this scheme is also presented. Our finding paves the way for revealing the role of a non-Hermitian Hamiltonian in physics.
Non-Hermitian Hamiltonians with a real spectrum and their physical applications
Indian Academy of Sciences (India)
Ali Mostafazadeh
2009-08-01
We present an evaluation of some recent attempts to understand the role of pseudo-Hermitian and $\\mathcal{PT}$-symmetric Hamiltonians in modelling unitary quantum systems and elaborate on a particular physical phenomenon whose discovery originated in the study of complex scattering potentials.
Non-Hermitian Hamiltonians with a Real Spectrum and Their Physical Applications
Mostafazadeh, Ali
2009-01-01
We present an evaluation of some recent attempts at understanding the role of pseudo-Hermitian and PT-symmetric Hamiltonians in modeling unitary quantum systems and elaborate on a particular physical phenomenon whose discovery originated in the study of complex scattering potentials.
Non-hermitian quantum thermodynamics
Gardas, Bartłomiej; Deffner, Sebastian; Saxena, Avadh
2016-03-01
Thermodynamics is the phenomenological theory of heat and work. Here we analyze to what extent quantum thermodynamic relations are immune to the underlying mathematical formulation of quantum mechanics. As a main result, we show that the Jarzynski equality holds true for all non-hermitian quantum systems with real spectrum. This equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium. In the quasistatic limit however, the second law leads to the Carnot bound which is fulfilled even if some eigenenergies are complex provided they appear in conjugate pairs. Furthermore, we propose two setups to test our predictions, namely with strongly interacting excitons and photons in a semiconductor microcavity and in the non-hermitian tight-binding model.
Non-hermitian quantum thermodynamics.
Gardas, Bartłomiej; Deffner, Sebastian; Saxena, Avadh
2016-03-22
Thermodynamics is the phenomenological theory of heat and work. Here we analyze to what extent quantum thermodynamic relations are immune to the underlying mathematical formulation of quantum mechanics. As a main result, we show that the Jarzynski equality holds true for all non-hermitian quantum systems with real spectrum. This equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium. In the quasistatic limit however, the second law leads to the Carnot bound which is fulfilled even if some eigenenergies are complex provided they appear in conjugate pairs. Furthermore, we propose two setups to test our predictions, namely with strongly interacting excitons and photons in a semiconductor microcavity and in the non-hermitian tight-binding model.
On Einstein, Hermitian 4-Manifolds
LeBrun, Claude
2010-01-01
Let (M,h) be a compact 4-dimensional Einstein manifold, and suppose that h is Hermitian with respect to some complex structure J on M. Then either (M,J,h) is Kaehler-Einstein, or else, up to rescaling and isometry, it is one of the following two exceptions: the Page metric on CP2 # (-CP2), or the Einstein metric on CP2 # 2 (-CP2) constructed in Chen-LeBrun-Weber.
EBQ code: transport of space-charge beams in axially symmetric devices
Energy Technology Data Exchange (ETDEWEB)
Paul, A.C.
1982-11-01
Such general-purpose space charge codes as EGUN, BATES, WOLF, and TRANSPORT do not gracefully accommodate the simulation of relativistic space-charged beams propagating a long distance in axially symmetric devices where a high degree of cancellation has occurred between the self-magnetic and self-electric forces of the beam. The EBQ code was written specifically to follow high current beam particles where space charge is important in long distance flight in axially symmetric machines possessing external electric and magnetic field. EBQ simultaneously tracks all trajectories so as to allow procedures for charge deposition based on inter-ray separations. The orbits are treated in Cartesian geometry (position and momentum) with z as the independent variable. Poisson's equation is solved in cylindrical geometry on an orthogonal rectangular mesh. EBQ can also handle problems involving multiple ion species where the space charge from each must be included. Such problems arise in the design of ion sources where different charge and mass states are present.
Kobayashi's and Teichmüller's Metrics on the Teichmüller Space of Symmetric Circle Homeomorphisms
Institute of Scientific and Technical Information of China (English)
Jun HU; Yun Ping JIANG; Zhe WANG
2011-01-01
We give a direct proof of a result of Earle, Gardiner and Lakic, that is, Kobayashi's metric and Teichmüller's metric coincide with each other on the Teichmüller space of symmetric circle homeomorphisms.
Hermitian Groups over Local Rings
Institute of Scientific and Technical Information of China (English)
Guoping Tang
2001-01-01
The general hermitian group GH2n(R,a1,... ,ar) of rank n and its elementary subgroup EH2n(R,a1,... ,ar) were introduced by Bak [1] and Tang [4], respectively. It is known that EH2n(R, a1,... , ar) is perfect whenever n ≥ r+ 3 and the stable elementary hermitian group EH(R,a1,... ,ar) is the commutator subgroup of the stable general hermitian group GH(R, a1,... , ar).In this paper, we prove that, when R is a local ring, EH2n(R, a1,... , ar) is a normal subgroup of GH2n (R, a1,... , ar) if n ≥ r+2, and is the commutator subgroup of GH2n(R, a1,... , ar) if n ≥ r + 3. In the special case that R is a division ring,we show that the quotient group of GH2n(R, a1,... , ar) by EH2n(R, a1,... , ar)is independent of the choice of a1,... , ar.
Detecting topological exceptional points in a parity-time symmetric system with cold atoms.
Xu, Jian; Du, Yan-Xiong; Huang, Wei; Zhang, Dan-Wei
2017-07-10
We reveal a novel topological property of the exceptional points in a two-level parity-time symmetric system and then propose a scheme to detect the topological exceptional points in the system, which is embedded in a larger Hilbert space constructed by a four-level cold atomic system. We show that a tunable parameter in the presented system for simulating the non-Hermitian Hamiltonian can be tuned to sweep the eigenstates through the exceptional points in parameter space. The non-trivial Berry phases of the eigenstates obtained in this loop from the exceptional points can be measured by the atomic interferometry. Since the proposed operations and detection are experimentally feasible, our scheme may pave a promising way to explore the novel properties of non-Hermitian systems.
Axion-photon conversion in space and in low symmetrical dielectric crystals
Gorelik, V. S.
2016-07-01
The opportunities of axions detection as the result of axion-photon conversion processes in the space and in low symmetrical dielectric crystals are discussed. In accordance with the modern theory predictions, axions are pseudoscalar vacuum particles having very small (0.001-1.0 meV) rest energy. The possibility of axions conversion into photons and vice-versa processes in vacuum at the presence of outer magnetic field has been analyzed before. Pseudoscalar (axion type) modes are existing in some types of crystals. Polar pseudoscalar lattice and exciton modes in low symmetrical crystals are strongly interacted with axions. In this work, optical excitation of axion-type modes in low symmetrical crystals is proposed for observation of axion - photon conversion processes. Instead of outer magnetic field, the crystalline field of such crystals may be used. The experimental schemes for axion-photon conversion processes observation with recording the secondary emission of luminescence, infrared or Stimulated Raman Scattering in some dielectric crystals are discussed.
Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory
Directory of Open Access Journals (Sweden)
Georgi G. Grahovski
2010-06-01
Full Text Available The algebraic structure and the spectral properties of a special class of multi-component NLS equations, related to the symmetric spaces of BD.I-type are analyzed. The focus of the study is on the spectral theory of the relevant Lax operators for different fundamental representations of the underlying simple Lie algebra g. Special attention is paid to the structure of the dressing factors in spinor representation of the orthogonal simple Lie algebras of B_r simeq so(2r+1,C type.
Coefficient multipliers of H~p spaces over bounded symmetric domains in C
Institute of Scientific and Technical Information of China (English)
肖建斌
1995-01-01
One way to give information about the Taylor coefficients of Hp functions is to describe the multipliers of Hp into various spaces. In the case of one complex variable, Duren and Shields described the multipliers of Hp into lq (0
symmetric domains in Cn are generalized. The results are sharp if q≥2. A sufficient condition of Hp into Hq is given for any p and q, 0
She, M.; Jiang, L. P.
2014-12-01
In this paper, an oscillating dark energy model is presented in an isotropic but inhomogeneous plane symmetric space-time by considering a time periodic varying deceleration parameter. We find three different types of new solutions which describe different scenarios of oscillating universe. The first two solutions show an oscillating universe with singularities. For the third one, the universe is singularity-free during the whole evolution. Moreover, the Hubble parameter oscillates and keeps positive which explores an interesting possibility to unify the early inflation and late time acceleration of the universe.
Moller Energy-Momentum Prescription for a Locally Rotationally Symmetric Space-Time
Aydogdu, O
2006-01-01
The energy distribution in the Locally Rotationally Symmetric (LRS) Bianchi type II space-time is obtained by considering the Moller energy-momentum definition in both Einstein's theory of general relativity and teleparallel theory of relativity. The energy distribution which includes both the matter and gravitational field is found to be zero in both of these different gravitation theories. This result agrees with previous works of Cooperstock and Israelit, Rosen, Johri et al., Banerjee and Sen, Vargas, and Aydogdu and Salti. Our result that the total energy of the universe is zero supports the view points of Albrow and Tryon.
Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
Directory of Open Access Journals (Sweden)
Derek K. Wise
2009-08-01
Full Text Available Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1 and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'.
Samsonov, Boris F
2013-04-28
One of the simplest non-Hermitian Hamiltonians, first proposed by Schwartz in 1960, that may possess a spectral singularity is analysed from the point of view of the non-Hermitian generalization of quantum mechanics. It is shown that the η operator, being a second-order differential operator, has supersymmetric structure. Asymptotic behaviour of the eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result, the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of a spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point, the equivalent Hermitian Hamiltonian becomes undetermined.
Shestakova, T P
2013-01-01
We construct Hamiltonian dynamics of the generalized spherically symmetric gravitational model in extended phase space. We start from the Faddeev - Popov effective action with gauge-fixing and ghost terms, making use of gauge conditions in differential form. It enables us to introduce missing velocities into the Lagrangian and then construct a Hamiltonian function according a usual rule which is applied for systems without constraints. The main feature of Hamiltonian dynamics in extended phase space is that it can be proved to be completely equivalent to Lagrangian dynamics derived from the effective action. The sets of Lagrangian and Hamiltonian equations are not gauge invariant in general. We demonstrate that solutions to the obtained equations include those of the gauge invariant Einstein equations, and also discuss a possible role of gauge-noninvariant terms. Then, we find a BRST invariant form of the effective action by adding terms not affecting Lagrangian equations. After all, we construct the BRST cha...
Supersymmetric Extension of Non-Hermitian su(2 Hamiltonian and Supercoherent States
Directory of Open Access Journals (Sweden)
Omar Cherbal
2010-12-01
Full Text Available A new class of non-Hermitian Hamiltonians with real spectrum, which are written as a real linear combination of su(2 generators in the form H=ωJ_3+αJ_−+βJ_+, α≠β, is analyzed. The metrics which allows the transition to the equivalent Hermitian Hamiltonian is established. A pseudo-Hermitian supersymmetic extension of such Hamiltonians is performed. They correspond to the pseudo-Hermitian supersymmetric systems of the boson-phermion oscillators. We extend the supercoherent states formalism to such supersymmetic systems via the pseudo-unitary supersymmetric displacement operator method. The constructed family of these supercoherent states consists of two dual subfamilies that form a bi-overcomplete and bi-normal system in the boson-phermion Fock space. The states of each subfamily are eigenvectors of the boson annihilation operator and of one of the two phermion lowering operators.
Pseudo-Hermitian Transition in Degenerate Nonlinear Four-Wave Mixing
Ge, Li
2016-01-01
We show that degenerate four-wave mixing (FWM) in nonlinear optics can be described by an effective Hamiltonian that is pseudo-Hermitian, which enables a transition between a pseudo-Hermitian phase with real eigenvalues and a broken pseudo-Hermitian phase with complex conjugate eigenvalues. While bearing certain similarity to that in Parity-Time symmetric systems, this transition is in stark contrast because of the absence of gain and loss in the effective Hamiltonian. The latter is real after factoring out the system decay, and the onset of non-Hermiticity in degenerate FWM is due to the total phase change of the signal wave and the idler wave. This property underlines the intrinsic coherence in FWM, which opens the door to probe quantum implications of exceptional points.
Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type
Directory of Open Access Journals (Sweden)
Nikolay A. Kostov
2008-03-01
Full Text Available New reductions for the multicomponent modified Korteweg-de Vries (MMKdV equations on the symmetric spaces of {f DIII}-type are derived using the approach based on the reduction group introduced by A.V. Mikhailov. The relevant inverse scattering problem is studied and reduced to a Riemann-Hilbert problem. The minimal sets of scattering data $mathcal{T}_i$, $i=1,2$ which allow one to reconstruct uniquely both the scattering matrix and the potential of the Lax operator are defined. The effect of the new reductions on the hierarchy of Hamiltonian structures of MMKdV and on $mathcal{T}_i$ are studied. We illustrate our results by the MMKdV equations related to the algebra $mathfrak{g}simeq so(8$ and derive several new MMKdV-type equations using group of reductions isomorphic to ${mathbb Z}_{2}$, ${mathbb Z}_{3}$, ${mathbb Z}_{4}$.
Institute of Scientific and Technical Information of China (English)
Chang-Jun Zheng; Hai-Bo Chen; Lei-Lei Chen
2013-01-01
This paper presents a novel wideband fast multipole boundary element approach to 3D half-space/planesymmetric acoustic wave problems.The half-space fundamental solution is employed in the boundary integral equations so that the tree structure required in the fast multipole algorithm is constructed for the boundary elements in the real domain only.Moreover,a set of symmetric relations between the multipole expansion coefficients of the real and image domains are derived,and the half-space fundamental solution is modified for the purpose of applying such relations to avoid calculating,translating and saving the multipole/local expansion coefficients of the image domain.The wideband adaptive multilevel fast multipole algorithm associated with the iterative solver GMRES is employed so that the present method is accurate and efficient for both lowand high-frequency acoustic wave problems.As for exterior acoustic problems,the Burton-Miller method is adopted to tackle the fictitious eigenfrequency problem involved in the conventional boundary integral equation method.Details on the implementation of the present method are described,and numerical examples are given to demonstrate its accuracy and efficiency.
Tedder, Sarah A.; Schoenholz, Bryan; Suddath, Shannon N.
2016-01-01
This paper describes the study of lateral misalignment tolerance of a symmetric high-rate free-space optical link (FSOL) for use between International Space Station (ISS) payload sites and the main cabin. The link will enable gigabit per second (Gbps) transmission of data, which is up to three orders of magnitude greater than the current capabilities. This application includes 10-20 meter links and requires minimum size, weight, and power (SWaP). The optical power must not present an eye hazard and must be easily integrated into the existing ISS infrastructure. On the ISS, rapid thermal changes and astronaut movement will cause flexure of the structure which will potentially misalign the free space transmit and receive optics 9 cm laterally and 0.2 degrees angularly. If this misalignment is not accounted for, a loss of the link or degradation of link performance will occur. Power measurements were collected to better understand the effect of various system design parameters on lateral misalignment. Parameters that were varied include: the type of small form pluggable (SFP) transceivers, type of fiber, and transmitted power level. A potential solution was identified that can reach the lateral misalignment tolerance (decenter span) required to create an FSOL on the ISS by using 105 m core fibers, a duplex SFP, two channels of light, and two fiber amplifiers.
Carrington, Connie; Fikes, John; Gerry, Mark; Perkinson, Don
2000-01-01
New energy sources are vital for the development of emerging nations, and the growth of industry in developed economies. Also vital is the need for these energy sources to be clean and renewable. For the past several years, NASA has been taking a new look at collecting solar energy in space and transmitting it to Earth, to planetary surfaces, and to orbiting spacecraft. Several innovative concepts are being studied for the space segment component of solar power beaming. One is the Abacus/Reflector, a large sun-oriented array structure fixed to the transmitter, and a rotating RF reflector that tracks a receiving rectenna on Earth. This concept eliminates the need for power-conducting slip rings in rotating joints between the solar collectors and the transmitter. Another concept is the Integrated Symmetrical Concentrator (ISC), composed of two very large segmented reflectors which rotate to collect and reflect the incident sunlight onto two centrally-located photovoltaic arrays. Adjacent to the PV arrays is the RF transmitter, which as a unit track the receiving rectenna, again eliminating power-conducting joints, and in addition reducing the cable lengths between the arrays and transmitter. The metering structure to maintain the position of the reflectors is a long mast, oriented perpendicular to the equatorial orbit plane. This paper presents a status of ongoing systems studies and configurations for the Abacus/Reflector and the ISC concepts, and a top-level study of packaging for launch and assembly.
Bruhat Order in the Full Symmetric sl(n) Toda Lattice on Partial Flag Space
Chernyakov, Yury B.; Sharygin, Georgy I.; Sorin, Alexander S.
2016-08-01
In our previous paper [Comm. Math. Phys. 330 (2014), 367-399] we described the asymptotic behaviour of trajectories of the full symmetric sl_n Toda lattice in the case of distinct eigenvalues of the Lax matrix. It turned out that it is completely determined by the Bruhat order on the permutation group. In the present paper we extend this result to the case when some eigenvalues of the Lax matrix coincide. In that case the trajectories are described in terms of the projection to a partial flag space where the induced dynamical system verifies the same properties as before: we show that when tto±∞ the trajectories of the induced dynamical system converge to a finite set of points in the partial flag space indexed by the Schubert cells so that any two points of this set are connected by a trajectory if and only if the corresponding cells are adjacent. This relation can be explained in terms of the Bruhat order on multiset permutations.
On the -Invariant of Hermitian Forms
Indian Academy of Sciences (India)
Sudeep S Parihar; V Suresh
2013-08-01
Let be a field of characteristic not 2 and a central simple algebra with an involution . A result of Mahmoudi provides an upper bound for the -invariants of hermitian forms and skew-hermitian forms over (,) in terms of the -invariant of . In this paper we give a different upper bound when is a tensor product of quaternion algebras and is a the tensor product of canonical involutions. We also show that our bounds are sharper than those of Mahmoudi.
Non-Hermitian bidirectional robust transport
Longhi, Stefano
2017-01-01
Transport of quantum or classical waves in open systems is known to be strongly affected by non-Hermitian terms that arise from an effective description of system-environment interaction. A simple and paradigmatic example of non-Hermitian transport, originally introduced by Hatano and Nelson two decades ago [N. Hatano and D. R. Nelson, Phys. Rev. Lett. 77, 570 (1996), 10.1103/PhysRevLett.77.570], is the hopping dynamics of a quantum particle on a one-dimensional tight-binding lattice in the presence of an imaginary vectorial potential. The imaginary gauge field can prevent Anderson localization via non-Hermitian delocalization, opening up a mobility region and realizing robust transport immune to disorder and backscattering. Like for robust transport of topologically protected edge states in quantum Hall and topological insulator systems, non-Hermitian robust transport in the Hatano-Nelson model is unidirectional. However, there is not any physical impediment to observe robust bidirectional non-Hermitian transport. Here it is shown that in a quasi-one-dimensional zigzag lattice, with non-Hermitian (imaginary) hopping amplitudes and a synthetic gauge field, robust transport immune to backscattering can occur bidirectionally along the lattice.
PT-symmetric quantum oscillator in an optical cavity
Longhi, Stefano
2016-01-01
The quantum harmonic oscillator with parity-time ($\\mathcal{PT}$) symmetry, obtained from the ordinary (Hermitian) quantum harmonic oscillator by an imaginary displacement of the spatial coordinate, provides an important and exactly-solvable model to investigate non-Hermitian extension of the Ehrenfest theorem. Here it is shown that transverse light dynamics in an optical resonator with off-axis longitudinal pumping can emulate a $\\mathcal{PT}$-symmetric quantum harmonic oscillator, providing an experimentally accessible system to investigate non-Hermitian coherent state propagation.
Quantum mechanics of $PT$ and non-$PT$ -symmetric potentials in three dimensions
Indian Academy of Sciences (India)
BHARDWAJ S B; SINGH RAM MEHAR; MISHRA S C
2016-07-01
With a view of exploring new vistas with regard to the nature of complex eigenspectra of a non-Hermitian Hamiltonian, the quasi-exact solutions of the Schrödinger equation are investigated for a shifted harmonic potential under the framework of extended complex phase-space approach. Analyticity property ofthe eigenfunction alone is found sufficient to throw light on the nature of the eigenvalues and eigenfunctions of a system. Explicit expressions of eigenvalues and eigenfunctions for the ground state as well as excited state including their $PT$-symmetric version are worked out.
Moduli spaces of polarised symplectic O'Grady varieties and Borcherds products
Gritsenko, V; Sankaran, G K
2010-01-01
We study moduli spaces of O'Grady's ten-dimensional irreducible symplectic manifolds. These moduli spaces are covers of modular varieties of dimension 21, namely quotients of hermitian symmetric domains by a suitable arithmetic group. The interesting and new aspect of this case is that the group in question is strictly bigger than the stable orthogonal group. This makes it different from both the K3 and the K3^[n] case, which are of dimension 19 and 20 respectively.
LETTER TO THE EDITOR: Remarks on the construction of a Hermitian phase operator
Roy, P.; Roy, B.
1997-12-01
We consider an oscillator system (whose states are realized in the space of Kravchuk polynomials) in a finite-dimensional Hilbert space. A Hermitian phase operator and the phase quanta raising and lowering operator are constructed using this framework. Also, we have constructed phase coherent states and examined their squeezing property.
Abelman, Herven; Abelman, Shirley
2014-01-01
Unacceptable principal powers in well-centred lenses may require a toric over-refraction which differs in nature from the one where correct powers have misplaced meridians. This paper calculates residual (over) refractions and their natures. The magnitude of the power of the over-refraction serves as a general, reliable, real scalar criterion for acceptance or tolerance of lenses whose surface relative curvatures change or whose meridians are rotated and cause powers to differ. Principal powers and meridians of lenses are analogous to eigenvalues and eigenvectors of symmetric matrices, which facilitates the calculation of powers and their residuals. Geometric paths in symmetric power space link intended refractive correction and these carefully chosen, undue refractive corrections. Principal meridians alone vary along an arc of a circle centred at the origin and corresponding powers vary autonomously along select diameters of that circle in symmetric power space. Depending on the path of the power change, residual lenses different from their prescription in principal powers and meridians are pure cross-cylindrical or spherocylindrical in nature. The location of residual power in symmetric dioptric power space and its optical cross-representation characterize the lens that must be added to the compensation to attain the power in the prescription.
Directory of Open Access Journals (Sweden)
Herven Abelman
2014-01-01
Full Text Available Unacceptable principal powers in well-centred lenses may require a toric over-refraction which differs in nature from the one where correct powers have misplaced meridians. This paper calculates residual (over refractions and their natures. The magnitude of the power of the over-refraction serves as a general, reliable, real scalar criterion for acceptance or tolerance of lenses whose surface relative curvatures change or whose meridians are rotated and cause powers to differ. Principal powers and meridians of lenses are analogous to eigenvalues and eigenvectors of symmetric matrices, which facilitates the calculation of powers and their residuals. Geometric paths in symmetric power space link intended refractive correction and these carefully chosen, undue refractive corrections. Principal meridians alone vary along an arc of a circle centred at the origin and corresponding powers vary autonomously along select diameters of that circle in symmetric power space. Depending on the path of the power change, residual lenses different from their prescription in principal powers and meridians are pure cross-cylindrical or spherocylindrical in nature. The location of residual power in symmetric dioptric power space and its optical cross-representation characterize the lens that must be added to the compensation to attain the power in the prescription.
Efficient numerical diagonalization of hermitian 3x3 matrices
Kopp, J
2006-01-01
A very common problem in science is the numerical diagonalization of symmetric or hermitian 3x3 matrices. Since standard "black box" packages may be very inefficient if the number of matrices is large, we study several alternatives. We consider optimized implementations of the Jacobi, QL, and Cuppen algorithms and compare them with a new, carefully designed analytical method relying on Cardano's formula for the eigenvalues and on vector cross products for the eigenvectors. This analytical algorithm outperforms the other algorithms by more than a factor of 2, but may be less accurate if the eigenvalues differ greatly in magnitude. Jacobi is the most accurate, but also the slowest method, while QL and Cuppen are good general purpose algorithms. For all algorithms, we give an overview of the underlying mathematical ideas, and present detailed benchmark results. C and Fortran implementations of our code are available for download from http://www.mpi-hd.mpg.de/~jkopp/3x3/ .
Zhu, Yi-Fan; Fan, Xu-Dong; Liang, Bin; Zou, Xin-Ye; Yang, Jing; Cheng, Jian-Chun
2016-01-01
Non-Hermitian systems always play a negative role in wave manipulations due to inherent non-conservation of energy as well as loss of information. Recently, however, there has been a paradigm shift on utilizing non-Hermitian systems to implement varied miraculous wave controlling. For example, parity-time symmetric media with well-designed loss and gain are presented to create a nontrivial effect of unidirectional diffraction, which is observed near the exceptional points (EPs) in the non-Hermitian systems. Here, we report the design and realization of non-Hermitian acoustic metamaterial (NHAM) and show that by judiciously tailoring the inherent loss, the phase and amplitude of reflection can possibly be tuned in a decoupled manner. Such decoupled tuning of phase and amplitude is closely related to the EPs. As a demonstration of functionality, we experimentally generate a high-quality acoustic hologram via NHAM. Our work may open a new degree of freedom for realizing the complete control of sound.
Krylov subspace methods for complex non-Hermitian linear systems. Thesis
Freund, Roland W.
1991-01-01
We consider Krylov subspace methods for the solution of large sparse linear systems Ax = b with complex non-Hermitian coefficient matrices. Such linear systems arise in important applications, such as inverse scattering, numerical solution of time-dependent Schrodinger equations, underwater acoustics, eddy current computations, numerical computations in quantum chromodynamics, and numerical conformal mapping. Typically, the resulting coefficient matrices A exhibit special structures, such as complex symmetry, or they are shifted Hermitian matrices. In this paper, we first describe a Krylov subspace approach with iterates defined by a quasi-minimal residual property, the QMR method, for solving general complex non-Hermitian linear systems. Then, we study special Krylov subspace methods designed for the two families of complex symmetric respectively shifted Hermitian linear systems. We also include some results concerning the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.
Barrios, Nahuel; Pullin, Jorge
2015-01-01
We consider a massive scalar field living on the recently found exact quantum space-time corresponding to vacuum spherically symmetric loop quantum gravity. The discreteness of the quantum space time naturally regularizes the scalar field, eliminating divergences. However, the resulting finite theory depends on the details of the micro physics. We argue that such dependence can be eliminated through a finite renormalization and discuss its nature. This is an example of how quantum field theories on quantum space times deal with the issues of divergences in quantum field theories.
Small zeros of hermitian forms over a quaternion algebra
Chan, Wai Kiu
2009-01-01
Let $D$ be a positive definite quaternion algebra over a totally real number field $K$, $F(X,Y)$ a hermitian form in 2N variables over $D$, and $Z$ a right $D$-vector space which is isotropic with respect to $F$. We prove the existence of a small-height basis for $Z$ over $D$, such that $F(X,X)$ vanishes at each of the basis vectors. This constitutes a non-commutative analogue of a theorem of Vaaler, and presents an extension of the classical theorem of Cassels on small zeros of rational quadratic forms to the context of quaternion algebras.
Geometry of 2×2 hermitian matrices
Institute of Scientific and Technical Information of China (English)
HUANG; Liping(黄礼平); WAN; Zhexian(万哲先)
2002-01-01
Let D be a division ring which possesses an involution a→ā. Assume that F = {a∈D|a=ā} is a proper subfield of D and is contained in the center of D. It is pointed out that if D is of characteristic not two, D is either a separable quadratic extension of F or a division ring of generalized quaternions over F and that if D is of characteristic two, D is a separable quadratic extension of F. Thus the trace map Tr: D→F,hermitian matrices over D when n≥3 and now can be deleted. When D is a field, the fundamental theorem of 2×2 hermitian matrices over D has already been proved. This paper proves the fundamental theorem of 2×2 hermitian matrices over any division ring of generalized quaternions of characteristic not two.
FEAST Eigensolver for non-Hermitian Problems
Kestyn, James; Tang, Ping Tak Peter
2015-01-01
A detailed new upgrade of the FEAST eigensolver targeting non-Hermitian eigenvalue problems is presented and thoroughly discussed. It aims at broadening the class of eigenproblems that can be addressed within the framework of the FEAST algorithm. The algorithm is ideally suited for computing selected interior eigenvalues and their associated right/left bi-orthogonal eigenvectors,located within a subset of the complex plane. It combines subspace iteration with efficient contour integration techniques that approximate the left and right spectral projectors. We discuss the various algorithmic choices that have been made to improve the stability and usability of the new non-Hermitian eigensolver. The latter retains the convergence property and multi-level parallelism of Hermitian FEAST, making it a valuable new software tool for the scientific community.
Indian Academy of Sciences (India)
G Santhanum
2007-08-01
Let be a closed hypersurface in a simply connected rank-1 symmetric space $\\overline{M}$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of in terms of the Ricci curvature of $\\overline{M}$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of .
Institute of Scientific and Technical Information of China (English)
Ananya Ghatak; Bhabani Prasad Mandal
2013-01-01
To develop a unitary quantum theory with probabilistic description for pseudo-Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator.There are different approaches to find such metric operators.We compare the different approaches of calculating positive definite metric operators in pseudo-Hermitian theories with the help of several explicit examples in non-relativistic as well as in relativistic situations.Exceptional points and spontaneous symmetry breaking are also discussed in these models.
CONSTRUCTION OF INDECOMPOSABLE DEFINITE HERMITIAN FORMS
Institute of Scientific and Technical Information of China (English)
ZHUFUZU
1994-01-01
This paper gives a method to construct indecomposable positive definite integral Hermitian forms over an imaginary qusadratic field Q(√-m)with given discriminant and given rank.It is shown that for any natural numbers n and a, there are n-ary indecompossble positive definite integral Hermitian lattices over Q(√-1)(resp.Q(√-2)with discriminant a1 except for four (resp.one) exceptions. In these exceptional cases there are no lattices with the desired properties.
Revisiting the Optical PT-Symmetric Dimer
Directory of Open Access Journals (Sweden)
José Delfino Huerta Morales
2016-08-01
Full Text Available Optics has proved a fertile ground for the experimental simulation of quantum mechanics. Most recently, optical realizations of PT -symmetric quantum mechanics have been shown, both theoretically and experimentally, opening the door to international efforts aiming at the design of practical optical devices exploiting this symmetry. Here, we focus on the optical PT -symmetric dimer, a two-waveguide coupler where the materials show symmetric effective gain and loss, and provide a review of the linear and nonlinear optical realizations from a symmetry-based point of view. We go beyond a simple review of the literature and show that the dimer is just the smallest of a class of planar N-waveguide couplers that are the optical realization of the Lorentz group in 2 + 1 dimensions. Furthermore, we provide a formulation to describe light propagation through waveguide couplers described by non-Hermitian mode coupling matrices based on a non-Hermitian generalization of the Ehrenfest theorem.
Revisiting the optical $PT$-symmetric dimer
Morales, J D Huerta; López-Aguayo, S; Rodríguez-Lara, B M
2016-01-01
Optics has proved a fertile ground for the experimental simulation of quantum mechanics. Most recently, optical realizations of $\\mathcal{PT}$-symmetric quantum mechanics have been shown, both theoretically and experimentally, opening the door to international efforts aiming at the design of practical optical devices exploiting this symmetry. Here, we focus on the optical $\\mathcal{PT}$-symmetric dimer, a two-waveguide coupler were the materials show symmetric effective gain and loss, and provide a review of the linear and nonlinear optical realizations from a symmetry based point of view. We go beyond a simple review of the literature and show that the dimer is just the smallest of a class of planar $N$-waveguide couplers that are the optical realization of Lorentz group in 2+1 dimensions. Furthermore, we provide a formulation to describe light propagation through waveguide couplers described by non-Hermitian mode coupling matrices based on a non-Hermitian generalization of Ehrenfest theorem.
Acik, O; Önder, M; Vercin, A
2008-01-01
Killing-Yano (KY) two and three forms of a class of spherically symmetric space-times that includes the well-known Minkowski, Schwarzschild, Reissner-Nordstrom, Robertson-Walker and six different forms of de Sitter space-times as special cases are derived in a unified and exhaustive manner. It is directly proved that while the Schwarzschild and Reissner-Nordstrom space-times do not accept any KY 3-form and they accept only one 2-form, the Robertson-Walker space-time admits four KY 2-forms and only one KY 3-form. Maximal number of KY-forms are obtained for Minkowski and all known forms of de Sitter space-times. Complete lists comprising explicit expressions of KY-forms are given.
Coulomb analogy for non-Hermitian degeneracies near quantum phase transitions.
Cejnar, Pavel; Heinze, Stefan; Macek, Michal
2007-09-07
Degeneracies near the real axis in a complex-extended parameter space of a Hermitian Hamiltonian are studied. We present a method to measure distributions of such degeneracies on the Riemann sheet of a selected level and apply it in classification of quantum phase transitions. The degeneracies are shown to behave similarly as complex zeros of a partition function.
Quantum entropy of non-Hermitian entangled systems
Zhang, Shi-Yang; Fang, Mao-Fa; Xu, Lan
2017-10-01
Non-Hermitian Hamiltonians are an effective tool for describing the dynamics of open quantum systems. Previous research shows that the restrictions of conventional quantum mechanics may be violated in the non-Hermitian cases. We studied the entropy of a system of entangled qubits governed by a local non-Hermitian Hamiltonian operator. We find that local non-Hermitian operation influences the entropies of the two subsystems equally and simultaneously. This indicates that non-Hermitian operators possess the property of non-locality, which makes information exchange possible between subsystems. These information exchanges reduce the uncertainty of outcomes associated with two incompatible quantum measurements.
Indian Academy of Sciences (India)
Fakir Chand; S C Mishra; Ram Mehar Singh
2009-08-01
We investigate the quasi-exact solutions of an analogous Schrödinger wave equation for two-dimensional non-Hermitian complex Hamiltonian systems within the framework of an extended complex phase space characterized by = 1 + 3, = 2 + 4, = 1 + 3, = 2 + 4. Explicit expressions for the energy eigenvalues and eigenfunctions for ground and first excited states of a two-dimensional $\\mathcal{PT}$-symmetric sextic potential and some of its variants are obtained. The eigenvalue spectra are found to be real within some parametric domains.
Einstein Hermitian Metrics of Positive Sectional Curvature
Koca, Caner
2011-01-01
In this paper we will prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is the complex projective plane CP^2, with its Fubini-Study metric.
Non-Hermitian Euclidean random matrix theory.
Goetschy, A; Skipetrov, S E
2011-07-01
We develop a theory for the eigenvalue density of arbitrary non-Hermitian Euclidean matrices. Closed equations for the resolvent and the eigenvector correlator are derived. The theory is applied to the random Green's matrix relevant to wave propagation in an ensemble of pointlike scattering centers. This opens a new perspective in the study of wave diffusion, Anderson localization, and random lasing.
On the subfield subcodes of Hermitian codes
DEFF Research Database (Denmark)
Pinero, Fernando; Janwa, Heeralal
2014-01-01
We present a fast algorithm using Gröbner basis to compute the dimensions of subfield subcodes of Hermitian codes. With these algorithms we are able to compute the exact values of the dimension of all subfield subcodes up to q ≤ 32 and length up to 215. We show that some of the subfield subcodes ...
Generalized scalar tensor theory in four and higher dimensional spherically symmetric space-time
Indian Academy of Sciences (India)
Subenoy Chakraborty; Arabinda Ghosh
2001-05-01
In this paper, we have studied generalized scalar tensor theory for spherically symmetric models, both in four and higher dimensions with a bulk viscous fluid. We have considered both exponential and power law solutions with some assumptions among the physical parameters and solutions have been discussed.
Institute of Scientific and Technical Information of China (English)
韩敬稳; 郑宝东
2007-01-01
Let Q be the quaternion division algebra over real field F.Denote by Hn(Q)the set of all n×n hermitian matrices over Q.We characterize the additive maps from Hn(Q)into Hm(Q)that preserve rank-1 matrices when the rank of the image of In is equal to n.Let QR be the quaternion division algebra over the field of real number R.The additive maps from Hn(QR)into Hm(QR)that preserve rank-1 matrices are also given.
Exact diagonalization of non-Hermitian so(3,2) models: Generalized two-mode boson systems
Zhang, Hong-Biao; Wang, Gangcheng
2016-12-01
We propose a unified approach to exactly diagonalize generalized non-Hermitian so(3,2) models. This approach is a series of similarity transformations, which is constructed by some similarity transformation operators associated with su(1,1) and su(2) subalgebras of so(3,2) Lie algebra. During this diagonalization, it is worth noting that a key step is to get rid of the terms E ˆ ± and F ˆ ± together via the proper similarity transformations first. In this way, exact solutions of the non-Hermitian so(3,2) models are obtained. Meanwhile we give the corresponding eigenstates, which are regarded as Lie algebra so(3,2) coherent-like number states. The results can cover the generic form of the eigenvalues and eigenstates to the generalized non-Hermitian two-mode boson systems with the discrete spectrum, including 2D PT-symmetric and non-PT-symmetric oscillators as the special cases. Also they are true for the Hermitian case.
Derivation of Klein-Gordon-Fock equation from General relativity in a time-space symmetrical model
Van Thuan, Vo
2016-01-01
Following a bi-cylindrical model of geometrical dynamics, in the present study we show that Einstein gravitational equation leads to bi-geodesic description in an extended symmetrical time-space which fit Hubble expansion in a "microscopic" cosmological model. As a duality, the geodesic solution is mathematically equivalent to the basic Klein-Gordon-Fock equations of free massive elementary particles, in particular, as the squared Dirac equations of leptons and as a sub-solution with pseudo-axion. This result would serve an explicit approach to consistency between quantum mechanics and general relativity.
Classifying spaces of degenerating polarized Hodge structures
Kato, Kazuya
2009-01-01
In 1970, Phillip Griffiths envisioned that points at infinity could be added to the classifying space D of polarized Hodge structures. In this book, Kazuya Kato and Sampei Usui realize this dream by creating a logarithmic Hodge theory. They use the logarithmic structures begun by Fontaine-Illusie to revive nilpotent orbits as a logarithmic Hodge structure. The book focuses on two principal topics. First, Kato and Usui construct the fine moduli space of polarized logarithmic Hodge structures with additional structures. Even for a Hermitian symmetric domain D, the present theory is a refinem
PT-Symmetric Quantum Electrodynamics
Bender, C M; Milton, K A; Shajesh, K V; Bender, Carl M.; Cavero-Pelaez, Ines; Milton, Kimball A.
2005-01-01
The Hamiltonian for quantum electrodynamics becomes non-Hermitian if the unrenormalized electric charge $e$ is taken to be imaginary. However, if one also specifies that the potential $A^\\mu$ in such a theory transforms as a pseudovector rather than a vector, then the Hamiltonian becomes PT symmetric. The resulting non-Hermitian theory of electrodynamics is the analog of a spinless quantum field theory in which a pseudoscalar field $\\phi$ has a cubic self-interaction of the form $i\\phi^3$. The Hamiltonian for this cubic scalar field theory has a positive spectrum, and it has recently been demonstrated that the time evolution of this theory is unitary. The proof of unitarity requires the construction of a new operator called C, which is then used to define an inner product with respect to which the Hamiltonian is self-adjoint. In this paper the corresponding C operator for non-Hermitian quantum electrodynamics is constructed perturbatively. This construction demonstrates the unitarity of the theory. Non-Hermit...
PT-Symmetric Cubic Anharmonic Oscillator as a Physical Model
Mostafazadeh, A
2004-01-01
We perform a perturbative calculation of the physical observables, in particular pseudo-Hermitian position and momentum operators, the equivalent Hermitian Hamiltonian operator, and the classical Hamiltonian for the PT-symmetric cubic anharmonic oscillator, $ H=p^1/(2m)+\\mu^2x^2/2+i\\epsilon x^3 $. Ignoring terms of order $ \\epsilon^4 $ and higher, we show that this system describes an ordinary quartic anharmonic oscillator with a position-dependent mass and real and positive coupling constants. This observation elucidates the classical origin of the reality and positivity of the energy spectrum. We also discuss the quantum-classical correspondence for this PT-symmetric system, compute the associated conserved probability density, and comment on the issue of factor-ordering in the pseudo-Hermitian canonical quantization of the underlying classical system.
Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains
Institute of Scientific and Technical Information of China (English)
Guang Bin REN; Ji Huai SHI
2002-01-01
Sharp estimates of the point-evaluation functional in weighted Bergman spaces Lpa(Ω, dva)and for the point-evaluation derivalive functional in Besov spaces BP(Ω) are obtained for boundedsymmetric domains Ω in Cn.
Decoding Hermitian Codes with Sudan's Algorithm
DEFF Research Database (Denmark)
Høholdt, Tom; Nielsen, Rasmus Refslund
1999-01-01
We present an efficient implementation of Sudan's algorithm for list decoding Hermitian codes beyond half the minimum distance. The main ingredients are an explicit method to calculate so-called increasing zero bases, an efficient interpolation algorithm for finding the Q-polynomial......, and a reduction of the problem of factoring the Q-polynomial to the problem of factoring a univariate polynomial over a large finite field....
Matrix Algebras in Non-Hermitian Quantum Mechanics
Institute of Scientific and Technical Information of China (English)
Alessandro Sergi
2011-01-01
In principle, non-Hermitian quantum equations of motion can be formulated using as a starting point either the Heisenberg's or the Schr(o)dinger's picture of quantum dynamics. Here it is shown in both cases how to map the algebra of commutators, defining the time evolution in terms of a non-Hermitian Hamiltonian, onto a non-Hamiltonian algebra with a Hermitian Hamiltonian. The logic behind such a derivation is reversible, so that any Hermitian Hamiltonian can be used in the formulation of non-Hermitian dynamics through a suitable algebra of generalized (non-Hamiltonian)commutators. These results provide a general structure (a template) for non-Hermitian equations of motion to be used in the computer simulation of open quantum systems dynamics.
Natural Diagonal Riemannian Almost Product and Para-Hermitian Cotangent Bundles
Druta-Romaniuc, Simona-Luiza
2011-01-01
We obtain the natural diagonal almost product and locally product structures on the total space of the cotangent bundle of a Riemannian manifold. We find the Riemannian almost product (locally product) and the (almost) para-Hermitian cotangent bundles of natural diagonal lift type. We prove the characterization theorem for the natural diagonal (almost) para-K\\"ahlerian structures on the total spaces of the cotangent bundle.
Institute of Scientific and Technical Information of China (English)
Hideo KUBO; K(o)ji KUBOTA
2006-01-01
This paper is concerned with a class of semilinear hyperbolic systems in odd space dimensions. Our main aim is to prove the existence of a small amplitude solution which is asymptotic to the free solution as t → -∞ in the energy norm, and to show it has a free profile as t → +∞. Our approach is based on the work of [11]. Namely we use a weighted L∞ norm to get suitable a priori estimates. This can be done by restricting our attention to radially symmetric solutions. Corresponding initial value problem is also considered in an analogous framework. Besides, we give an extended result of [14] for three space dimensional case in Section 5, which is prepared independently of the other parts of the paper.
Several splittings for non-Hermitian linear systems
Institute of Scientific and Technical Information of China (English)
2008-01-01
For large sparse non-Hermitian positive definite system of linear equations,we present several variants of the Hermitian and skew-Hermitian splitting(HSS)about the coefficient matrix and establish correspondingly several HSS-based iterative schemes.Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations,and they may show advantages on problems that the HSS method is ineffiective.
Several splittings for non-Hermitian linear systems
Institute of Scientific and Technical Information of China (English)
BAI Zhong-Zhi
2008-01-01
For large sparse non-Hermitian positive definite system of linear equations, we present several variants of the Hermitian and skew-Hermitian splitting (HSS) about the coefficient matrix and establish correspondingly several HSS-based iterative schemes. Theoretical analyses show that these methods are convergent unconditionally to the exact solution of the referred system of linear equations,and they may show advantages on problems that the HSS method is ineffective.
Ornstein-Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links
Blaizot, Jean-Paul; Grela, Jacek; Nowak, Maciej A.; Tarnowski, Wojciech; Warchoł, Piotr
2016-05-01
We compare the Ornstein-Uhlenbeck process for the Gaussian unitary ensemble to its non-hermitian counterpart—for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green’s functions, which involves a new, hidden complex variable, in comparison to the standard treatment of the resolvents. This new variable turns out to be crucial to understand the pattern of the evolution of non-hermitian systems. The new feature is the emergence of the coupling between the flow of eigenvalues and that of left/right eigenvectors. We analyze local and global equilibria for both systems. Finally, we highlight some unexpected links between both ensembles.
Exceptional contours and band structure design in parity-time symmetric photonic crystals
Cerjan, Alexander; Fan, Shanhui
2016-01-01
We investigate the properties of multidimensional parity-time symmetric periodic systems whose non-Hermitian periodicity is an integer multiple of the underlying Hermitian system's periodicity. This creates a natural set of degeneracies which can undergo thresholdless $\\mathcal{PT}$ transitions. We derive a $\\mathbf{k} \\cdot \\mathbf{p}$ perturbation theory suited to the continuous eigenvalues of such systems in terms of the modes of the underlying Hermitian system. In photonic crystals, such thresholdless $\\mathcal{PT}$ transitions are shown to yield significant control over the band structure of the system, and can result in all-angle supercollimation, a $\\mathcal{PT}$-superprism effect, and unidirectional behavior.
Exceptional Contours and Band Structure Design in Parity-Time Symmetric Photonic Crystals.
Cerjan, Alexander; Raman, Aaswath; Fan, Shanhui
2016-05-20
We investigate the properties of two-dimensional parity-time symmetric periodic systems whose non-Hermitian periodicity is an integer multiple of the underlying Hermitian system's periodicity. This creates a natural set of degeneracies that can undergo thresholdless PT transitions. We derive a k·p perturbation theory suited to the continuous eigenvalues of such systems in terms of the modes of the underlying Hermitian system. In photonic crystals, such thresholdless PT transitions are shown to yield significant control over the band structure of the system, and can result in all-angle supercollimation, a PT-superprism effect, and unidirectional behavior.
Institute of Scientific and Technical Information of China (English)
罗少盈; 刘琦
2014-01-01
In this article, we concern the motion of relativistic membranes and null mem-branes in the Reissner-Nordstr¨om space-time. The equation of relativistic membranes moving in the Reissner-Nordstr¨om space-time is derived and some properties are discussed. Spherical symmetric solutions for the motion are illustrated and some interesting physical phenomena are discovered. The equations of the null membranes are derived and the exact solutions are also given. Spherical symmetric solutions for null membranes are just the two horizons of Reissner-Nordstr¨om space-time.
Joglekar, Yogesh N
2010-01-01
We study the properties of a parity- and time-reversal- (PT) symmetric tight-binding chain of size N with position-dependent hopping amplitude. In contrast to the fragile PT-symmetric phase of a chain with constant hopping and imaginary impurity potentials, we show that, under very general conditions, our model is {\\it always} in the PT-symmetric phase. We numerically obtain the energy spectrum and the density of states of such a chain, and show that they are widely tunable. By studying the size-dependence of inverse participation ratios, we show that although the chain is not translationally invariant, most of its eigenstates are extended. Our results indicate that tight-binding models with non-Hermitian PT-symmetric hopping have a robust PT-symmetric phase and rich dynamics.
Holographic CFTs on maximally symmetric spaces: Correlators, integral transforms, and applications
Hinterbichler, Kurt; Stokes, James; Trodden, Mark
2015-09-01
We study one- and two-point functions of conformal field theories (CFTs) on spaces of maximal symmetry with and without boundaries and investigate their spectral representations. Integral transforms are found, relating the spectral decomposition to renormalized position-space correlators. Several applications are presented, including the holographic boundary CFTs as well as spacelike boundary CFTs, which provide realizations of the pseudoconformal universe.
Anomalous edge state in a non-Hermitian lattice
Lee, Tony E
2016-01-01
We show that the bulk-boundary correspondence for topological insulators can be modified in the presence of non-Hermiticity. We consider a one-dimensional tight-binding model with gain and loss as well as long-range hopping. The system is described by a non-Hermitian Hamiltonian that encircles an exceptional point in momentum space. The winding number has a fractional value of 1/2. There is only one dynamically stable zero-energy edge state due to the defectiveness of the Hamiltonian. This edge state is robust to disorder due to protection by a chiral symmetry. We also discuss experimental realization with arrays of coupled resonator optical waveguides.
Sublattice signatures of transitions in a $\\mathcal{PT}$-symmetric dimer lattice
Harter, Andrew K
2016-01-01
Lattice models with non-hermitian, parity and time-reversal ($\\mathcal{PT}$) symmetric Hamiltonians, realized most readily in coupled optical systems, have been intensely studied in the past few years. A $\\mathcal{PT}$-symmetric dimer lattice consists of dimers with intra-dimer coupling $\
Qureshi, Muhammad Amer; Mahomed, K S
2016-01-01
A study of proper teleparallel conformal vector field in spherically symmetric static space-times is given using the direct integration technique and diagonal tetrads. In this study we show that the above space-times do not admit proper teleparallel conformal vector fields.
Fring, Andreas
2016-01-01
We propose a procedure to obtain exact analytical solutions to the time-dependent Schr\\"{o}dinger equations involving explicit time-dependent Hermitian Hamitonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.
Fring, Andreas; Frith, Thomas
2017-01-01
We propose a procedure to obtain exact analytical solutions to the time-dependent Schrödinger equations involving explicit time-dependent Hermitian Hamiltonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation, together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.
Nonlinear waves in $\\cal PT$-symmetric systems
Konotop, Vladimir V.; Yang, Jianke; Zezyulin, Dmitry A.
2016-01-01
Recent progress on nonlinear properties of parity-time ($\\cal PT$-) symmetric systems is comprehensively reviewed in this article. $\\cal PT$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\\cal PT$ symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a $\\cal PT$-symmetric system. The nat...
Axially Symmetric Null Dust Space-Time, Naked Singularity, and Cosmic Time Machine
National Research Council Canada - National Science Library
Faizuddin Ahmed
2017-01-01
... the cylinder which has closed orbits. The space-time admits closed timelike curves (CTCs) which develop at some particular moment in a causally well-behaved manner and may represent a Cosmic Time Machine...
Efficient Numerical Diagonalization of Hermitian 3 × 3 Matrices
Kopp, Joachim
A very common problem in science is the numerical diagonalization of symmetric or hermitian 3 × 3 matrices. Since standard "black box" packages may be too inefficient if the number of matrices is large, we study several alternatives. We consider optimized implementations of the Jacobi, QL, and Cuppen algorithms and compare them with an alytical method relying on Cardano's formula for the eigenvalues and on vector cross products for the eigenvectors. Jacobi is the most accurate, but also the slowest method, while QL and Cuppen are good general purpose algorithms. The analytical algorithm outperforms the others by more than a factor of 2, but becomes inaccurate or may even fail completely if the matrix entries differ greatly in magnitude. This can mostly be circumvented by using a hybrid method, which falls back to QL if conditions are such that the analytical calculation might become too inaccurate. For all algorithms, we give an overview of the underlying mathematical ideas, and present detailed benchmark results. C and Fortran implementations of our code are available for download from .
Hermitian 随机矩阵特征值%Eigenvalues of the Hermitian Random Matrices
Institute of Scientific and Technical Information of China (English)
马明玥; 付志慧
2016-01-01
利用随机矩阵理论中的矩方法研究一类 Hermitian 随机矩阵极端特征值的极限性质。结果表明,Hermitian 随机矩阵的极端特征值几乎处处有界；特别地,对任意固定整数 m,有lim infn→∞λm 1 n H æèçöø÷ n ≥2σ,lim supn→∞λn-m 1 n H æèçöø÷ n ≤-2σ,其中σ2=mink,l σ2kl 。%Using the moment methods of the random matrix theory,we studied the limit properties of extreme eigenvalues of a class of Hermitian random matrices.The result shows that the extreme eigenvalue of the Hermitian random matrices H n is bounded almost everywhere,especially for any fixed integer m,lim inf n→∞λm 1 n H æ è ç ö ø ÷ n ≥ 2σ,lim sup n→∞λn-m 1 n H æ è ç ö ø ÷ n ≤- 2σ,whereσ2 =min k,lσ2kl .
DÍaz, R.; Rivas, M.
2010-01-01
In order to study Boolean algebras in the category of vector spaces we introduce a prop whose algebras in set are Boolean algebras. A probabilistic logical interpretation for linear Boolean algebras is provided. An advantage of defining Boolean algebras in the linear category is that we are able to study its symmetric powers. We give explicit formulae for products in symmetric and cyclic Boolean algebras of various dimensions and formulate symmetric forms of the inclusion-exclusion principle.
Tang, Jian-Shun; Wang, Yi-Tao; Yu, Shang; He, De-Yong; Xu, Jin-Shi; Liu, Bi-Heng; Chen, Geng; Sun, Yong-Nan; Sun, Kai; Han, Yong-Jian; Li, Chuan-Feng; Guo, Guang-Can
2016-10-01
The experimental progress achieved in parity-time () symmetry in classical optics is the most important accomplishment in the past decade and stimulates many new applications, such as unidirectional light transport and single-mode lasers. However, in the quantum regime, some controversial effects are proposed for -symmetric theory, for example, the potential violation of the no-signalling principle. It is therefore important to understand whether -symmetric theory is consistent with well-established principles. Here, we experimentally study this no-signalling problem related to the -symmetric theory using two space-like separated entangled photons, with one of them passing through a post-selected quantum gate, which effectively simulates a -symmetric evolution. Our results suggest that the superluminal information transmission can be simulated when the successfully -symmetrically evolved subspace is solely considered. However, considering this subspace is only a part of the full Hermitian system, additional information regarding whether the -symmetric evolution is successful is necessary, which transmits to the receiver at maximally light speed, maintaining the no-signalling principle.
Continuous block-symmetric polynomials of degree at most two on the space $(L_\\infty^2$
Directory of Open Access Journals (Sweden)
T. V. Vasylyshyn
2016-06-01
Full Text Available We introduce block-symmetric polynomials on $(L_\\infty^2$ and prove that every continuous block-symmetric polynomial of degree at most two on $(L_\\infty^2$ can be uniquely represented by some ``elementary'' block-symmetric polynomials.
An optimum Hamiltonian for non-Hermitian quantum evolution and the complex Bloch sphere
Energy Technology Data Exchange (ETDEWEB)
Nesterov, Alexander I., E-mail: nesterov@cencar.udg.m [Departamento de Fisica, CUCEI, Universidad de Guadalajara, Av. Revolucion 1500, Guadalajara, CP 44420, Jalisco (Mexico)
2009-09-28
For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time tau. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.
Shikakhwa, M. S.; Chair, N.
2017-01-01
We construct the Hermitian Schrödinger Hamiltonian of spin-less particles and the gauge-covariant Pauli Hamiltonian of spin one-half particles in a magnetic field, which are confined to cylindrical and spherical surfaces. The approach does not require the use of involved differential-geometrical methods and is intuitive and physical, relying on the general requirements of Hermicity and gauge-covariance. The surfaces are embedded in the full three-dimensional space and confinement to the surfaces is achieved by strong radial potentials. We identify the Hermitian and gauge-covariant (in the presence of a magnetic field) physical radial momentum in each case and set it to zero upon confinement to the surfaces. The resulting surface Hamiltonians are seen to be automatically Hermitian and gauge-covariant. The well-known geometrical kinetic energy also emerges naturally.
Directory of Open Access Journals (Sweden)
Marwan Amin Kutbi
2014-01-01
weakly compatible mappings in symmetric spaces satisfying generalized (ψ,φ-contractive conditions employing the common limit range property. We furnish some interesting examples which support our main theorems. Our results generalize and extend some recent results contained in Imdad et al. (2013 to symmetric spaces. Consequently, a host of metrical common fixed theorems are generalized and improved. In the process, we also derive a fixed point theorem for four finite families of mappings which can be utilized to derive common fixed point theorems involving any number of finite mappings.
Numerical relativity for D dimensional axially symmetric space-times: Formalism and code tests
Zilhão, Miguel; Witek, Helvi; Sperhake, Ulrich; Cardoso, Vitor; Gualtieri, Leonardo; Herdeiro, Carlos; Nerozzi, Andrea
2010-04-01
The numerical evolution of Einstein’s field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modeling black hole production in TeV gravity scenarios, to analysis of the stability of exact solutions, and to tests of cosmic censorship. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of D dimensional vacuum space-times with an SO(D-2) isometry group for D≥5, or SO(D-3) for D≥6. Performing a dimensional reduction on a (D-4) sphere, the D dimensional vacuum Einstein equations are rewritten as a 3+1 dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata, and Nakamura formulation. This allows the use of existing 3+1 dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in D dimensions and a procedure to match them to our 3+1 dimensional evolution equations is given. We have implemented our framework by adapting the Lean code and perform a variety of simulations of nonspinning black hole space-times. Specifically, we present a modified moving puncture gauge, which facilitates long-term stable simulations in D=5. We further demonstrate the internal consistency of the code by studying convergence and comparing numerical versus analytic results in the case of geodesic slicing for D=5, 6.
Numerical relativity for D dimensional axially symmetric space-times: formalism and code tests
Zilhao, Miguel; Sperhake, Ulrich; Cardoso, Vitor; Gualtieri, Leonardo; Herdeiro, Carlos; Nerozzi, Andrea
2010-01-01
The numerical evolution of Einstein's field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modelling black hole production in TeV gravity scenarios, analysis of the stability of exact solutions and tests of Cosmic Censorship. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of D dimensional vacuum space-times with an SO(D-2) isometry group for D\\ge 5, or SO(D-3) for D\\ge 6. Performing a dimensional reduction on a (D-4)-sphere, the D dimensional vacuum Einstein equations are rewritten as a 3+1 dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata and Nakamura (BSSN) formulation. This allows the use of existing 3+1 dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in D dimensions an...
Harada, T; Iguchi, H; Harada, Tomohiro; Nakao, Ken-ichi; Iguchi, Hideo
1999-01-01
It was recently shown that the metric functions which describe a spherically symmetric space-time with vanishing radial pressure can be explicitly integrated. We investigate the nakedness and curvature strength of the shell-focusing singularity in that space-time. If the singularity is naked, the relation between the circumferential radius and the Misner-Sharp mass is given by $R\\approx 2y_{0} m^{\\beta}$ with $ 1/3<\\beta\\le 1$ along the first radial null geodesic from the singularity. The $\\beta$ is closely related to the curvature strength of the naked singularity. For example, for the outgoing or ingoing null geodesic, if the strong curvature condition (SCC) by Tipler holds, then $\\beta$ must be equal to 1. We define the ``gravity dominance condition'' (GDC) for a geodesic. If GDC is satisfied for the null geodesic, both SCC and the limiting focusing condition (LFC) by Królak hold for $\\beta=1$ and $y_{0}\
Topologically protected bound states in photonic parity-time-symmetric crystals.
Weimann, S; Kremer, M; Plotnik, Y; Lumer, Y; Nolte, S; Makris, K G; Segev, M; Rechtsman, M C; Szameit, A
2017-04-01
Parity-time (PT)-symmetric crystals are a class of non-Hermitian systems that allow, for example, the existence of modes with real propagation constants, for self-orthogonality of propagating modes, and for uni-directional invisibility at defects. Photonic PT-symmetric systems that also support topological states could be useful for shaping and routing light waves. However, it is currently debated whether topological interface states can exist at all in PT-symmetric systems. Here, we show theoretically and demonstrate experimentally the existence of such states: states that are localized at the interface between two topologically distinct PT-symmetric photonic lattices. We find analytical closed form solutions of topological PT-symmetric interface states, and observe them through fluorescence microscopy in a passive PT-symmetric dimerized photonic lattice. Our results are relevant towards approaches to localize light on the interface between non-Hermitian crystals.
Quantization of massive scalar fields over axis symmetric space-time backgrounds
Piedra, O P F; Oca, Alejandro Cabo Montes de; Piedra, Owen Pavel Fernandez
2007-01-01
The renormalized mean value of the quantum Lagrangian and the Energy-Momentum tensor for scalar fields coupled to an arbitrary gravitational field configuration are analytically evaluated in the Schwinger-DeWitt approximation, up to second order in the inverse mass value. The cylindrical symmetry situation is considered. The results furnish the starting point for investigating iterative solutions of the back-reaction problem related with the quantization of cylindrical scalar field configurations. Due to the homogeneity of the equations of motion of the Klein-Gordon field, the general results are also valid for performing the quantization over either vanishing or non-vanishing mean field configurations. As an application, compact analytical expressions are derived here for the quantum mean Lagrangian and Energy-Momentum tensor in the particular background given by the Black-String space-time.
Divide and conquer the Hilbert space of translation-symmetric spin systems.
Weisse, Alexander
2013-04-01
Iterative methods that operate with the full Hamiltonian matrix in the untrimmed Hilbert space of a finite system continue to be important tools for the study of one- and two-dimensional quantum spin models, in particular in the presence of frustration. To reach sensible system sizes such numerical calculations heavily depend on the use of symmetries. We describe a divide-and-conquer strategy for implementing translation symmetries of finite spin clusters, which efficiently uses and extends the "sublattice coding" of H. Q. Lin [Phys. Rev. B 42, 6561 (1990)]. With our method, the Hamiltonian matrix can be generated on-the-fly in each matrix vector multiplication, and problem dimensions beyond 10^{11} become accessible.
Mapping of error cells in clinical measure to symmetric power space.
Abelman, H; Abelman, S
2007-09-01
During the refraction procedure, the power of the nearest equivalent sphere lens, known as the scalar power, is conserved within upper and lower bounds in the sphere (and cylinder) lens powers. Bounds are brought closer together while keeping the circle of least confusion on the retina. The sphere and cylinder powers and changes in these powers are thus dependent. Changes are depicted in the cylinder-sphere plane by error cells with one pair of parallel sides of negative gradient and the other pair aligned with the graph axis of cylinder power. Scalar power constitutes a vector space, is a meaningful ophthalmic quantity and is represented by the semi-trace of the dioptric power matrix. The purpose of this article is to map to error cells for the following: coordinates of the dioptric power matrix, its principal powers and meridians and its entries from error cells surrounding powers in sphere, cylinder and axis. Error cells in clinical measure for conserved scalar power now contain more compensatory lens powers. Such cells and their respective mappings in terms of most scientific and alternate clinical quantities now image consistently not only to the cells from where they originate but also to each other.
Pseudo-Hermitian continuous-time quantum walks
Energy Technology Data Exchange (ETDEWEB)
Salimi, S; Sorouri, A, E-mail: shsalimi@uok.ac.i, E-mail: a.sorouri@uok.ac.i [Department of Physics, University of Kurdistan, PO Box 66177-15175, Sanandaj (Iran, Islamic Republic of)
2010-07-09
In this paper we present a model exhibiting a new type of continuous-time quantum walk (as a quantum-mechanical transport process) on networks, which is described by a non-Hermitian Hamiltonian possessing a real spectrum. We call it pseudo-Hermitian continuous-time quantum walk. We introduce a method to obtain the probability distribution of walk on any vertex and then study a specific system. We observe that the probability distribution on certain vertices increases compared to that of the Hermitian case. This formalism makes the transport process faster and can be useful for search algorithms.
Hermitian Self-Orthogonal Constacyclic Codes over Finite Fields
Directory of Open Access Journals (Sweden)
Amita Sahni
2014-01-01
Full Text Available Necessary and sufficient conditions for the existence of Hermitian self-orthogonal constacyclic codes of length n over a finite field Fq2, n coprime to q, are found. The defining sets and corresponding generator polynomials of these codes are also characterised. A formula for the number of Hermitian self-orthogonal constacyclic codes of length n over a finite field Fq2 is obtained. Conditions for the existence of numerous MDS Hermitian self-orthogonal constacyclic codes are obtained. The defining set and the number of such MDS codes are also found.
Eigensolver for a Sparse, Large Hermitian Matrix
Tisdale, E. Robert; Oyafuso, Fabiano; Klimeck, Gerhard; Brown, R. Chris
2003-01-01
A parallel-processing computer program finds a few eigenvalues in a sparse Hermitian matrix that contains as many as 100 million diagonal elements. This program finds the eigenvalues faster, using less memory, than do other, comparable eigensolver programs. This program implements a Lanczos algorithm in the American National Standards Institute/ International Organization for Standardization (ANSI/ISO) C computing language, using the Message Passing Interface (MPI) standard to complement an eigensolver in PARPACK. [PARPACK (Parallel Arnoldi Package) is an extension, to parallel-processing computer architectures, of ARPACK (Arnoldi Package), which is a collection of Fortran 77 subroutines that solve large-scale eigenvalue problems.] The eigensolver runs on Beowulf clusters of computers at the Jet Propulsion Laboratory (JPL).
A Boundary Value Problem for Hermitian Monogenic Functions
Directory of Open Access Journals (Sweden)
Reyes JuanBory
2008-01-01
Full Text Available Abstract We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in . Necessary and sufficient conditions for the solvability of this problem are obtained.
Partial Hermitian Conjugate Separability Criteria for Pure Quantum States
Institute of Scientific and Technical Information of China (English)
ZHAO Xin; WU Hua; LI Yan-Song; LONG Gui-Lu
2009-01-01
We propose a criterion for the separability of quantum pure states using the concept of a partial Hermitian conjugate.It is equivalent to the usual positive partial transposition criteria,with a simple physical interpretation.
Numerical Optimization of Eigenvalues of Hermitian Matrix Functions
Mengi, Emre; Yıldırım, Emre Alper; Kılıç, Mustafa
2011-01-01
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. MATRIX ANAL. APPL. c 2014 Society for Industrial and Applied Mathematics Vol. 35, No. 2, pp. 699–724 NUMERICAL OPTIMIZATION OF EIGENVALUES OF HERMITIAN MATRIX FUNCTIONS∗ EMRE MENGI†, E. ALPER YILDIRIM ‡ , AND MUSTAFA KILIC¸ † Abstract. This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued func...
关于柯西sn-对称空间的一个注记%A Note on Cauchy sn-Symmetric Spaces
Institute of Scientific and Technical Information of China (English)
陈内萍
2013-01-01
It is proved that a space is a Cauchy sn-symmetric space if and only if it has a sn-devel-opment consisting of cs-covers ( or sn-covers) if and only if it is a sequence-covering π-image of a metric space .%证明了一个空间是柯西sn-对称空间当且仅当它有一个由cs-复盖或sn-复盖组成sn-展开当且仅当它是度量空间的序列复盖π映射。
New quasi-exactly solvable Hermitian as well as non-Hermitian $\\mathcal{PT}$ -invariant potentials
Indian Academy of Sciences (India)
Avinash Khare; Bhabani Prasad Mandal
2009-08-01
We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex $\\mathcal{PT}$ -invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as $\\mathcal{PT}$ - invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender–Dunne polynomials.
Directory of Open Access Journals (Sweden)
Manish Jain
2013-01-01
Full Text Available We establish the existence and uniqueness of coupled common fixed point for symmetric (φ,ψ-contractive mappings in the framework of ordered G-metric spaces. Present work extends, generalize, and enrich the recent results of Choudhury and Maity (2011, Nashine (2012, and Mohiuddine and Alotaibi (2012, thereby, weakening the involved contractive conditions. Our theoretical results are accompanied by suitable examples and an application to integral equations.
Anomalous Light Scattering by Topological PT-symmetric Particle Arrays
Ling, C. W.; Choi, Ka Hei; Mok, T. C.; Zhang, Zhao-Qing; Fung, Kin Hung
2016-12-01
Robust topological edge modes may evolve into complex-frequency modes when a physical system becomes non-Hermitian. We show that, while having negligible forward optical extinction cross section, a conjugate pair of such complex topological edge modes in a non-Hermitian -symmetric system can give rise to an anomalous sideway scattering when they are simultaneously excited by a plane wave. We propose a realization of such scattering state in a linear array of subwavelength resonators coated with gain media. The prediction is based on an analytical two-band model and verified by rigorous numerical simulation using multiple-multipole scattering theory. The result suggests an extreme situation where leakage of classical information is unnoticeable to the transmitter and the receiver when such a -symmetric unit is inserted into the communication channel.
Super Bloch Oscillation in a PT symmetric system
Turker, Z
2016-01-01
Wannier-Stark ladder in a PT symmetric system is generally complex that leads to amplified/damped Bloch oscillation. We show that a non-amplified wave packet oscillation with very large amplitude can be realized in a non-Hermitian tight binding lattice if certain conditions are satisfied. We show that pseudo PT symmetry guarantees the reality of the quasi energy spectrum in our system.
Fock model and Segal-Bargmann transform for minimal representations of Hermitian Lie groups
DEFF Research Database (Denmark)
Hilgert, Joachim; Kobayashi, Toshiyuki; Möllers, Jan;
2012-01-01
For any Hermitian Lie group G of tube type we construct a Fock model of its minimal representation. The Fock space is defined on the minimal nilpotent K_C-orbit X in p_C and the L^2-inner product involves a K-Bessel function as density. Here K is a maximal compact subgroup of G, and g_C=k_C+p_C i...... intertwines the Schroedinger and Fock model. Its kernel involves the same I-Bessel function. Using the Segal--Bargmann transform we also determine the integral kernel of the unitary inversion operator in the Schroedinger model which is given by a J-Bessel function....
Snyder noncommutativity and pseudo-Hermitian Hamiltonians from a Jordanian twist
Energy Technology Data Exchange (ETDEWEB)
Castro, P.G., E-mail: pgcastro@cbpf.b [Universidade Federal de Juiz de Fora (DM/ICE/UFJF), Juiz de Fora, MG (Brazil). Inst. de Ciencias Exatas. Dept. de Matematica; Kullock, R.; Toppan, F., E-mail: ricardokl@cbpf.b, E-mail: toppan@cbpf.b [Centro Brasileiro de Pesquisas Fisicas (TEO/CBPF), Rio de Janeiro, RJ (Brazil). Coordenacao de Fisica Teorica
2011-07-01
Nonrelativistic quantum mechanics and conformal quantum mechanics are de- formed through a Jordanian twist. The deformed space coordinates satisfy the Snyder noncommutativity. The resulting deformed Hamiltonians are pseudo-Hermitian Hamiltonians of the type discussed by Mostafazadeh. The quantization scheme makes use of the so-called 'unfolded formalism' discussed in previous works. A Hopf algebra structure, compatible with the physical interpretation of the coproduct, is introduced for the Universal Enveloping Algebra of a suitably chosen dynamical Lie algebra (the Hamiltonian is contained among its generators). The multi-particle sector, uniquely determined by the deformed 2-particle Hamiltonian, is composed of bosonic particles. (author)
On the singular sector of the Hermitian random matrix model in the large N limit
Energy Technology Data Exchange (ETDEWEB)
Konopelchenko, B. [Dipartimento di Fisica, Universita del Salento and Sezione INFN, 73100 Lecce (Italy); Martinez Alonso, L. [Departamento de Fisica Teorica II, Universidad Complutense, E28040 Madrid (Spain); Medina, E., E-mail: elena.medina@uca.e [Departamento de Matematicas, Universidad de Cadiz, E11510 Puerto Real, Cadiz (Spain)
2011-01-31
The one-cut case of the Hermitian random matrix model in the large N limit is considered. Its singular sector in the space of coupling constants is analyzed from the point of view of the hodograph equations of the underlying dispersionless Toda hierarchy. A deep connection with the singular sector of the hodograph equations of the 1-layer Benney (classical long wave equation) hierarchy is stablished. This property is a consequence of the fact that the hodograph equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux equations.
Non-Hermitian localization in biological networks
Amir, Ariel; Hatano, Naomichi; Nelson, David R.
2016-04-01
We explore the spectra and localization properties of the N -site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N , the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes but also with respect to 90∘ rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias-dependent shape of this hole tracks the bias-independent contours of constant localization length. We treat the large-N limit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the "charges" embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey "Dale's law" (each site is purely excitatory or inhibitory) and conclude with perturbation theory results that describe the limit of large directional bias, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.
CONVERGENCE DOMAINS OF AOR TYPE ITERATIVE MATRICES FOR SOLVING NON-HERMITIAN LINEAR SYSTEMS
Institute of Scientific and Technical Information of China (English)
Li Wang
2004-01-01
We discuss AOR type iterative methods for solving non-Hermitian linear systems based on Hermitian splitting and skew-Hermitian splitting. Convergence domains of iterative matrices are given and optimal parameters are investigated for skew-Hermitian splitting.Numerical examples are presented to compare the effectiveness of the iterative methods in different points in the domain. In addition, a model problem of three-dimensional convection-diffusion equation is used to illustrated the application of our results.
Non-Hermitian dynamics in the quantum Zeno limit
Kozlowski, W.; Caballero-Benitez, S. F.; Mekhov, I. B.
2016-07-01
We show that weak measurement leads to unconventional quantum Zeno dynamics with Raman-like transitions via virtual states outside the Zeno subspace. We extend this concept into the realm of non-Hermitian dynamics by showing that the stochastic competition between measurement and a system's own dynamics can be described by a non-Hermitian Hamiltonian. We obtain a solution for ultracold bosons in a lattice and show that a dark state of tunneling is achieved as a steady state in which the observable's fluctuations are zero and tunneling is suppressed by destructive matter-wave interference.
Pseudo-Hermitian Systems, Involutive Symmetries and Pseudofermions
Cherbal, O.; Trifonov, D.; Zenad, M.
2016-12-01
We have briefly analyzed the existence of the pseudofermionic structure of multilevel pseudo-Hermitian systems with odd time-reversal and higher order involutive symmetries. We have shown that 2 N-level Hamiltonians with N- order eigenvalue degeneracy can be represented in the oscillator-like form in terms of pseudofermionic creation and annihilation operators for both real and complex eigenvalues. The example of most general four-level traceless Hamiltonian with odd time-reversal symmetry, which is an extension of the SO(5) Hermitian Hamiltonian, is considered in greater and explicit detail.
Non-Hermitian oscillator Hamiltonians and multiple Charlier polynomials
Energy Technology Data Exchange (ETDEWEB)
Miki, Hiroshi, E-mail: miki@amp.i.kyoto-u.ac.jp [Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-Ku, Kyoto 606 8501 (Japan); Vinet, Luc, E-mail: luc.vinet@umontreal.ca [Centre de recherches mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal (Québec), H3C 3J7 (Canada); Zhedanov, Alexei, E-mail: zhedanov@fti.dn.ua [Donetsk Institute for Physics and Technology, Donetsk 83 114 (Ukraine)
2011-12-05
A set of r non-Hermitian oscillator Hamiltonians in r dimensions is shown to be simultaneously diagonalizable. Their spectra are real and the common eigenstates are expressed in terms of multiple Charlier polynomials. An algebraic interpretation of these polynomials is thus achieved and the model is used to derive some of their properties. -- Highlights: ► A set of r non-Hermitian oscillator Hamiltonians in r dimensions is presented. ► Their spectra are real. ► The common eigenstates are expressed in terms of multiple Charlier polynomials (MCP). ► This “integrable” model allows to interpret structural formulas of the MCPs.
Non-Hermitian interferometer: Unidirectional amplification without distortion
Li, C.; Jin, L.; Song, Z.
2017-02-01
A non-Hermitian interferometer can realize asymmetric transmission in the presence of imaginary potential and magnetic flux. Here, we propose a non-Hermitian dimer with an unequal hopping rate by an interferometerlike cluster in the framework of a tight-binding model. The intriguing features of this design are the wave-vector independence and unidirectionality of scattering, which amplify the wave packet without distortion and absorb the incoherent wave without reflection. The absorption relates to the system spectral singularities. The dynamical behaviors of the spectral singularities are also investigated analytically and numerically.
IS PT -SYMMETRIC QUANTUM THEORY FALSE AS A FUNDAMENTAL THEORY?
Directory of Open Access Journals (Sweden)
Miloslav Znojil
2016-06-01
Full Text Available Yi-Chan Lee et al. claim (cf. Phys. Rev. Lett. 112, 130404 (2014 that the “recent extension of quantum theory to non-Hermitian Hamiltonians” (which is widely known under the nickname of “PT-symmetric quantum theory” is “likely false as a fundamental theory”. By their opinion their results “essentially kill any hope of PT-symmetric quantum theory as a fundamental theory of nature”. In our present text we explain that their toy-model-based considerations are misleading and that they do not imply any similar conclusions.
Non-Hermitian extension of gauge theories and implications for neutrino physics
Alexandre, Jean; Millington, Peter
2015-01-01
An extension of QED is considered in which the Dirac fermion has both Hermitian and anti-Hermitian mass terms, as well as both vector and axial-vector couplings to the gauge field. Gauge invariance is restored when the Hermitian and anti-Hermitian masses are of equal magnitude, and the theory reduces to that of a single massless Weyl fermion. An analogous non-Hermitian Yukawa theory is considered and it is shown that this model can explain the smallness of the light-neutrino masses and provide an additional source of leptonic CP violation.
Physics counterpart of the PT non-hermitian tight-binding chain
Jin, L
2010-01-01
We explore an alternative way of finding the link between a PT non-Hermitian Hamiltonian and a Hermitian one. Based on the analysis of the scattering problem for an imaginary potential and its time reversal process, it is shown that any real-energy eigenstate of a PT tight-binding lattice with on-site imaginary potentials shares the same wave function with a resonant transmission state of the corresponding Hermitian lattice embedded in a chain. It indicates that the PT eigenstate of a PT non-Hermitian Hamiltonian has connection to the resonance transmission state of the extended Hermitian Hamiltonian.
Institute of Scientific and Technical Information of China (English)
YANG Shu-Zheng; JIANG Qing-Quan; LI Hui-Ling
2006-01-01
Applying Parikh-Wilzcek's semi-classical quantum tunneling model, we study the Hawking radiation of charged particles as tunneling from the event horizon of a cylindrically symmetric black hole in anti-de Sitter space-time.The derived result shows that the tunneling rate of charged particles is related to the change of Bekenstein-Hawking entropy and that the radiation spectrum is not strictly pure thermal after taking the black hole background dynamical and self-gravitation interaction into account, but is consistent with the underlying unitary theory.
Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems
Directory of Open Access Journals (Sweden)
Cui-Xia Li
2014-01-01
Full Text Available Based on the modified Hermitian and skew-Hermitian splitting (MHSS and preconditioned MHSS (PMHSS methods, a generalized preconditioned MHSS (GPMHSS method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.
A Boundary Value Problem for Hermitian Monogenic Functions
Directory of Open Access Journals (Sweden)
Ricardo Abreu Blaya
2008-02-01
Full Text Available We study the problem of finding a Hermitian monogenic function with a given jump on a given hypersurface in Ã¢Â„Âm,Ã¢Â€Â‰m=2n. Necessary and sufficient conditions for the solvability of this problem are obtained.
Average Density of States for Hermitian Wigner Matrices
Maltsev, Anna
2010-01-01
We consider ensembles of $N \\times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on {\\it arbitrarily} small intervals converges to the semicircle law, as $N$ tends to infinity.
On nondecomposable positive definite Hermitian forms over imaginary quadratic fields
Institute of Scientific and Technical Information of China (English)
ZHU; Fuzu
2001-01-01
［1］Mordell, L. J., The representation of a definite quadratic form as a sum of two others, Ann. of Math., 937, 38: 75.［2］Erds, P., Ko Chao, On definite quadratic forms, which are not the sum of two definite or semidefinite forms, Acta Arith., 939, 3: 02.［3］Erds, P., Ko Chao, Some results on definite quadratic forms, J. London Math. Soc., 938, 3: 27.［4］Zhu Fu-zu, Construction of nondecomposable positive definite quadratic forms, Sci. Sinica, Ser. A, 987, 30(): 9.［5］Zhu Fuzu, On nondecomposability and indecomposability of quadratic forms, Sci. Sinica, Ser. A, 988, 3(3): 265.［6］Pleskin, W., Additively indecomposable positive integral quadratic forms, J. Number Theory, 994, 47: 273.［7］Zhu Fuzu, An existence theorem on positive definite unimodular even Hermitian forms, Chinese Ann. of Math., Ser. A, 984, 5: 33.［8］Zhu Fu-Zu, On the construction of positive definite indecomposable unimodular even Hermitian forms, J. Number Theory, 995, 30: 38.［9］O'Meara, O. T., Introduction to Quadratic Forms, Berlin, New York: Springer-Verlag, 973.［10］Zhu Fuzu, Construction of indecomposable definite Hermitian forms, Chinese Ann. of Math., Ser. B, 994, 5: 349.［11］Zhu Fuzu, On nondecomposable Hermitian forms over Gaussian domain, Acta Math. Sinica, New Ser., 998, 4: 447.
Sub-quadratic decoding of one-point hermitian codes
DEFF Research Database (Denmark)
Nielsen, Johan Sebastian Rosenkilde; Beelen, Peter
2015-01-01
We present the first two sub-quadratic complexity decoding algorithms for one-point Hermitian codes. The first is based on a fast realization of the Guruswami-Sudan algorithm using state-of-the-art algorithms from computer algebra for polynomial-ring matrix minimization. The second is a power...
Pure states, positive matrix polynomials and sums of hermitian squares
Klep, Igor
2009-01-01
Let M be an archimedean quadratic module of real t-by-t matrix polynomials in n variables, and let S be the set of all real n-tuples where each element of M is positive semidefinite. Our key finding is a natural bijection between the set of pure states of M and the cartesian product of S with the real projective (t-1)-space. This leads us to conceptual proofs of positivity certificates for matrix polynomials, including the recent seminal result of Hol and Scherer: If a symmetric matrix polynomial is positive definite on S, then it belongs to M. We also discuss what happens for non-symmetric matrix polynomials or in the absence of the archimedean assumption, and review some of the related classical results. The methods employed are both algebraic and functional analytic.
Symmetric relations of finite negativity
Kaltenbaeck, M.; Winkler, H.; Woracek, H.; Forster, KH; Jonas, P; Langer, H
2006-01-01
We construct and investigate a space which is related to a symmetric linear relation S of finite negativity on an almost Pontryagin space. This space is the indefinite generalization of the completion of dom S with respect to (S.,.) for a strictly positive S on a Hilbert space.
Floquet control of the gain and loss in a PT-symmetric optical coupler
Wu, Yi; Zhu, Bo; Hu, Shu-Fang; Zhou, Zheng; Zhong, Hong-Hua
2017-02-01
Controlling the balanced gain and loss in a PT-symmetric system is a rather challenging task. Utilizing Floquet theory, we explore the constructive role of periodic modulation in controlling the gain and loss of a PT-symmetric optical coupler. It is found that the gain and loss of the system can be manipulated by applying a periodic modulation. Further, such an original non-Hermitian system can even be modulated into an effective Hermitian system derived by the high-frequency Floquet method. Therefore, compared with other PT symmetry control schemes, our protocol can modulate the unbroken PT-symmetric range to a wider parameter region. Our results provide a promising approach for controlling the gain and loss of a realistic system.
Singular Mapping for a $PT$-Symmetric Sinusoidal Optical Lattice at the Symmetry-Breaking Threshold
Jones, H F
2014-01-01
A popular $PT$-symmetric optical potential (variation of the refractive index) that supports a variety of interesting and unusual phenomena is the imaginary exponential, the limiting case of the potential $V_0[\\cos(2\\pi x/a)+i\\lambda\\sin(2\\pi x/a)]$ as $\\lambda \\to 1$, the symmetry-breaking point. For $\\lambda<1$, when the spectrum is entirely real, there is a well-known mapping by a similarity transformation to an equivalent Hermitian potential. However, as $\\lambda \\to 1$, the spectrum, while remaining real, contains Jordan blocks in which eigenvalues and the corresponding eigenfunctions coincide. In this limit the similarity transformation becomes singular. Nonetheless, we show that the mapping from the original potential to its Hermitian counterpart can still be implemented; however, the inverse mapping breaks down. We also illuminate the role of Jordan associated functions in the original problem, showing that they map onto eigenfunctions in the associated Hermitian problem.
Guven, Jemal; Murchadha, Niall O.'
1995-07-01
This is the first of a series of papers in which we examine the constraints of spherically symmetric general relativity with one asymptotically flat region. Our approach is manifestly invariant under spatial diffeomorphisms, exploiting both traditional metric variables as well as the optical scalar variables introduced recently in this context. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints one of which is obtained from the other by time reversal. Boundary conditions on the spherically symmetric three-geometries and extrinsic curvature tensors are discussed. We introduce a one-parameter family of foliations of spacetime involving a linear combination of the two scalars characterizing a spherically symmetric extrinsic curvature tensor. We can exploit this gauge to express one of these scalars in terms of the other and thereby solve the radial momentum constraint uniquely in terms of the radial current. The values of the parameter yielding potentially globally regular gauges corresponding to the vanishing of a timelike vector in the superspace of spherically symmetric geometries. We define a quasilocal mass (QLM) on spheres of fixed proper radius which provides observables of the theory. When the constraints are satisfied the QLM can be expressed as a volume integral over the sources and is positive. We provide two proofs of the positivity of the QLM. If the dominant energy condition (DEC) and the constraints are satisfied positivity can be established in a manifestly gauge-invariant way. This is most easily achieved exploiting the optical scalars. In the second proof we specify the foliation. The payoff is that the weak energy condition replaces the DEC and the Hamiltonian constraint replaces the full constraints. Underpinning this proof is a bound on the derivative of the circumferential radius of the geometry with respect to its proper radius. We show that, when the DEC is satisfied, analogous
Vasiljević, Gorazd
2014-01-01
This BSc thesis deals with certain topics from graph theory. When we talk about studying graphs, we usually mean studying their structure and their structural properties. By doing that, we are often interested in automorphisms of a graph (symmetries), which are permutations of its vertex set, preserving adjacency. There exist graphs, which are symmetric enough, so that automorhism group acts transitively on their vertex set. This means that for any pair of vertices of the graph, there is an a...
Quantum centrality testing on directed graphs via P T -symmetric quantum walks
Izaac, J. A.; Wang, J. B.; Abbott, P. C.; Ma, X. S.
2017-09-01
Various quantum-walk-based algorithms have been proposed to analyze and rank the centrality of graph vertices. However, issues arise when working with directed graphs: the resulting non-Hermitian Hamiltonian leads to nonunitary dynamics, and the total probability of the quantum walker is no longer conserved. In this paper, we discuss a method for simulating directed graphs using P T -symmetric quantum walks, allowing probability-conserving nonunitary evolution. This method is equivalent to mapping the directed graph to an undirected, yet weighted, complete graph over the same vertex set, and can be extended to cover interdependent networks of directed graphs. Previous work has shown centrality measures based on the continuous-time quantum walk provide an eigenvectorlike quantum centrality; using the P T -symmetric framework, we extend these centrality algorithms to directed graphs with a significantly reduced Hilbert space compared to previous proposals. In certain cases, this centrality measure provides an advantage over classical algorithms used in network analysis, for example, by breaking vertex rank degeneracy. Finally, we perform a statistical analysis over ensembles of random graphs, and show strong agreement with the classical PageRank measure on directed acyclic graphs.
Higher genus correlators from the hermitian one-matrix model
Energy Technology Data Exchange (ETDEWEB)
Ambjoern, J. (Niels Bohr Inst., Copenhagen (Denmark)); Chekhov, L. (Steklov Mathematical Inst., Moscow (Russia)); Makeenko, Yu. (Inst. of Theoretical and Experimental Physics, Moscow (Russia))
1992-05-28
We develop an iterative algorithm for the genus expansion of the hermitian NxN one-matrix model (is the Penner model in an external field). By introducing moments of the external field, we prove that the genus g contribution to the m-loop correlator depends only on 3g-2+m lower moments (3g-2 for the partition function). We present the explicit results for the partition function and the one-loop correlator in genus one. We compare the correlators for the hermitian one-matrix model with those at zero momenta for c=1 CFT and show an agreement of the one-loop correlators for genus zero. (orig.).
Optical realization of relativistic non-Hermitian quantum mechanics.
Longhi, Stefano
2010-07-02
Light propagation in distributed-feedback optical structures with gain or loss regions is shown to provide an accessible laboratory tool to visualize in optics the spectral properties of the one-dimensional Dirac equation with non-Hermitian interactions. Spectral singularities and PT symmetry breaking of the Dirac Hamiltonian are shown to correspond to simple observable physical quantities and are related to well-known physical phenomena such as resonance narrowing and laser oscillation.
Spectral properties of sums of Hermitian matrices and algebraic geometry
Chau Huu-Tai, P.; Van Isacker, P.
2016-04-01
It is shown that all the eigenvectors of a sum of Hermitian matrices belong to the same algebraic variety. A polynomial system characterizing this variety is given and a set of nonlinear equations is derived which allows the construction of the variety. Moreover, in some specific cases, explicit expressions for the eigenvectors and eigenvalues can be obtained. Explicit solutions of selected models are also derived.
Analyzing the spectrum of general, non-hermitian Dirac operators
Gattringer, C R; Gattringer, Christof; Hip, Ivan
1999-01-01
We discuss the computational problems when analyzing general, non-hermitian matrices and in particular the un-modified Wilson lattice Dirac operator. We report on our experiences with the Implicitly Restarted Arnoldi Method. The eigenstates of the Wilson-Dirac operator which have real eigenvalues and correspond to zero modes in the continuum are analyzed by correlating the size of the eigenvalues with the chirality of the eigenstates.
Edge Modes, Degeneracies, and Topological Numbers in Non-Hermitian Systems
Leykam, Daniel; Huang, Chunli; Chong, Y D; Nori, Franco
2016-01-01
We analyze chiral topological edge modes in a non-Hermitian variant of the 2D Dirac equation. Such modes appear at interfaces between media with different "masses" and/or signs of the "non-Hermitian charge". The appearance and regions of existence of the edge modes are intimately related to exceptional points, i.e., degeneracies in the bulk spectra of the media. We find that the topological edge modes can be divided into three classes ("Hermitian-like", "non-Hermitian", and "mixed"), and these are described by two winding numbers corresponding to a pair of half-integer charges carried by exceptional points of the bulk Hamiltonian. We also show that the non-Hermitian topological edge modes can be realized in a honeycomb-like lattice of ring resonators with non-Hermitian couplings.
Shestakova, Tatyana P.
2015-01-01
Among theoretical issues in General Relativity the problem of constructing its Hamiltonian formulation is still of interest. The most of attempts to quantize Gravity are based upon Dirac generalization of Hamiltonian dynamics for system with constraints. At the same time there exists another way to formulate Hamiltonian dynamics for constrained systems guided by the idea of extended phase space. We have already considered some features of this approach in the previous MG12 Meeting by the example of a simple isotropic model. Now we apply the approach to a generalized spherically symmetric model which imitates the structure of General Relativity much better. In particular, making use of a global BRST symmetry and the Noether theorem, we construct the BRST charge that generates correct gauge transformations for all gravitational degrees of freedom.
Shestakova, T P
2013-01-01
Among theoretical issues in General Relativity the problem of constructing its Hamiltonian formulation is still of interest. The most of attempts to quantize Gravity are based upon Dirac generalization of Hamiltonian dynamics for system with constraints. At the same time there exists another way to formulate Hamiltonian dynamics for constrained systems guided by the idea of extended phase space. We have already considered some features of this approach in the previous MG12 Meeting by the example of a simple isotropic model. Now we apply the approach to a generalized spherically symmetric model which imitates the structure of General Relativity much better. In particular, making use of a global BRST symmetry and the Noether theorem, we construct the BRST charge that generates correct gauge transformations for all gravitational degrees of freedom.
Sasakian quiver gauge theories and instantons on cones over lens 5-spaces
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Olaf Lechtenfeld
2015-10-01
Full Text Available We consider SU(3-equivariant dimensional reduction of Yang–Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki–Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki–Einstein structure, which are projective planes. We describe the Higgs branches of these quiver gauge theories as moduli spaces of spherically symmetric instantons which are SU(3-equivariant solutions to the Hermitian Yang–Mills equations on the associated Calabi–Yau cones, and further compare them to moduli spaces of translationally-invariant instantons on the cones. We provide an explicit unified construction of these moduli spaces as Kähler quotients and show that they have the same cyclic orbifold singularities as the cones over the lens 5-spaces.
Aldrovandi, Ettore
2004-01-01
We introduce a model for Hermitian holormorphic Deligne cohomology on a projective algebraic manifold which allows to incorporate singular hermitian structures along a normal crossing divisor. In the case of a projective curve, the cup-product in cohomology is shown to correspond to a generalization of the Deligne pairing to line bundles with "good" hermitian metrics in the sense of Mumford and others. A particular case is that of the tangent bundle of the curve twisted by the negative of the...
Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic
Douglas, M R; Lukic, S; Reinbacher, R; Douglas, Michael R.; Karp, Robert L.; Lukic, Sergio; Reinbacher, Rene
2007-01-01
We develop an iterative method for finding solutions to the hermitian Yang-Mills equation on stable holomorphic vector bundles, following ideas recently developed by Donaldson. As illustrations, we construct numerically the hermitian Einstein metrics on the tangent bundle and a rank three vector bundle on P^2. In addition, we find a hermitian Yang-Mills connection on a stable rank three vector bundle on the Fermat quintic.
Nonlinear waves in $\\cal PT$-symmetric systems
Konotop, Vladimir V; Zezyulin, Dmitry A
2016-01-01
Recent progress on nonlinear properties of parity-time ($\\cal PT$-) symmetric systems is comprehensively reviewed in this article. $\\cal PT$ symmetry started out in non-Hermitian quantum mechanics, where complex potentials obeying $\\cal PT$ symmetry could exhibit all-real spectra. This concept later spread out to optics, Bose-Einstein condensates, electronic circuits, and many other physical fields, where a judicious balancing of gain and loss constitutes a $\\cal PT$-symmetric system. The natural inclusion of nonlinearity into these $\\cal PT$ systems then gave rise to a wide array of new phenomena which have no counterparts in traditional dissipative systems. Examples include the existence of continuous families of nonlinear modes and integrals of motion, stabilization of nonlinear modes above $\\cal PT$-symmetry phase transition, symmetry breaking of nonlinear modes, distinctive soliton dynamics, and many others. In this article, nonlinear $\\cal PT$-symmetric systems arising from various physical disciplines ...
Morozov, A.
2017-03-01
Racah matrices and higher j-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the recent proposal of [1] we obtain the exclusive Racah matrix S bar for the currently-front-line case of representation R = [ 3 , 1 ] with non-trivial multiplicities, where it is actually operator-valued, i.e. depends on the choice of bases in the intertwiner spaces. Effective field theory for arborescent knots in this case possesses gauge invariance, which is not yet properly described and understood. Because of this lack of knowledge a big part (about a half) of S bar needs to be reconstructed from orthogonality conditions. Therefore we discuss the abundance of symmetric orthogonal matrices, to which S bar belongs, and explain that dimension of their moduli space is also about a half of that for the ordinary orthogonal matrices. Thus the knowledge approximately matches the freedom and this explains why the method can work - with some limited addition of educated guesses. A similar calculation for R = [ r , 1 ] for r > 3 should also be doable.
Finite Projective Spaces, Geometric Spreads of Lines and Multi-Qubits
Saniga, Metod
2010-01-01
Given a (2N - 1)-dimensional projective space over GF(2), PG(2N - 1, 2), and its geometric spread of lines, there exists a remarkable mapping of this space onto PG(N - 1, 4) where the lines of the spread correspond to the points and subspaces spanned by pairs of lines to the lines of PG(N - 1, 4). Under such mapping, a non-degenerate quadric surface of the former space has for its image a non-singular Hermitian variety in the latter space, this quadric being {\\it hyperbolic} or {\\it elliptic} in dependence on N being {\\it even} or {\\it odd}, respectively. We employ this property to show that generalized Pauli groups of N-qubits also form two distinct families according to the parity of N and to put the role of symmetric operators into a new perspective. The N=4 case is taken to illustrate the issue.
Sasakian quiver gauge theories and instantons on cones over lens 5-spaces
Lechtenfeld, Olaf; Sperling, Marcus; Szabo, Richard J
2015-01-01
We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory over certain cyclic orbifolds of the 5-sphere which are Sasaki-Einstein manifolds. We obtain new quiver gauge theories extending those induced via reduction over the leaf spaces of the characteristic foliation of the Sasaki-Einstein structure, which are projective planes. We describe the Higgs branches of these quiver gauge theories as moduli spaces of spherically symmetric instantons which are SU(3)-equivariant solutions to the Hermitian Yang-Mills equations on the associated Calabi-Yau cones, and further compare them to moduli spaces of translationally-invariant instantons on the cones. We provide an explicit unified construction of these moduli spaces as K\\"ahler quotients and show that they have the same cyclic orbifold singularities as that of the cones.
Anomalous Light Scattering by Topological ${\\mathcal{PT}}$-symmetric Particle Arrays
Ling, C W; Mok, T C; Zhang, Z Q; Fung, Kin Hung
2016-01-01
Robust topological edge modes may evolve into complex-frequency modes when a physical system becomes non-Hermitian. We show that, while having negligible forward optical extinction cross section, a conjugate pair of such complex topological edge modes in a non-Hermitian $\\mathcal{PT}$-symmetric system can give rise to an anomalous sideway scattering when they are simultaneously excited by a plane wave. We propose a realization of such scattering state in a linear array of subwavelength resonators coated with gain media. The prediction is based on an analytical two-band model and verified by rigorous numerical simulation using multiple-multipole scattering theory. The result suggests an extreme situation where leakage of classical information is unnoticeable to the transmitter and the receiver when such a $\\mathcal{PT}$-symmetric unit is inserted into the communication channel.
Poirier, Bill; Salam, A
2004-07-22
In a previous paper [J. Theo. Comput. Chem. 2, 65 (2003)], one of the authors (B.P.) presented a method for solving the multidimensional Schrodinger equation, using modified Wilson-Daubechies wavelets, and a simple phase space truncation scheme. Unprecedented numerical efficiency was achieved, enabling a ten-dimensional calculation of nearly 600 eigenvalues to be performed using direct matrix diagonalization techniques. In a second paper [J. Chem. Phys. 121, 1690 (2004)], and in this paper, we extend and elaborate upon the previous work in several important ways. The second paper focuses on construction and optimization of the wavelength functions, from theoretical and numerical viewpoints, and also examines their localization. This paper deals with their use in representations and eigenproblem calculations, which are extended to 15-dimensional systems. Even higher dimensionalities are possible using more sophisticated linear algebra techniques. This approach is ideally suited to rovibrational spectroscopy applications, but can be used in any context where differential equations are involved.
Pseudo-Hermitian Hamiltonians Generating Waveguide Mode Evolution
Chen, Penghua
2016-01-01
We study the properties of Hamiltonians defined as the generators of transfer matrices in quasi- one-dimensional waveguides. For single- or multi-mode waveguides obeying flux conservation and time-reversal invariance, the Hamiltonians defined in this way are non-Hermitian, but satisfy sym- metry properties that have previously been identified in the literature as "pseudo Hermiticity" and "spectral anti-PT symmetry". We show how simple one-channel and two-channel models exhibit symmetry-breaking transitions between real, imaginary, and complex eigenvalue pairs.
FAST PARALLELIZABLE METHODS FOR COMPUTING INVARIANT SUBSPACES OF HERMITIAN MATRICES
Institute of Scientific and Technical Information of China (English)
Zhenyue Zhang; Hongyuan Zha; Wenlong Ying
2007-01-01
We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix.Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition.We present an accurate convergence analysis of the algorithm without using the big O notation.We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates.Several numerical examples are given which compare some aspects of the existing algorithms and the new Mgorithms.
(Anti-Hermitian Generalized (Anti-Hamiltonian Solution to a System of Matrix Equations
Directory of Open Access Journals (Sweden)
Juan Yu
2014-01-01
Full Text Available We mainly solve three problems. Firstly, by the decomposition of the (anti-Hermitian generalized (anti-Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-Hermitian generalized (anti-Hamiltonian solutions to the system of matrix equations AX=B,XC=D are derived, respectively. Secondly, the optimal approximation solution minX∈K∥X^-X∥ is obtained, where K is the (anti-Hermitian generalized (anti-Hamiltonian solution set of the above system and X^ is the given matrix. Thirdly, the least squares (anti-Hermitian generalized (anti-Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-Hermitian generalized (anti-Hamiltonian solution and the corresponding numerical examples are presented.
Modified stochastic variational approach to non-Hermitian quantum systems
Kraft, Daniel; Plessas, Willibald
2016-08-01
The stochastic variational method has proven to be a very efficient and accurate tool to calculate especially bound states of quantum-mechanical few-body systems. It relies on the Rayleigh-Ritz variational principle for minimizing real eigenenergies of Hermitian Hamiltonians. From molecular to atomic, nuclear, and particle physics there is actually a great demand of describing also resonant states to a high degree of reliance. This is especially true with regard to hadron resonances, which have to be treated in a relativistic framework. So far standard methods of dealing with quantum chromodynamics have not yet succeeded in describing hadron resonances in a realistic manner. Resonant states can be handled by non-Hermitian quantum Hamiltonians. These states correspond to poles in the lower half of the unphysical sheet of the complex energy plane and are therefore intimately connected with complex eigenvalues. Consequently the Rayleigh-Ritz variational principle cannot be employed in the usual manner. We have studied alternative selection principles for the choice of test functions to treat resonances along the stochastic variational method. We have found that a stationarity principle for the complex energy eigenvalues provides a viable method for selecting test functions for resonant states in a constructive manner. We discuss several variants thereof and exemplify their practical efficiencies.
Non-Hermitian Dynamics in the Quantum Zeno Limit
Kozlowski, Wojciech; Mekhov, Igor B
2015-01-01
Measurement is one of the most counter-intuitive aspects of quantum physics. Frequent measurements of a quantum system lead to quantum Zeno dynamics where time evolution becomes confined to a subspace defined by the projections. However, weak measurement performed at a finite rate is also capable of locking the system into such a Zeno subspace in an unconventional way: by Raman-like transitions via virtual intermediate states outside this subspace, which are not forbidden. Here, we extend this concept into the realm of non-Hermitian dynamics by showing that the stochastic competition between measurement and a system's own dynamics can be described by a non-Hermitian Hamiltonian. We obtain an analytic solution for ultracold bosons in a lattice and show that a dark state of the tunnelling operator is a steady state in which the observable's fluctuations are zero and tunnelling is suppressed by destructive matter-wave interference. This opens a new venue of investigation beyond the canonical quantum Zeno dynamic...
Noncommutative spaces and matrix embeddings on flat ℝ{sup 2n+1}
Energy Technology Data Exchange (ETDEWEB)
Karczmarek, Joanna L.; Yeh, Ken Huai-Che [Department of Physics and Astronomy, University of British Columbia,6224 Agricultural Road, Vancouver (Canada)
2015-11-23
We conjecture an embedding operator which assigns, to any 2n+1 hermitian matrices, a 2n-dimensional hypersurface in flat (2n+1)-dimensional Euclidean space. This corresponds to precisely defining a fuzzy D(2n)-brane corresponding to N D0-branes. Points on the emergent hypersurface correspond to zero eigenstates of the embedding operator, which have an interpretation as coherent states underlying the emergent noncommutative geometry. Using this correspondence, all physical properties of the emergent D(2n)-brane can be computed. We apply our conjecture to noncommutative flat and spherical spaces. As a by-product, we obtain a construction of a rotationally symmetric flat noncommutative space in 4 dimensions.
Longhi, Stefano
2017-01-01
We consider wave transport phenomena in a PT -symmetric extension of the periodically kicked quantum rotator model and reveal that dynamical localization assists the unbroken PT phase. In the delocalized (quantum resonance) regime, PT symmetry is always in the broken phase and ratchet acceleration arises as a signature of unidirectional non-Hermitian transport. An optical implementation of the periodically kicked PT -symmetric Hamiltonian, based on transverse beam propagation in a passive optical resonator with combined phase and loss gratings, is suggested to visualize acceleration modes in fractional Talbot cavities.
Giusteri, Giulio G.; Mattiotti, Francesco; Celardo, G. Luca
2015-03-01
We investigate the validity of the non-Hermitian Hamiltonian approach in describing quantum transport in disordered tight-binding networks connected to external environments, acting as sinks. Usually, non-Hermitian terms are added, on a phenomenological basis, to such networks to summarize the effects of the coupling to the sinks. Here, we consider a paradigmatic model of open quantum network for which we derive a non-Hermitian effective model, discussing its limit of validity by a comparison with the analysis of the full Hermitian model. Specifically, we consider a ring of sites connected to a central one-dimensional lead. The lead acts as a sink that absorbs the excitation initially present in the ring. The coupling strength to the lead controls the opening of the ring system. This model has been widely discussed in literature in the context of light-harvesting systems. We analyze the effectiveness of the non-Hermitian description both in absence and in presence of static disorder on the ring. In both cases, the non-Hermitian model is valid when the energy range determined by the eigenvalues of the ring Hamiltonian is smaller than the energy band in the lead. Under such condition, we show that results about the interplay of opening and disorder, previously obtained within the non-Hermitian Hamiltonian approach, remain valid when the full Hermitian model in presence of disorder is considered. The results of our analysis can be extended to generic networks with sinks, leading to the conclusion that the non-Hermitian approach is valid when the energy dependence of the coupling to the external environments is sufficiently smooth in the energy range spanned by the eigenstates of the network.
EXCEPTIONAL POINTS IN OPEN AND PT-SYMMETRIC SYSTEMS
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Hichem Eleuch
2014-04-01
Full Text Available Exceptional points (EPs determine the dynamics of open quantum systems and cause also PT symmetry breaking in PT symmetric systems. From a mathematical point of view, this is caused by the fact that the phases of the wavefunctions (eigenfunctions of a non-Hermitian Hamiltonian relative to one another are not rigid when an EP is approached. The system is therefore able to align with the environment to which it is coupled and, consequently, rigorous changes of the system properties may occur. We compare analytically as well as numerically the eigenvalues and eigenfunctions of a 2 × 2 matrix that is characteristic either of open quantum systems at high level density or of PT symmetric optical lattices. In both cases, the results show clearly the influence of the environment on the system in the neighborhood of EPs. Although the systems are very different from one another, the eigenvalues and eigenfunctions indicate the same characteristic features.
(Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations
Juan Yu; Qing-Wen Wang; Chang-Zhou Dong
2014-01-01
We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equations AX=B,XC=D are derived, respectively. Secondly, the optimal approximation solution minX∈K∥X^-X∥ is obtained, where K is the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of t...
Blind Source Separation in Farsi Language by Using Hermitian Angle in Convolutive Enviroment
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Atefeh Soltani
2013-04-01
Full Text Available This paper presents a T-F masking method for convolutive blind source separation based on hermitian angle concept. The hermitian angle is calculated between T-F domain mixture vector and reference vector. Two different reference vectors are assumed for calculating two different hermitian angles, and then these angles are clustered with k-means or FCM method to estimate unmixing masks. The well-known permutation problem is solved based on k-means clustering of estimated masks which are partitioned to small groups. The experimental results show an improvement in performance when using two different reference vectors compared to only one.
Kordy, M.; Wannamaker, P.; Maris, V.; Cherkaev, E.; Hill, G.
2016-01-01
Following the creation described in Part I of a deformable edge finite-element simulator for 3-D magnetotelluric (MT) responses using direct solvers, in Part II we develop an algorithm named HexMT for 3-D regularized inversion of MT data including topography. Direct solvers parallelized on large-RAM, symmetric multiprocessor (SMP) workstations are used also for the Gauss-Newton model update. By exploiting the data-space approach, the computational cost of the model update becomes much less in both time and computer memory than the cost of the forward simulation. In order to regularize using the second norm of the gradient, we factor the matrix related to the regularization term and apply its inverse to the Jacobian, which is done using the MKL PARDISO library. For dense matrix multiplication and factorization related to the model update, we use the PLASMA library which shows very good scalability across processor cores. A synthetic test inversion using a simple hill model shows that including topography can be important; in this case depression of the electric field by the hill can cause false conductors at depth or mask the presence of resistive structure. With a simple model of two buried bricks, a uniform spatial weighting for the norm of model smoothing recovered more accurate locations for the tomographic images compared to weightings which were a function of parameter Jacobians. We implement joint inversion for static distortion matrices tested using the Dublin secret model 2, for which we are able to reduce nRMS to ˜1.1 while avoiding oscillatory convergence. Finally we test the code on field data by inverting full impedance and tipper MT responses collected around Mount St Helens in the Cascade volcanic chain. Among several prominent structures, the north-south trending, eruption-controlling shear zone is clearly imaged in the inversion.
Bulk Universality and Related Properties of Hermitian Matrix Models
Pastur, L
2007-01-01
We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \\ref{t:U.t1}). The proof as our previous proof in \\cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \\cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.
Optomechanical interactions in non-Hermitian photonic molecules
Schönleber, David W; El-Ganainy, Ramy
2016-01-01
We study optomechanical interactions in non-Hermitian photonic molecules that support two photonic states and one acoustic mode. The nonlinear steady-state solutions and their linear stability landscapes are investigated as a function of the system's parameters and excitation power levels. We also examine the temporal evolution of the system and uncover different regimes of nonlinear dynamics. Our analysis reveals several important results: 1) Parity-time ($\\mathcal{PT}$) symmetry is not necessarily the optimum choice for maximum optomechanical interaction. 2) Stable steady-state solutions are not always reached under continuous wave (CW) optical excitations. 3) Accounting for gain saturation effects can regulate the behavior of the otherwise unbounded oscillation amplitudes. Our study provides a deeper insight into the interplay between optical non-Hermiticity and optomechanical coupling and can thus pave the way for new device applications.
Robust light transport in non-Hermitian photonic lattices
Longhi, Stefano; Della Valle, Giuseppe
2015-01-01
Combating the effects of disorder on light transport in micro- and nano-integrated photonic devices is of major importance from both fundamental and applied viewpoints. In ordinary waveguides, imperfections and disorder cause unwanted back-reflections, which hinder large-scale optical integration. Topological photonic structures, a new class of optical systems inspired by quantum Hall effect and topological insulators, can realize robust transport via topologically-protected unidirectional edge modes. Such waveguides are realized by the introduction of synthetic gauge fields for photons in a two-dimensional structure, which break time reversal symmetry and enable one-way guiding at the edge of the medium. Here we suggest a different route toward robust transport of light in lower-dimensional (1D) photonic lattices, in which time reversal symmetry is broken because of the non-Hermitian nature of transport. While a forward propagating mode in the lattice is amplified, the corresponding backward propagating mode...
Magnified imaging based on non-Hermitian nonlocal cylindrical metasurfaces
Savoia, Silvio; Valagiannopoulos, Constantinos A.; Monticone, Francesco; Castaldi, Giuseppe; Galdi, Vincenzo; Alà, Andrea
2017-03-01
We show that a cylindrical lensing system composed of two metasurfaces with suitably tailored non-Hermitian (i.e., with distributed gain and loss) and nonlocal (i.e., spatially dispersive) properties can perform magnified imaging with reduced aberrations. More specifically, we analytically derive the idealized surface-impedance values that are required for "perfect" magnification and imaging and elucidate the role and implications of non-Hermiticity and nonlocality in terms of spatial resolution and practical implementation. For a basic demonstration, we explore some proof-of-principle quasilocal and multilayered implementations and independently validate the outcomes via full-wave numerical simulations. We also show that the metasurface frequency-dispersion laws can be chosen so as to ensure unconditional stability with respect to arbitrary temporal excitations. These results, which extend previous studies on planar configurations, may open intriguing venues in the design of metastructures for field imaging and processing.
PT-symmetric deformations of integrable models.
Fring, Andreas
2013-04-28
We review recent results on new physical models constructed as PT-symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero-Moser-Sutherland type and nonlinear integrable field equations of Korteweg-de Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero-Moser-Sutherland models, we provide three alternative deformations: a complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real-valued field equations of nonlinear integrable systems; and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of Korteweg-de Vries type are studied with regard to different kinds of PT-symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.
The fuzzy space construction kit
Sykora, Andreas
2016-01-01
Fuzzy spaces like the fuzzy sphere or the fuzzy torus have received remarkable attention, since they appeared as objects in string theory. Although there are higher dimensional examples, the most known and most studied fuzzy spaces are realized as matrix algebras defined by three Hermitian matrices, which may be seen as fuzzy membrane or fuzzy surface. We give a mapping between directed graphs and matrix algebras defined by three Hermitian matrices and show that the matrix algebras of known two-dimensional fuzzy spaces are associated with unbranched graphs. By including branchings into the graphs we find matrix algebras that represent fuzzy spaces associated with surfaces having genus 2 and higher.
Potluri, Shobha; Yan, Anthony K; Chou, James J; Donald, Bruce R; Bailey-Kellogg, Chris
2006-10-01
Structural studies of symmetric homo-oligomers provide mechanistic insights into their roles in essential biological processes, including cell signaling and cellular regulation. This paper presents a novel algorithm for homo-oligomeric structure determination, given the subunit structure, that is both complete, in that it evaluates all possible conformations, and data-driven, in that it evaluates conformations separately for consistency with experimental data and for quality of packing. Completeness ensures that the algorithm does not miss the native conformation, and being data-driven enables it to assess the structural precision possible from data alone. Our algorithm performs a branch-and-bound search in the symmetry configuration space, the space of symmetry axis parameters (positions and orientations) defining all possible C(n) homo-oligomeric complexes for a given subunit structure. It eliminates those symmetry axes inconsistent with intersubunit nuclear Overhauser effect (NOE) distance restraints and then identifies conformations representing any consistent, well-packed structure to within a user-defined similarity level. For the human phospholamban pentamer in dodecylphosphocholine micelles, using the structure of one subunit determined from a subset of the experimental NMR data, our algorithm identifies a diverse set of complex structures consistent with the nine intersubunit NOE restraints. The distribution of determined structures provides an objective characterization of structural uncertainty: backbone RMSD to the previously determined structure ranges from 1.07 to 8.85 A, and variance in backbone atomic coordinates is an average of 12.32 A(2). Incorporating vdW packing reduces structural diversity to a maximum backbone RMSD of 6.24 A and an average backbone variance of 6.80 A(2). By comparing data consistency and packing quality under different assumptions of oligomeric number, our algorithm identifies the pentamer as the most likely oligomeric state
Repeatability of measurements: Non-Hermitian observables and quantum Coriolis force
Gardas, Bartłomiej; Deffner, Sebastian; Saxena, Avadh
2016-08-01
A noncommuting measurement transfers, via the apparatus, information encoded in a system's state to the external "observer." Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restricts the recording process to orthogonal states as only those are distinguishable by measurements. Therefore, even a possibility to describe physical reality by means of non-Hermitian operators should volens nolens be excluded as their eigenstates are not orthogonal. Here, we show that non-Hermitian operators with real spectra can be treated within the standard framework of quantum mechanics. Furthermore, we propose a quantum canonical transformation that maps Hermitian systems onto non-Hermitian ones. Similar to classical inertial forces this map is accompanied by an energetic cost, pinning the system on the unitary path.
Integral Transforms and a Class of Singular S-Hermitian Eigenvalue Problems
Dijksma, A.; Snoo, H.S.V. de
1973-01-01
For a class of singular S-hermitian eigenvalue problems we show that the corresponding integral transforms are surjective. This class was discussed by us earlier and is more restricted than the one, which has been considered by others.
Symmetric Powers of Symmetric Bilinear Forms
Institute of Scientific and Technical Information of China (English)
Se(a)n McGarraghy
2005-01-01
We study symmetric powers of classes of symmetric bilinear forms in the Witt-Grothendieck ring of a field of characteristic not equal to 2, and derive their basic properties and compute their classical invariants. We relate these to earlier results on exterior powers of such forms.
Complex {PT}-symmetric extensions of the nonlinear ultra-short light pulse model
Yan, Zhenya
2012-11-01
The short pulse equation u_{xt}=u+\\frac{1}{2}(u^2u_x)_x is PT symmetric, which arises in nonlinear optics for the ultra-short pulse case. We present a family of new complex PT-symmetric extensions of the short pulse equation, i[(iu_x)^{\\sigma }]_t=au+bu^m+ic[u^n(iu_x)^{\\epsilon }]_x \\,\\, (\\sigma ,\\, \\epsilon ,\\,a,\\,b,\\,c,\\,m,\\,n \\in {R}), based on the complex PT-symmetric extension principle. Some properties of these equations with some chosen parameters are studied including the Hamiltonian structures and exact solutions such as solitary wave solutions, doubly periodic wave solutions and compacton solutions. Our results may be useful to understand complex PT-symmetric nonlinear physical models. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
Least-Squares Solutions of the Equation AX = B Over Anti-Hermitian Generalized Hamiltonian Matrices
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Upon using the denotative theorem of anti-Hermitian generalized Hamiltonian matrices, we solve effectively the least-squares problem min ‖AX - B‖ over anti-Hermitian generalized Hamiltonian matrices. We derive some necessary and sufficient conditions for solvability of the problem and an expression for general solution of the matrix equation AX = B. In addition, we also obtain the expression for the solution of a relevant optimal approximate problem.
Holographic Spherically Symmetric Metrics
Petri, Michael
The holographic principle (HP) conjectures, that the maximum number of degrees of freedom of any realistic physical system is proportional to the system's boundary area. The HP has its roots in the study of black holes. It has recently been applied to cosmological solutions. In this article we apply the HP to spherically symmetric static space-times. We find that any regular spherically symmetric object saturating the HP is subject to tight constraints on the (interior) metric, energy-density, temperature and entropy-density. Whenever gravity can be described by a metric theory, gravity is macroscopically scale invariant and the laws of thermodynamics hold locally and globally, the (interior) metric of a regular holographic object is uniquely determined up to a constant factor and the interior matter-state must follow well defined scaling relations. When the metric theory of gravity is general relativity, the interior matter has an overall string equation of state (EOS) and a unique total energy-density. Thus the holographic metric derived in this article can serve as simple interior 4D realization of Mathur's string fuzzball proposal. Some properties of the holographic metric and its possible experimental verification are discussed. The geodesics of the holographic metric describe an isotropically expanding (or contracting) universe with a nearly homogeneous matter-distribution within the local Hubble volume. Due to the overall string EOS the active gravitational mass-density is zero, resulting in a coasting expansion with Ht = 1, which is compatible with the recent GRB-data.
Directory of Open Access Journals (Sweden)
Giuseppe Di Maio
2008-04-01
Full Text Available The subject of hyperspace topologies on closed or closed and compact subsets of a topological space X began in the early part of the last century with the discoveries of Hausdorff metric and Vietoris hit-and-miss topology. In course of time, several hyperspace topologies were discovered either for solving some problems in Applied or Pure Mathematics or as natural generalizations of the existing ones. Each hyperspace topology can be split into a lower and an upper part. In the upper part the original set inclusion of Vietoris was generalized to proximal set inclusion. Then the topologization of the Wijsman topology led to the upper Bombay topology which involves two proximities. In all these developments the lower topology, involving intersection of finitely many open sets, was generalized to locally finite families but intersection was left unchanged. Recently the authors studied symmetric proximal topology in which proximity was used for the first time in the lower part replacing intersection with its generalization: nearness. In this paper we use two proximities also in the lower part and we obtain the lower Bombay hypertopology. Consequently, a new hypertopology arises in a natural way: the symmetric Bombay topology which is the join of a lower and an upper Bombay topology.
Morfonios, C V; Diakonos, F K; Schmelcher, P
2016-01-01
A nonlocal discrete continuity formalism is developed which relates spatial symmetries in subparts of Hermitian or non-Hermitian lattice systems to the properties of adapted nonlocal currents. Broken local symmetries thereby act as current sources or sinks, and the time evolution of the associated nonlocal charge is governed by the nonlocal currents at the boundaries of domains with local symmetry. We apply the framework to locally inversion-(time-) and translation-(time-) symmetric one-dimensional photonic waveguide arrays effectively described by Schr\\"odinger's equation with a tight-binding Hamiltonian. The nonlocal currents of stationary states are shown to be translationally invariant within local symmetry domains for arbitrary wavefunction profiles, and cases of complete, overlapping, and gapped local symmetry are demonstrated for model setups. Two distinct versions of the nonlocal invariant currents enable a mapping between wave amplitudes of symmetry-related sites, thereby generalizing the global Bloc...
Jiang, Haiyong
2016-04-11
We present an automatic algorithm for symmetrizing facade layouts. Our method symmetrizes a given facade layout while minimally modifying the original layout. Based on the principles of symmetry in urban design, we formulate the problem of facade layout symmetrization as an optimization problem. Our system further enhances the regularity of the final layout by redistributing and aligning boxes in the layout. We demonstrate that the proposed solution can generate symmetric facade layouts efficiently. © 2015 IEEE.
Symmetrization of Facade Layouts
Jiang, Haiyong
2016-02-26
We present an automatic approach for symmetrizing urban facade layouts. Our method can generate a symmetric layout through minimally modifying the original input layout. Based on the principles of symmetry in urban design, we formulate facade layout symmetrization as an optimization problem. Our method further enhances the regularity of the final layout by redistributing and aligning elements in the layout. We demonstrate that the proposed solution can effectively generate symmetric facade layouts.
Chambler, A F; Chapman-Sheath, P J; Pearse, M F; Hollingdale, J
1997-10-01
Chronic recurrent multifocal osteomyelitis is often confused with symmetrical Brodie's abscess as it has a similar pathogenesis. We report an otherwise healthy 17-year-old boy presenting with a true symmetrical Brodie's abscess. We conclude that a symmetrical Brodie's abscess presenting in an otherwise healthy patient is a separate clinical condition with a different management protocol.
Probing non-Hermitian physics with flying atoms
Wen, Jianming; Xiao, Yanhong; Peng, Peng; Cao, Wanxia; Shen, Ce; Qu, Weizhi; Jiang, Liang
2016-05-01
Non-Hermtian optical systems with parity-time (PT) symmetry provide new means for light manipulation and control. To date, most of experimental demonstrations on PT symmetry rely on advanced nanotechnologies and sophisticated fabrication techniques to manmade solid-state materials. Here, we report the first experimental realization of optical anti-PT symmetry, a counterpart of conventional PT symmetry, in a warm atomic-vapor cell. By exploiting rapid coherence transport via flying atoms, we observe essential features of anti-PT symmetry with an unprecedented precision on phase-transition threshold. Moreover, our system allows nonlocal interference of two spatially-separated fields as well as anti-PT assisted four-wave mixing. Besides, another intriguing feature offered by the system is refractionless (or unit-refraction) light propagation. Our results thus represent a significant advance in non-Hermitian physics by bridging a firm connection with the AMO field, where novel phenomena and applications in quantum and nonlinear optics aided by (anti-)PT symmetry can be anticipated.
High-Performance Solvers for Dense Hermitian Eigenproblems
Petschow, Matthias; Bientinesi, Paolo
2012-01-01
We introduce a new collection of solvers - subsequently called EleMRRR - for large-scale dense Hermitian eigenproblems. EleMRRR solves various types of problems: generalized, standard, and tridiagonal eigenproblems. Among these, the last is of particular importance as it is a solver on its own right, as well as the computational kernel for the first two; we present a fast and scalable tridiagonal solver based on the Algorithm of Multiple Relatively Robust Representations - referred to as PMRRR. Like the other EleMRRR solvers, PMRRR is part of the freely available Elemental library, and is designed to fully support both message-passing (MPI) and multithreading parallelism (SMP). As a result, the solvers can equally be used in pure MPI or in hybrid MPI-SMP fashion. We conducted a thorough performance study of EleMRRR and ScaLAPACK's solvers on two supercomputers. Such a study, performed with up to 8,192 cores, provides precise guidelines to assemble the fastest solver within the ScaLAPACK framework; it also ind...
The complex Laguerre symplectic ensemble of non-Hermitian matrices
Energy Technology Data Exchange (ETDEWEB)
Akemann, G. [Department of Mathematical Sciences, School of Information Systems, Computing and Mathematics, Brunel University West London, Uxbridge UB8 3PH (United Kingdom)]. E-mail: gernot.akemann@brunel.ac.uk
2005-12-12
We solve the complex extension of the chiral Gaussian symplectic ensemble, defined as a Gaussian two-matrix model of chiral non-Hermitian quaternion real matrices. This leads to the appearance of Laguerre polynomials in the complex plane and we prove their orthogonality. Alternatively, a complex eigenvalue representation of this ensemble is given for general weight functions. All k-point correlation functions of complex eigenvalues are given in terms of the corresponding skew orthogonal polynomials in the complex plane for finite-N, where N is the matrix size or number of eigenvalues, respectively. We also allow for an arbitrary number of complex conjugate pairs of characteristic polynomials in the weight function, corresponding to massive quark flavours in applications to field theory. Explicit expressions are given in the large-N limit at both weak and strong non-Hermiticity for the weight of the Gaussian two-matrix model. This model can be mapped to the complex Dirac operator spectrum with non-vanishing chemical potential. It belongs to the symmetry class of either the adjoint representation or two colours in the fundamental representation using staggered lattice fermions.
The Determinant Bundle on the Moduli Space of Stable Triples over a Curve
Indian Academy of Sciences (India)
Indranil Biswas; N Raghavendra
2002-08-01
We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (1, 2, ), where 1 and 2 are holomorphic vector bundles over a fixed compact Riemann surface , and : 2 → 1 is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed ∞ Hermitian vector bundle over a compact Riemann surface.
对称算子空间上的Jordan环同构%Jordan Ring Isomorphism on the Space of Symmetric Operators
Institute of Scientific and Technical Information of China (English)
安润玲; 侯晋川
2012-01-01
设H为无限维的复Hilbert空间,J（H）是H上全体对称算子构成的Jordan代数,Φ：J（H）→J（H）为双射且Φ（I）=I.证明下列条件等价：（1）Φ（ABA）=Φ（A）Φ（B）Φ（A）,A,B∈J（H）;（2）Φ（1/2（AB＋BA））=1/2Φ（A）Φ（B）＋1/2Φ（B）Φ（A）,A,B∈J（H）;（3）Φ（ABC＋CBA）=Φ（A）Φ（B）Φ（C）＋Φ（C）Φ（B）Φ（A）,A,B,C∈J（H）;（4）Φ（1/2（ABC＋CBA））=1/2Φ（A）Φ（B）Φ（C）＋1/2Φ（C）Φ（B）Φ（A）,A,B,C∈J（H）;（5）Φ是J（H）上的Jordan环同构;（6）存在有界可逆的线性或共轭线性算子A：H→H,A~t=A~（-1）,使得Φ（X）=AXA~t,X∈J（H）.得到了J（H）上Jordan环同构的新刻画.%Let H be an infinite dimensional complex Hilbert space and I(H) bethe Jordan algebra of all symmetric operators in B(H).We show that if bijectivemapsΦ:I(H)→I(H) withΦ(I) = I,then the following conditions are equivalent:(1)Φ(ABA) =Φ(A)Φ(B)Φ(A),A,B∈I(H);(2)Φ((1/2)(AB + BA)) =(1/2)Φ(A)Φ(B) +(1/2)(B)Φ(A),A,B∈I(H);(3)Φ(ABC + CBA) =Φ(A)Φ(B)Φ(C) +Φ(C)Φ(B)Φ(A),A,B,C∈I(H);(4)Φ((1/2)(ABC + CBA)) =(1/2)Φ(A)Φ(B)Φ(C) +(1/2)Φ(C)Φ(B)Φ(A),A,B,C∈I(H);(5)Φis a Jordan ring isomorphism on I(H);(6) there exists a bounded invertible linear or conjugate linear operator A:H→Hwith A~t = A~(-1) such thatΦ(X) = AX A~t for every X∈I(H).New characterizationsof Jordan ring isomorphism on I(H) were got.
Longhi, Stefano
2016-01-01
We consider wave transport phenomena in a $\\mathcal{PT}$-symmetric extension of the periodically-kicked quantum rotator model and reveal that dynamical localization assists the unbroken $\\mathcal{PT}$ phase. In the delocalized (quantum resonance) regime, $\\mathcal{PT}$ symmetry is always in the broken phase and ratchet acceleration arises as a signature of unidirectional non-Hermitian transport. An optical implementation of the periodically-kicked $\\mathcal{PT}$-symmetric Hamiltonian, based on transverse beam propagation in a passive optical resonator with combined phase and loss gratings, is suggested to visualize acceleration modes in fractional Talbot cavities.
Energy Technology Data Exchange (ETDEWEB)
Fadin, V.S. [Budker Institute of Nuclear Physics, SD RAS, Novosibirsk (Russian Federation); Novosibirsk State University, Novosibirsk (Russian Federation); Fiore, R. [Universita della Calabria, Dipartimento di Fisica, Cosenza (Italy); Istituto Nazionale di Fisica, Nucleare Gruppo Collegato di Cosenza (Italy)
2016-05-15
We analyze a modification of the BFKL kernel for the adjoint representation of the color group in the maximally supersymmetric (N = 4) Yang-Mills theory in the limit of a large number of colors, related to the modification of the eigenvalues of the kernel suggested by Bondarenko and Prygarin in order to obtain Hermitian separability of the eigenvalues. We restore the modified kernel in the momentum space. It turns out that the modification is related only to the real part of the kernel and that the correction to the kernel cannot be presented by a single analytic function in the entire momentum region, which contradicts the known properties of the kernel. (orig.)
Canteaut, Anne; Videau, Marion
2005-01-01
http://www.ieee.org/; We present an extensive study of symmetric Boolean functions, especially of their cryptographic properties. Our main result establishes the link between the periodicity of the simplified value vector of a symmetric Boolean function and its degree. Besides the reduction of the amount of memory required for representing a symmetric function, this property has some consequences from a cryptographic point of view. For instance, it leads to a new general bound on the order of...
Pseudo-Hermitian ensemble of random Gaussian matrices.
Marinello, G; Pato, M P
2016-07-01
It is shown how pseudo-Hermiticity, a necessary condition satisfied by operators of PT symmetric systems can be introduced in the three Gaussian classes of random matrix theory. The model describes transitions from real eigenvalues to a situation in which, apart from a residual number, the eigenvalues are complex conjugate.
Pseudo-Hermitian ensemble of random Gaussian matrices
Marinello, G.; Pato, M. P.
2016-07-01
It is shown how pseudo-Hermiticity, a necessary condition satisfied by operators of PT symmetric systems can be introduced in the three Gaussian classes of random matrix theory. The model describes transitions from real eigenvalues to a situation in which, apart from a residual number, the eigenvalues are complex conjugate.
Energy Technology Data Exchange (ETDEWEB)
Trigub, R M [Donetsk National University, Donetsk (Ukraine)
2009-08-31
We prove a general direct theorem on the simultaneous pointwise approximation of smooth periodic functions and their derivatives by trigonometric polynomials and their derivatives with Hermitian interpolation. We study the order of approximation by polynomials whose graphs lie above or below the graph of the function on certain intervals. We prove several inequalities for Hermitian interpolation with absolute constants (for any system of nodes). For the first time we get a theorem on the best-order approximation of functions by polynomials with interpolation at a given system of nodes. We also provide a construction of Hermitian interpolating trigonometric polynomials for periodic functions (in the case of one node, these are trigonometric Taylor polynomials)
Direct measurement of large-scale quantum states via expectation values of non-Hermitian matrices
Bolduc, Eliot; Gariepy, Genevieve; Leach, Jonathan
2016-01-01
In quantum mechanics, predictions are made by way of calculating expectation values of observables, which take the form of Hermitian operators. Non-Hermitian operators, however, are not necessarily devoid of physical significance, and they can play a crucial role in the characterization of quantum states. Here we show that the expectation values of a particular set of non-Hermitian matrices, which we call column operators, directly yield the complex coefficients of a quantum state vector. We provide a definition of the state vector in terms of measurable quantities by decomposing these column operators into observables. The technique we propose renders very-large-scale quantum states significantly more accessible in the laboratory, as we demonstrate by experimentally characterizing a 100,000-dimensional entangled state. This represents an improvement of two orders of magnitude with respect to previous phase-and-amplitude characterizations of discrete entangled states.
ON HERMITIAN POSITIVE DEFINITE SOLUTION OF NONLINEAR MATRIX EQUATION X + A*X-2A = Q
Institute of Scientific and Technical Information of China (English)
Xiao-xia Guo
2005-01-01
Based on the fixed-point theory, we study the existence and the uniqueness of the maximal Hermitian positive definite solution of the nonlinear matrix equation X + A* X-2A =Q, where Q is a square Hermitian positive definite matrix and A* is the conjugate transpose of the matrix A. We also demonstrate some essential properties and analyze the sensitivity of this solution. In addition, we derive computable error bounds about the approximations to the maximal Hermitian positive definite solution of the nonlinear matrix equation X + A*X-2A = Q. At last, we further generalize these results to the nonlinear matrix equation X + A*X-nA = Q, where n ≥ 2 is a given positive integer.
Application of Lie transform perturbation method for multidimensional non-Hermitian systems
Indian Academy of Sciences (India)
Asiri Nanayakkara
2011-01-01
Three-dimensional non-Hermitian systems are investigated using classical perturbation theory based on Lie transformations. Analytic expressions for total energy in terms of action variables are derived. Both real and complex semiclassical eigenvalues are obtained by quantizing the action variables. It was found that semiclassical energy eigenvalues calculated with the classical perturbation theory are in very good agreement with exact energies and for certain non-Hermitian systems second-order classical perturbation theory performed better than the secondorder Rayleigh–Schroedinger perturbation theory.
Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
Directory of Open Access Journals (Sweden)
Andrey V. Sokolov
2011-12-01
Full Text Available This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.
Inverse Symmetric Inflationary Attractors
Odintsov, S D
2016-01-01
We present a class of inflationary potentials which are invariant under a special symmetry, which depends on the parameters of the models. As we show, in certain limiting cases, the inverse symmetric potentials are qualitatively similar to the $\\alpha$-attractors models, since the resulting observational indices are identical. However, there are some quantitative differences which we discuss in some detail. As we show, some inverse symmetric models always yield results compatible with observations, but this strongly depends on the asymptotic form of the potential at large $e$-folding numbers. In fact when the limiting functional form is identical to the one corresponding to the $\\alpha$-attractors models, the compatibility with the observations is guaranteed. Also we find the relation of the inverse symmetric models with the Starobinsky model and we highlight the differences. In addition, an alternative inverse symmetric model is studied and as we show, not all the inverse symmetric models are viable. Moreove...
Symmetric cryptographic protocols
Ramkumar, Mahalingam
2014-01-01
This book focuses on protocols and constructions that make good use of symmetric pseudo random functions (PRF) like block ciphers and hash functions - the building blocks for symmetric cryptography. Readers will benefit from detailed discussion of several strategies for utilizing symmetric PRFs. Coverage includes various key distribution strategies for unicast, broadcast and multicast security, and strategies for constructing efficient digests of dynamic databases using binary hash trees. • Provides detailed coverage of symmetric key protocols • Describes various applications of symmetric building blocks • Includes strategies for constructing compact and efficient digests of dynamic databases
Experimental demonstration of PT-symmetric stripe lasers
Gu, Zhiyuan; Lyu, Quan; Li, Meng; Xiao, Shumin; Song, Qinghai
2015-01-01
Recently, the coexistence of parity-time (PT) symmetric laser and absorber has gained tremendous research attention. While the PT symmetric absorber has been observed in microwave metamaterials, the experimental demonstration of PT symmetric laser is still absent. Here we experimentally study PT-symmetric laser absorber in stripe waveguide. Using the concept of PT symmetry to exploit the light amplification and absorption, PT-symmetric laser absorbers have been successfully obtained. Different from the single-mode PT symmetric lasers, the PT-symmetric stripe lasers have been experimentally confirmed by comparing the relative wavelength positions and mode spacing under different pumping conditions. When the waveguide is half pumped, the mode spacing is doubled and the lasing wavelengths shift to the center of every two initial lasing modes. All these observations are consistent with the theoretical predictions and confirm the PT-symmetry breaking well.
Entangling capabilities of symmetric two-qubit gates
Indian Academy of Sciences (India)
Swarnamala Sirsi; Veena Adiga; Subramanya Hegde
2014-08-01
Our work addresses the problem of generating maximally entangled two spin-1/2 (qubit) symmetric states using NMR, NQR, Lipkin–Meshkov–Glick Hamiltonians. Time evolution of such Hamiltonians provides various logic gates which can be used for quantum processing tasks. Pairs of spin-1/2s have modelled a wide range of problems in physics. Here, we are interested in two spin-1/2 symmetric states which belong to a subspace spanned by the angular momentum basis $\\{|j = 1,\\langle; = + 1, 0, -12\\}$. Our technique relies on the decomposition of a Hamiltonian in terms of (3) basis matrices. In this context, we define a set of linearly independent, traceless, Hermitian operators which provides an alternate set of () generators. These matrices are constructed out of angular momentum operators J$_x$, J$_y$, J$_z$. We construct and study the properties of perfect entanglers acting on a symmetric subspace, i.e., spin-1 operators that can generate maximally entangled states from some suitably chosen initial separable states in terms of their entangling power.
An integral conservative gridding--algorithm using Hermitian curve interpolation.
Volken, Werner; Frei, Daniel; Manser, Peter; Mini, Roberto; Born, Ernst J; Fix, Michael K
2008-11-07
The problem of re-sampling spatially distributed data organized into regular or irregular grids to finer or coarser resolution is a common task in data processing. This procedure is known as 'gridding' or 're-binning'. Depending on the quantity the data represents, the gridding-algorithm has to meet different requirements. For example, histogrammed physical quantities such as mass or energy have to be re-binned in order to conserve the overall integral. Moreover, if the quantity is positive definite, negative sampling values should be avoided. The gridding process requires a re-distribution of the original data set to a user-requested grid according to a distribution function. The distribution function can be determined on the basis of the given data by interpolation methods. In general, accurate interpolation with respect to multiple boundary conditions of heavily fluctuating data requires polynomial interpolation functions of second or even higher order. However, this may result in unrealistic deviations (overshoots or undershoots) of the interpolation function from the data. Accordingly, the re-sampled data may overestimate or underestimate the given data by a significant amount. The gridding-algorithm presented in this work was developed in order to overcome these problems. Instead of a straightforward interpolation of the given data using high-order polynomials, a parametrized Hermitian interpolation curve was used to approximate the integrated data set. A single parameter is determined by which the user can control the behavior of the interpolation function, i.e. the amount of overshoot and undershoot. Furthermore, it is shown how the algorithm can be extended to multidimensional grids. The algorithm was compared to commonly used gridding-algorithms using linear and cubic interpolation functions. It is shown that such interpolation functions may overestimate or underestimate the source data by about 10-20%, while the new algorithm can be tuned to
PREFACE: 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics
Fring, Andreas; Jones, Hugh; Znojil, Miloslav
2008-06-01
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS5 x S5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the
On projective invariants of the complex Finsler spaces
Aldea, Nicoleta
2011-01-01
In this paper the projective curvature invariants of a complex Finsler space are obtained. By means of these invariants the notion of complex Douglas space is then defined. A special approach is devoted to obtain the equivalence conditions that a complex Finsler space should be Douglas. It is shown that any weakly K\\"{a}hler Douglas space is a complex Berwald space. A projective curvature invariant of Weyl type characterizes the complex Berwald spaces. They must be either purely Hermitian of constant holomorphic curvature or non purely Hermitian of vanish holomorphic curvature. The locally projectively flat complex Finsler metrics are also studied
Sturm, Erica J.
Strong correlation due to multi-referenced electronic states of quantum chemical systems are crucial for a proper understanding of important phenomena including excited states, bond breakage and formation, singlet fission and biological transport. By solving for the 2-electron reduced density matrix (2-RDM) directly via the anti-Hermitian contracted Schrodinger equation (ACSE) we provide a balanced treatment of single and multi-referenced correlation effects without utilizing the N-electron wave function. This significantly reduces the computational expense while still maintaining near full configuration interaction accuracy when available. When provided with an initial 2-RDM guess from an active-space multi-configuration self consistent field wave function the ACSE scales as [special characters omitted] where ra is the number of active molecular orbitals (MOs) and ra is the number of external MOs. This work demonstrates the energetic accuracy of ACSE calculations with several small multi-referenced systems and presents a novel approach for investigating intermolecular interactions, using a simple dimer test case. In this monomer-optimized basis set approach we compute each monomer's properties in isolation and obtain a set of natural orbitals that best describe the monomer. We then remove or truncate orbitals deemed excessive as a function of occupation number, defining a monomer molecular orbital basis uniquely suited to that monomer. Combining two such monomers yields a super-system expressed in the monomer basis which we then rotate to a dimer basis at a desired geometry before creating a new initial 2-RDM for the final optimization by an ACSE calculation. It is found that the intermolecular properties calculated in this fashion from larger atomic basis sets maintain their high accuracy but at a fraction of the computational cost. Furthermore this basis set optimization is free of basis set superposition error, circumventing the need for an expensive
Energy Technology Data Exchange (ETDEWEB)
Tipler, F.J.
1977-08-01
Causally symmetric spacetimes are spacetimes with J/sup +/(S) isometric to J/sup -/(S) for some set S. We discuss certain properties of these spacetimes, showing for example that, if S is a maximal Cauchy surface with matter everywhere on S, then the spacetime has singularities in both J/sup +/(S) and J/sup -/(S). We also consider totally vicious spacetimes, a class of causally symmetric spacetimes for which I/sup +/(p) =I/sup -/(p) = M for any point p in M. Two different notions of stability in general relativity are discussed, using various types of causally symmetric spacetimes as starting points for perturbations.
Symmetrization and Applications
Kesavan, S
2006-01-01
The study of isoperimetric inequalities involves a fascinating interplay of analysis, geometry and the theory of partial differential equations. Several conjectures have been made and while many have been resolved, a large number still remain open.One of the principal tools in the study of isoperimetric problems, especially when spherical symmetry is involved, is Schwarz symmetrization, which is also known as the spherically symmetric and decreasing rearrangement of functions. The aim of this book is to give an introduction to the theory of Schwarz symmetrization and study some of its applicat
Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem
Raman, Aaswath
2010-02-26
We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general. © 2010 The American Physical Society.
Krylov subspace methods and the sign function: multishifts and deflation in the non-Hermitian case
Bloch, Jacques C R; Frommer, Andreas; Heybrock, Simon; Schäfer, Katrin; Wettig, Tilo
2009-01-01
Rational approximations of the matrix sign function lead to multishift methods. For non-Hermitian matrices long recurrences can cause storage problems, which can be circumvented with restarts. Together with deflation we obtain efficient iterative methods, as we show in numerical experiments for the overlap Dirac operator at non-vanishing quark chemical potential for lattices up to size 10^4.
Non-Hermitian Swanson model with a time-dependent metric
Fring, Andreas; Moussa, Miled H. Y.
2016-10-01
We provide further nontrivial solutions to the recently proposed time-dependent Dyson and quasi-Hermiticity relation. Here, we solve them for the generalized version of the non-Hermitian Swanson Hamiltonian with time-dependent coefficients. We construct time-dependent solutions by employing the Lewis-Riesenfeld method of invariants and discuss concrete physical applications of our results.
Convergence of TTS Iterative Method for Non-Hermitian Positive Definite Linear Systems
Directory of Open Access Journals (Sweden)
Cheng-Yi Zhang
2014-01-01
Full Text Available The TTS iterative method is proposed to solve non-Hermitian positive definite linear systems and some convergence conditions are presented. Subsequently, these convergence conditions are applied to the ALUS method proposed by Xiang et al. in 2012 and comparison of some convergence theorems is made. Furthermore, an example is given to demonstrate the results obtained in this paper.
Semiclassical expansions in the Toda hierarchy and the Hermitian matrix model
Energy Technology Data Exchange (ETDEWEB)
Alonso, L MartInez [Departamento de Fisica Teorica II, Universidad Complutense, E28040 Madrid (Spain); Medina, E [Departamento de Matematicas, Universidad de Cadiz, E11510 Puerto Real, Cadiz (Spain)
2007-11-23
An iterative algorithm for determining a type of solutions of the dispersionful 2-Toda hierarchy characterized by string equations is developed. This type includes the solution which underlies the large-N limit of the Hermitian matrix model in the one-cut case. It is also shown how the double scaling limit can be naturally formulated in this scheme.
Invariant Hermitian Operator and Density Operator for the Adiabatically Time-Dependent System
Institute of Scientific and Technical Information of China (English)
YAN Feng-Li; YANG Lin-Guang
2001-01-01
The density operator is approximately expressed as a function of the invariant Hermitian operator for the adiabatically time-dependent system. Using this method, the calculation of the density operator for the Heisenberg spin system in a weakly time-dependent magnetic field is exemplified. By virtue of the density operator, we obtain equilibrium.``
Dunajewski, Adam; Dusza, Jacek J.; Rosado Muñoz, Alfredo
2014-11-01
The article presents a proposal for the description of human gait as a periodic and symmetric process. Firstly, the data for researches was obtained in the Laboratory of Group SATI in the School of Engineering of University of Valencia. Then, the periodical model - Mean Double Step (MDS) was made. Finally, on the basis of MDS, the symmetrical models - Left Mean Double Step and Right Mean Double Step (LMDS and RMDS) could be created. The method of various functional extensions was used. Symmetrical gait models can be used to calculate the coefficients of asymmetry at any time or phase of the gait. In this way it is possible to create asymmetry, function which better describes human gait dysfunction. The paper also describes an algorithm for calculating symmetric models, and shows exemplary results based on the experimental data.
Spherically symmetric brane spacetime with bulk gravity
Chakraborty, Sumanta; SenGupta, Soumitra
2015-01-01
Introducing term in the five-dimensional bulk action we derive effective Einstein's equation on the brane using Gauss-Codazzi equation. This effective equation is then solved for different conditions on dark radiation and dark pressure to obtain various spherically symmetric solutions. Some of these static spherically symmetric solutions correspond to black hole solutions, with parameters induced from the bulk. Specially, the dark pressure and dark radiation terms (electric part of Weyl curvature) affect the brane spherically symmetric solutions significantly. We have solved for one parameter group of conformal motions where the dark radiation and dark pressure terms are exactly obtained exploiting the corresponding Lie symmetry. Various thermodynamic features of these spherically symmetric space-times are studied, showing existence of second order phase transition. This phenomenon has its origin in the higher curvature term with gravity in the bulk.
Immanant Conversion on Symmetric Matrices
Directory of Open Access Journals (Sweden)
Purificação Coelho M.
2014-01-01
Full Text Available Letr Σn(C denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σn(C -> Σn (C satisfying for any fixed irre- ducible characters X, X' -SC the condition dx(A +aB = dχ·(Φ(Α + αΦ(Β for all matrices A,В ε Σ„(С and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on ΣИ(С.
Time-Symmetric Approach to Gravity
Chu, S Y
1998-01-01
Quantization of the time symmetric system of interacting strings requires that gravity, just as electromagnetism in Wheeler-Feynman's time symmetric electro- dynamics, also be an "adjunct field" instead of an independent entity. The "adjunct field" emerges, at a scale large compared to that of the strings, as a "statistic" that summarizes how the string positions in the underlying space- time are "compactified" into those in Minkowski space. We are able to show, WITHOUT adding a scalar curvature term to the string action, that the "adjunct gravitational field" satisfies Einstein's equation with no cosmological term.
Moduli Space of Integrable Dirac Structures
Milani, Vida
2009-01-01
In this paper we introduce the notion of integrable Dirac structures on Hermitian modules. The moduli space of the space of integrable Dirac structures is studied. Then a necessary and sufficient condition for the integrability of a Dirac structure is obtained as the solution of a certain partial differential equation.
Sinha, Debdeep; Ghosh, Pijush K
2015-04-01
A class of nonlocal nonlinear Schrödinger equations (NLSEs) is considered in an external potential with a space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable nonlocal NLSE with self-induced potential that is parity-time-symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or nonautonomous nonlocal NLSE by using similarity transformation, and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the nonlocal NLSE without the external potential and a (d+1)-dimensional generalization of it admits all the symmetries of the (d+1)-dimensional Schrödinger group. The conserved Noether charges associated with the time translation, dilatation, and special conformal transformation are shown to be real-valued in spite of being non-Hermitian. Finally, the dynamics of different moments are studied with an exact description of the time evolution of the "pseudowidth" of the wave packet for the special case in which the system admits a O(2,1) conformal symmetry.
Sinha, Debdeep; Ghosh, Pijush K.
2015-04-01
A class of nonlocal nonlinear Schrödinger equations (NLSEs) is considered in an external potential with a space-time modulated coefficient of the nonlinear interaction term as well as confining and/or loss-gain terms. This is a generalization of a recently introduced integrable nonlocal NLSE with self-induced potential that is parity-time-symmetric in the corresponding stationary problem. Exact soliton solutions are obtained for the inhomogeneous and/or nonautonomous nonlocal NLSE by using similarity transformation, and the method is illustrated with a few examples. It is found that only those transformations are allowed for which the transformed spatial coordinate is odd under the parity transformation of the original one. It is shown that the nonlocal NLSE without the external potential and a (d +1 )-dimensional generalization of it admits all the symmetries of the (d +1 )-dimensional Schrödinger group. The conserved Noether charges associated with the time translation, dilatation, and special conformal transformation are shown to be real-valued in spite of being non-Hermitian. Finally, the dynamics of different moments are studied with an exact description of the time evolution of the "pseudowidth" of the wave packet for the special case in which the system admits a O (2 ,1 ) conformal symmetry.
Jackson's Theorem on Bounded Symmetric Domains
Institute of Scientific and Technical Information of China (English)
Ming Zhi WANG; Guang Bin REN
2007-01-01
Polynomial approximation is studied on bounded symmetric domain Ω in C n for holo-morphic function spaces X ,such as Bloch-type spaces,Bergman-type spaces,Hardy spaces,Ω algebra and Lipschitz space.We extend the classical Jackson ’s theorem to several complex variables:E k f,X ) ω (1 /k,f,X ),where E k f,X )is the deviation of the best approximation of f ∈X by polynomials of degree at mostk with respect to the X -metric and ω (1/k,f,X )is the corresponding modulus of continuity.
Observation of Bloch oscillations in complex PT-symmetric photonic lattices
Wimmer, Martin; Christodoulides, Demetrios; Peschel, Ulf
2016-01-01
Light propagation in periodic environments is often associated with a number of interesting and potentially useful processes. If a crystalline optical potential is also linearly ramped, light can undergo periodic Bloch oscillations, a direct outcome of localized Wannier-Stark states and their equidistant eigenvalue spectrum. Even though these effects have been extensively explored in conservative settings, this is by no means the case in non-Hermitian photonic lattices encompassing both amplification and attenuation. Quite recently, Bloch oscillations have been predicted in parity-time-symmetric structures involving gain and loss in a balanced fashion. While in a complex bulk medium, one intuitively expects that light will typically follow the path of highest amplification, in a periodic system this behavior can be substantially altered by the underlying band structure. Here, we report the first experimental observation of Bloch oscillations in parity-time-symmetric mesh lattices. We show that these revivals ...
A selection rule for transitions in PT-symmetric quantum theory
Mead, Lawrence R
2016-01-01
Carl Bender and collaborators have developed a quantum theory governed by Hamiltonians that are PT-symmetric rather than Hermitian. To implement this theory, the inner product was redefined to guarantee positive norms of eigenstates of the Hamiltonian. In the general case, which includes arbitrary time-dependence in the Hamiltonian, a modification of the Schrodinger equation is necessary as shown by Gong and Wang to conserve probability. In this paper, we derive the following selection rule: transitions induced by time dependence in a PT-symmetric Hamiltonian cannot occur between normalized states of differing PT-norm. We show three examples of this selection rule in action: two matrix models and one in the continuum.
N>=2 symmetric superpolynomials
Alarie-Vézina, L; Mathieu, P
2015-01-01
The theory of symmetric functions has been extended to the case where each variable is paired with an anticommuting one. The resulting expressions, dubbed superpolynomials, provide the natural N=1 supersymmetric version of the classical bases of symmetric functions. Here we consider the case where two independent anticommuting variables are attached to each ordinary variable. The N=2 super-version of the monomial, elementary, homogeneous symmetric functions, as well as the power sums, are then constructed systematically (using an exterior-differential formalism for the multiplicative bases), these functions being now indexed by a novel type of superpartitions. Moreover, the scalar product of power sums turns out to have a natural N=2 generalization which preserves the duality between the monomial and homogeneous bases. All these results are then generalized to an arbitrary value of N. Finally, for N=2, the scalar product and the homogenous functions are shown to have a one-parameter deformation, a result that...
Counting with symmetric functions
Mendes, Anthony
2015-01-01
This monograph provides a self-contained introduction to symmetric functions and their use in enumerative combinatorics. It is the first book to explore many of the methods and results that the authors present. Numerous exercises are included throughout, along with full solutions, to illustrate concepts and also highlight many interesting mathematical ideas. The text begins by introducing fundamental combinatorial objects such as permutations and integer partitions, as well as generating functions. Symmetric functions are considered in the next chapter, with a unique emphasis on the combinatorics of the transition matrices between bases of symmetric functions. Chapter 3 uses this introductory material to describe how to find an assortment of generating functions for permutation statistics, and then these techniques are extended to find generating functions for a variety of objects in Chapter 4. The next two chapters present the Robinson-Schensted-Knuth algorithm and a method for proving Pólya’s enu...
Symmetric tensor decomposition
Brachat, Jerome; Mourrain, Bernard; Tsigaridas, Elias
2009-01-01
We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on th...
Multiparty Symmetric Sum Types
DEFF Research Database (Denmark)
Nielsen, Lasse; Yoshida, Nobuko; Honda, Kohei
2010-01-01
This paper introduces a new theory of multiparty session types based on symmetric sum types, by which we can type non-deterministic orchestration choice behaviours. While the original branching type in session types can represent a choice made by a single participant and accepted by others...... determining how the session proceeds, the symmetric sum type represents a choice made by agreement among all the participants of a session. Such behaviour can be found in many practical systems, including collaborative workflow in healthcare systems for clinical practice guidelines (CPGs). Processes...... with the symmetric sums can be embedded into the original branching types using conductor processes. We show that this type-driven embedding preserves typability, satisfies semantic soundness and completeness, and meets the encodability criteria adapted to the typed setting. The theory leads to an efficient...
Institute of Scientific and Technical Information of China (English)
ZHANG Zhong-zhi; HAN Xu-li
2005-01-01
By using the characteristic properties of the anti-Hermitian generalized anti-Hamiltonian matrices, we prove some necessary and sufficient conditions of the solvability for algebra inverse eigenvalue problem of anti-Hermitian generalized anti-Hamiltonian matrices, and obtain a general expression of the solution to this problem. By using the properties of the orthogonal projection matrix, we also obtain the expression of the solution to optimal approximate problem of an n× n complex matrix under spectral restriction.
A Non-Hermitian Approach to Non-Linear Switching Dynamics in Coupled Cavity-Waveguide Systems
DEFF Research Database (Denmark)
Heuck, Mikkel; Kristensen, Philip Trøst; Mørk, Jesper
2012-01-01
We present a non-Hermitian perturbation theory employing quasi-normal modes to investigate non-linear all-optical switching dynamics in a photonic crystal coupled cavity-waveguide system and compare with finite-difference-time-domain simulations.......We present a non-Hermitian perturbation theory employing quasi-normal modes to investigate non-linear all-optical switching dynamics in a photonic crystal coupled cavity-waveguide system and compare with finite-difference-time-domain simulations....
Progressive symmetric erythrokeratoderma
Directory of Open Access Journals (Sweden)
Gharpuray Mohan
1990-01-01
Full Text Available Four patients had symmetrically distributed hyperkeratotic plaques on the trunk and extremities; The lesions in all of them had appeared during infancy, and after a brief period of progression, had remained static, All of them had no family history of similar skin lesions. They responded well to topical applications of 6% salicylic acid in 50% propylene glycol. Unusual features in these cases of progressive symmetric erythrokeratoderma were the sparing of palms and soles, involvement of the trunk and absence of erythema.
Fields, Strings, Matrices and Symmetric Products
Dijkgraaf, R.
1999-01-01
In these notes we review the role played by the quantum mechanics and sigma models of symmetric product spaces in the light-cone quantization of quantum field theories, string theory and matrix theory. Lectures given at the Institute for Theoretical Physics, UC Santa Barbara, January 1998 and the Spring School on String Theory and Mathematics, Harvard University, May 1998.
Solving Quantum-Nonautonomous System with Non-Hermitian Hamiltonians by Algebraic Method
Institute of Scientific and Technical Information of China (English)
WEI Lian-Fu; WANG Shun-Jin
2001-01-01
A convenient method to exactly solve the quantum-nonautonomous systems with non-Hermitian Hamiltonians is proposed. It is shown that a nonadiabatic complete biorthonormal set can be easily obtained by the gauge transformation method in which the algebraic structure of systems has been used. The nonunitary evolution operator is also found by choosing a special gauge function. All auxiliary parameters introduced in the present approach are only determined by some algebraic equations. The dynamics of two quantum-nonautonomous systems ruled by non-Hermitian Hamiltonians, including a two-photon ionization process involving two-state only and a mesoscopic RLC circuit with a source, are treated as the demonstration of our general approach.``
Numerical Hermitian Yang-Mills connections and vector bundle stability in heterotic theories
Anderson, Lara B.; Braun, Volker; Karp, Robert L.; Ovrut, Burt A.
2010-06-01
A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the Hermitian Yang-Mills equations, including field-strength singularities, can be accurately reproduced numerically. These results make it possible to numerically determine whether or not a vector bundle is slope-stable, thus providing an important new tool in the exploration of heterotic vacua.
Numerical Hermitian Yang-Mills Connections and Vector Bundle Stability in Heterotic Theories
Anderson, Lara B; Karp, Robert L; Ovrut, Burt A
2010-01-01
A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. We then extend our results by investigating the behavior of non slope-stable bundles. In a variety of examples, it is shown that the failure of these bundles to satisfy the Hermitian Yang-Mills equations, including field-strength singularities, can be accurately reproduced numerically. These results make it possible to numerically determine whether or not a vector bundle is slope-stable, thus providing an important new tool in the exploration of heterotic vacua.
Geometry of 2×2 Hermitian matrices over any division ring
Institute of Scientific and Technical Information of China (English)
HUANG LiPing
2009-01-01
Let D be a division ring with an involution ~-, H_2(D) be the set of 2×2 Hermitian matrices over D. Let ad(A, B) = rank(A- B) be the arithmetic distance between A, B ∈ H2(D). In this paper, the fundamental theorem of the geometry of 2×2 Hermitian matrices over D (char(D) ≠2) is proved: if ψ : H_2(D) → H_2(D) is the adjacency preserving bijective map, then ψ is of the form ψ(X) = ~tPX~σP+ψ(0), where P ∈GL_2(D), σ is a quasi-automorphism of D. The quasi-automorphism of D is studied, and further results are obtained.
Geometry of 2×2 Hermitian matrices over any division ring
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
Let D be a division ring with an involution-,H2(D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A-B) be the arithmetic distance between A,B ∈ H2(D) . In this paper,the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D(char(D) = 2) is proved:if :H2(D) → H2(D) is the adjacency preserving bijective map,then is of the form (X) = tP XσP +(0) ,where P ∈ GL2(D) ,σ is a quasi-automorphism of D. The quasi-automorphism of D is studied,and further results are obtained.
Non-Hermitian approach of edge states and quantum transport in a magnetic field
Ostahie, B.; NiÅ£a, M.; Aldea, A.
2016-11-01
We develop a manifest non-Hermitian approach of spectral and transport properties of two-dimensional mesoscopic systems in a strong magnetic field. The finite system to which several terminals are attached constitutes an open system that can be described by an effective Hamiltonian. The lifetime of the quantum states expressed by the energy imaginary part depends specifically on the lead-system coupling and makes the difference among three regimes: resonant, integer quantum Hall effect, and superradiant. The discussion is carried on in terms of edge state lifetime in different gaps, channel formation, role of hybridization, and transmission coefficients quantization. A toy model helps in understanding non-Hermitian aspects in open systems.
A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES
Institute of Scientific and Technical Information of China (English)
Zhong-zhi Bai; Jun-feng Yin; Yang-feng Su
2006-01-01
A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.
Yi, Xingwen; Chen, Xuemei; Sharma, Dinesh; Li, Chao; Luo, Ming; Yang, Qi; Li, Zhaohui; Qiu, Kun
2014-06-02
Digital coherent superposition (DCS) provides an approach to combat fiber nonlinearities by trading off the spectrum efficiency. In analogy, we extend the concept of DCS to the optical OFDM subcarrier pairs with Hermitian symmetry to combat the linear and nonlinear phase noise. At the transmitter, we simply use a real-valued OFDM signal to drive a Mach-Zehnder (MZ) intensity modulator biased at the null point and the so-generated OFDM signal is Hermitian in the frequency domain. At receiver, after the conventional OFDM signal processing, we conduct DCS of the optical OFDM subcarrier pairs, which requires only conjugation and summation. We show that the inter-carrier-interference (ICI) due to phase noise can be reduced because of the Hermitain symmetry. In a simulation, this method improves the tolerance to the laser phase noise. In a nonlinear WDM transmission experiment, this method also achieves better performance under the influence of cross phase modulation (XPM).
Distributed Searchable Symmetric Encryption
Bösch, Christoph; Peter, Andreas; Leenders, Bram; Lim, Hoon Wei; Tang, Qiang; Wang, Huaxiong; Hartel, Pieter; Jonker, Willem
2014-01-01
Searchable Symmetric Encryption (SSE) allows a client to store encrypted data on a storage provider in such a way, that the client is able to search and retrieve the data selectively without the storage provider learning the contents of the data or the words being searched for. Practical SSE schemes
Energy Technology Data Exchange (ETDEWEB)
Amore, Paolo, E-mail: paolo.amore@gmail.com [Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima (Mexico); Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar [INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica, Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina); Garcia, Javier [INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica, Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina); Gutierrez, German [Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima (Mexico)
2014-04-15
We study both analytically and numerically the spectrum of inhomogeneous strings with PT-symmetric density. We discuss an exactly solvable model of PT-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules Z(p)≡∑{sub n=1}{sup ∞}1/E{sub n}{sup p}, with p=1,2,… and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex. -- Highlights: •PT-symmetric Hamiltonians exhibit real eigenvalues when PT symmetry is unbroken. •We study PT-symmetric strings with complex density. •They exhibit regions of unbroken PT symmetry. •We calculate the critical parameters at the boundaries of those regions. •There are exact real sum rules for some particular complex densities.
Brunner, T.; Mueller, A. R.; O'Sullivan, K.; Simon, M. C.; Kossick, M.; Ettenauer, S.; Gallant, A. T.; Mané, E.; Bishop, D.; Good, M.; Gratta, G.; Dilling, J.
2012-01-01
Bradbury Nielsen gates are well known devices used to switch ion beams and are typically applied in mass or mobility spectrometers for separating beam constituents by their different flight or drift times. A Bradbury Nielsen gate consists of two interleaved sets of electrodes. If two voltages of the same amplitude but opposite polarity are applied the gate is closed, and for identical (zero) potential the gate is open. Whereas former realizations of the device employ actual wires resulting in difficulties with winding, fixing and tensioning them, our approach is to use two grids photo-etched from a metallic foil. This design allows for simplified construction of gates covering large beam sizes up to at least 900\\,mm$^2$ with variable wire spacing down to 250\\,\\textmu m. By changing the grids the wire spacing can be varied easily. A gate of this design was installed and systematically tested at TRIUMF's ion trap facility, TITAN, for use with radioactive beams to separate ions with different mass-to-charge ratios by their time-of-flight.
Spectrum of the Hermitian Wilson-Dirac Operator for a Uniform Magnetic Field in Two Dimensions
Kurokawa, H
2003-01-01
It is shown that the eigenvalue problem for the hermitian Wilson-Dirac operator of for a uniform magnetic field in two dimensions can be reduced to one-dimensional problem described by a relativistic analog of the Harper equation. An explicit formula for the secular equations is given in term of a set of polynomials. The spectrum exhibits a fractal structure in the infinite volume limit. An exact result concerning the index theorem for the overlap Dirac operator is obtained.
Directory of Open Access Journals (Sweden)
Hjalmar Rosengren
2006-12-01
Full Text Available We study multivariable Christoffel-Darboux kernels, which may be viewed as reproducing kernels for antisymmetric orthogonal polynomials, and also as correlation functions for products of characteristic polynomials of random Hermitian matrices. Using their interpretation as reproducing kernels, we obtain simple proofs of Pfaffian and determinant formulas, as well as Schur polynomial expansions, for such kernels. In subsequent work, these results are applied in combinatorics (enumeration of marked shifted tableaux and number theory (representation of integers as sums of squares.
The Optimization on Ranks and Inertias of a Quadratic Hermitian Matrix Function and Its Applications
Directory of Open Access Journals (Sweden)
Yirong Yao
2013-01-01
Full Text Available We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function subject to a consistent system of matrix equations and . As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities , and in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.
Non-Hermitian Acoustic Metamaterials: the role of Exceptional Points in sound absorption
Achilleos, V.; Theocharis, G.; Richoux, O.; Pagneux, V.
2016-01-01
Effective non-Hermitian Hamiltonians are obtained to describe coherent perfect absorbing and lasing boundary conditions. PT -symmetry of the Hamiltonians enables to design configurations which perfectly absorb at multiple frequencies. Broadened and flat perfect absorption is predicted at the exceptional point of PT -symmetry breaking while, for a particular case, absorption is enhanced with the use of gain. The aforementioned phenomena are illustrated for acoustic scattering through Helmholtz...
Discrete and Continuum Virasoro Constraints in Two-Cut Hermitian Matrix Models
Ogura, W
1993-01-01
Continuum Virasoro constraints in the two-cut hermitian matrix models are derived from the discrete Ward identities by means of the mapping from the $GL(\\infty )$ Toda hierarchy to the nonlinear Schr\\"odinger (NLS) hierarchy. The invariance of the string equation under the NLS flows is worked out. Also the quantization of the integration constant $\\alpha$ reported by Hollowood et al. is explained by the analyticity of the continuum limit.
Extension of the $CPT$ Theorem to non-Hermitian Hamiltonians and Unstable States
Mannheim, Philip D
2016-01-01
We extend the $CPT$ theorem to quantum field theories with non-Hermitian Hamiltonians and unstable states. Our derivation is a quite minimal one as it requires only the time independent evolution of scalar products, invariance under complex Lorentz transformations, and a non-standard but nonetheless perfectly legitimate interpretation of charge conjugation as an anti-linear operator. The first of these requirements does not force the Hamiltonian to be Hermitian. Rather, it forces its eigenvalues to either be real or to appear in complex conjugate pairs, forces the eigenvectors of such conjugate pairs to be conjugates of each other, and forces the Hamiltonian to admit of an anti-linear symmetry. The latter two requirements then force this anti-linear symmetry to be $CPT$, while forcing the Hamiltonian to be real rather than Hermitian. Our work justifies the use of the $CPT$ theorem in establishing the equality of the lifetimes of unstable particles that are charge conjugates of each other. We show that the Euc...
Znojil, Miloslav
2017-07-01
The phenomenon of the birth of an isolated quantum bound state at the lower edge of the continuum is studied for a particle moving along a discrete real line of coordinates x ∈Z . The motion is controlled by a weakly nonlocal 2 J -parametric external potential V which is non-Hermitian but P T symmetric. The model is found exactly solvable. The bound states are interpreted as Sturmians. Their closed-form definitions are presented and discussed up to J =7 .
Quantization of Big Bang in crypto-Hermitian Heisenberg picture
Znojil, Miloslav
2015-01-01
A background-independent quantization of the Universe near its Big Bang singularity is considered using a drastically simplified toy model. Several conceptual issues are addressed. (1) The observable spatial-geometry characteristics of our empty-space expanding Universe is sampled by the time-dependent operator $Q=Q(t)$ of the distance between two space-attached observers (``Alice and Bob''). (2) For any pre-selected guess of the simple, non-covariant time-dependent observable $Q(t)$ one of the Kato's exceptional points (viz., $t=\\tau_{(EP)}$) is postulated {\\em real-valued}. This enables us to treat it as the time of Big Bang. (3) During our ``Eon'' (i.e., at all $t>\\tau_{(EP)}$) the observability status of operator $Q(t)$ is mathematically guaranteed by its self-adjoint nature with respect to an {\\em ad hoc} Hilbert-space metric $\\Theta(t) \
Generating functions for symmetric and shifted symmetric functions
Jing, Naihuan; Rozhkovskaya, Natasha
2016-01-01
We describe generating functions for several important families of classical symmetric functions and shifted Schur functions. The approach is originated from vertex operator realization of symmetric functions and offers a unified method to treat various families of symmetric functions and their shifted analogues.
Generating functions for symmetric and shifted symmetric functions
Jing, Naihuan; Rozhkovskaya, Natasha
2016-01-01
We describe generating functions for several important families of classical symmetric functions and shifted Schur functions. The approach is originated from vertex operator realization of symmetric functions and offers a unified method to treat various families of symmetric functions and their shifted analogues.
Symmetric States on the Octonionic Bloch Ball
Graydon, Matthew
2012-02-01
Finite-dimensional homogeneous self-dual cones arise as natural candidates for convex sets of states and effects in a variety of approaches towards understanding the foundations of quantum theory in terms of information-theoretic concepts. The positive cone of the ten-dimensional Jordan-algebraic spin factor is one particular instantiation of such a convex set in generalized frameworks for quantum theory. We consider a projection of the regular 9-simplex onto the octonionic projective line to form a highly symmetric structure of ten octonionic quantum states on the surface of the octonionic Bloch ball. A uniform subnormalization of these ten symmetric states yields a symmetric informationally complete octonionic quantum measurement. We discuss a Quantum Bayesian reformulation of octonionic quantum formalism for the description of two-dimensional physical systems. We also describe a canonical embedding of the octonionic Bloch ball into an ambient space for states in usual complex quantum theory.
Homogenous finitary symmetric groups
Directory of Open Access Journals (Sweden)
Otto. H. Kegel
2015-03-01
Full Text Available We characterize strictly diagonal type of embeddings of finitary symmetric groups in terms of cardinality and the characteristic. Namely, we prove the following. Let kappa be an infinite cardinal. If G=underseti=1stackrelinftybigcupG i , where G i =FSym(kappan i , (H=underseti=1stackrelinftybigcupH i , where H i =Alt(kappan i , is a group of strictly diagonal type and xi=(p 1 ,p 2 ,ldots is an infinite sequence of primes, then G is isomorphic to the homogenous finitary symmetric group FSym(kappa(xi (H is isomorphic to the homogenous alternating group Alt(kappa(xi , where n 0 =1,n i =p 1 p 2 ldotsp i .
Chen, Yan; Feng, Huijuan; Ma, Jiayao; Peng, Rui; You, Zhong
2016-06-01
The traditional waterbomb origami, produced from a pattern consisting of a series of vertices where six creases meet, is one of the most widely used origami patterns. From a rigid origami viewpoint, it generally has multiple degrees of freedom, but when the pattern is folded symmetrically, the mobility reduces to one. This paper presents a thorough kinematic investigation on symmetric folding of the waterbomb pattern. It has been found that the pattern can have two folding paths under certain circumstance. Moreover, the pattern can be used to fold thick panels. Not only do the additional constraints imposed to fold the thick panels lead to single degree of freedom folding, but the folding process is also kinematically equivalent to the origami of zero-thickness sheets. The findings pave the way for the pattern being readily used to fold deployable structures ranging from flat roofs to large solar panels.
Classification of symmetric toroidal orbifolds
Energy Technology Data Exchange (ETDEWEB)
Fischer, Maximilian; Ratz, Michael; Torrado, Jesus [Technische Univ. Muenchen, Garching (Germany). Physik-Department; Vaudrevange, Patrick K.S. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2012-09-15
We provide a complete classification of six-dimensional symmetric toroidal orbifolds which yield N{>=}1 supersymmetry in 4D for the heterotic string. Our strategy is based on a classification of crystallographic space groups in six dimensions. We find in total 520 inequivalent toroidal orbifolds, 162 of them with Abelian point groups such as Z{sub 3}, Z{sub 4}, Z{sub 6}-I etc. and 358 with non-Abelian point groups such as S{sub 3}, D{sub 4}, A{sub 4} etc. We also briefly explore the properties of some orbifolds with Abelian point groups and N=1, i.e. specify the Hodge numbers and comment on the possible mechanisms (local or non-local) of gauge symmetry breaking.
Classification of symmetric toroidal orbifolds
Energy Technology Data Exchange (ETDEWEB)
Fischer, Maximilian; Ratz, Michael; Torrado, Jesus [Technische Univ. Muenchen, Garching (Germany). Physik-Department; Vaudrevange, Patrick K.S. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2012-09-15
We provide a complete classification of six-dimensional symmetric toroidal orbifolds which yield N{>=}1 supersymmetry in 4D for the heterotic string. Our strategy is based on a classification of crystallographic space groups in six dimensions. We find in total 520 inequivalent toroidal orbifolds, 162 of them with Abelian point groups such as Z{sub 3}, Z{sub 4}, Z{sub 6}-I etc. and 358 with non-Abelian point groups such as S{sub 3}, D{sub 4}, A{sub 4} etc. We also briefly explore the properties of some orbifolds with Abelian point groups and N=1, i.e. specify the Hodge numbers and comment on the possible mechanisms (local or non-local) of gauge symmetry breaking.
Symmetric Extended Ockham Algebras
Institute of Scientific and Technical Information of China (English)
T.S. Blyth; Jie Fang
2003-01-01
The variety eO of extended Ockham algebras consists of those algealgebra with an additional endomorphism k such that the unary operations f and k commute. Here, we consider the cO-algebras which have a property of symmetry. We show that there are thirty two non-isomorphic subdirectly irreducible symmetric extended MS-algebras and give a complete description of them.2000 Mathematics Subject Classification: 06D15, 06D30
Symmetrization Selection Rules, 1
Page, P R
1996-01-01
We introduce a category of strong and electromagnetic interaction selection rules for the two-body connected decay and production of exotic J^{PC} = 0^{+-}, 1^{-+}, 2^{+-}, 3^{-+}, ... hybrid and four-quark mesons. The rules arise from symmetrization in states in addition to Bose symmetry and CP invariance. Examples include various decays to \\eta'\\eta, \\eta\\pi, \\eta'\\pi and four-quark interpretations of a 1^{-+} signal.
Symmetrization Selection Rules, 2
Page, P R
1996-01-01
We introduce strong interaction selection rules for the two-body decay and production of hybrid and conventional mesons coupling to two S-wave hybrid or conventional mesons. The rules arise from symmetrization in states in the limit of non-relativistically moving quarks. The conditions under which hybrid coupling to S-wave states is suppressed are determined by the rules, and the nature of their breaking is indicated.
Symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions
Institute of Scientific and Technical Information of China (English)
2008-01-01
Based on a linear finite element space,two symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions are constructed and analyzed.Some relationships between the finite element method and the finite difference method are addressed,too.
Estimation of Time-Varying Autoregressive Symmetric Alpha Stable
National Aeronautics and Space Administration — In the last decade alpha-stable distributions have become a standard model for impulsive data. Especially the linear symmetric alpha-stable processes have found...
Estimation of Time Varying Autoregressive Symmetric Alpha Stable
National Aeronautics and Space Administration — In this work, we present a novel method for modeling time-varying autoregressive impulsive signals driven by symmetric alpha stable distributions. The proposed...
Symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions
Institute of Scientific and Technical Information of China (English)
DAI Xiaoying; YANG Zhang; ZHOU Aihui
2008-01-01
Based on a linear finite element space, two symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions are constructed and analyzed. Some relationships between the finite element method and the finite difference method are addressed, too.
Super-symmetric informationally complete measurements
Energy Technology Data Exchange (ETDEWEB)
Zhu, Huangjun, E-mail: hzhu@pitp.ca
2015-11-15
Symmetric informationally complete measurements (SICs in short) are highly symmetric structures in the Hilbert space. They possess many nice properties which render them an ideal candidate for fiducial measurements. The symmetry of SICs is intimately connected with the geometry of the quantum state space and also has profound implications for foundational studies. Here we explore those SICs that are most symmetric according to a natural criterion and show that all of them are covariant with respect to the Heisenberg–Weyl groups, which are characterized by the discrete analog of the canonical commutation relation. Moreover, their symmetry groups are subgroups of the Clifford groups. In particular, we prove that the SIC in dimension 2, the Hesse SIC in dimension 3, and the set of Hoggar lines in dimension 8 are the only three SICs up to unitary equivalence whose symmetry groups act transitively on pairs of SIC projectors. Our work not only provides valuable insight about SICs, Heisenberg–Weyl groups, and Clifford groups, but also offers a new approach and perspective for studying many other discrete symmetric structures behind finite state quantum mechanics, such as mutually unbiased bases and discrete Wigner functions.
Symmetric Tensor Decomposition
DEFF Research Database (Denmark)
Brachat, Jerome; Comon, Pierre; Mourrain, Bernard
2010-01-01
of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation...... of total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view...
Symmetrically Constrained Compositions
Beck, Matthias; Lee, Sunyoung; Savage, Carla D
2009-01-01
Given integers $a_1, a_2, ..., a_n$, with $a_1 + a_2 + ... + a_n \\geq 1$, a symmetrically constrained composition $\\lambda_1 + lambda_2 + ... + lambda_n = M$ of $M$ into $n$ nonnegative parts is one that satisfies each of the the $n!$ constraints ${\\sum_{i=1}^n a_i \\lambda_{\\pi(i)} \\geq 0 : \\pi \\in S_n}$. We show how to compute the generating function of these compositions, combining methods from partition theory, permutation statistics, and lattice-point enumeration.
Construction of the moduli space of Spin (7)-instantons
Muñoz, Vicente
2016-01-01
We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.
Hermitian hat wavelet design for singularity detection in the Paraguay river-level data analyses
Szu, Harold H.; Hsu, Charles C.; Sa, Leonardo D.; Li, Weigang
1997-04-01
The direct differentiation of a noisy signal ds/dt is known to be inaccurate. Differentiation can be improved by employing the Dirac (delta) -function introduced into a convolution product denoted by (direct product) and then integrated by parts: ds/dt equals ds/dt (direct product) (delta) equals - s (direct product) d(delta) /dt. The Schwartz Gaussian representation of the delta function is then explicitly used in the differentiation. It turns out that such a convolution approach to the first and the second derivatives produces a pair of mother wavelets the combination of which is the complex generalization of the Mexican hat called a Hermitian hat wavelet. It is shown that the Hermitian filter is a single oscillation wavelet having much lower frequency bandwidth than the Mortlet or Gabor wavelet. As a result of Nyquist theorem, a fewer number of grid points would be needed for the discrete convolution operation. Therefore, the singularity characteristic will not be overly smeared and the noise can be smoothed away. The phase plot of the Hermitian wavelet transform in terms of the time scale and frequency domains reveal a bifurcation discontinuity of a noisy cusp singularity at the precise location of the singularity as well as the scale nature of the underlying dynamics. This phase plot is defined as (theta) (t/a) equals tan-1 [(ds/dt)/(-d2s/dt2] equals tan-1 [((d(delta) (t/a)dt) (direct product) s)/((d2(delta) (t/a)/dt2) (direct product) s)] applied to a real world data of the Paraguay river levels.
Chan, Garnet Kin-Lic; Van Voorhis, Troy
2005-05-22
We describe the theory and implementation of two extensions to the density-matrix renormalization-group (DMRG) algorithm in quantum chemistry: (i) to work with an underlying nonorthogonal one-particle basis (using a biorthogonal formulation) and (ii) to use non-Hermitian and complex operators and complex wave functions, which occur naturally in biorthogonal formulations. Using these developments, we carry out ground-state calculations on ethene, butadiene, and hexatriene, in a polarized atomic-orbital basis. The description of correlation in these systems using a localized nonorthogonal basis is improved over molecular-orbital DMRG calculations, and comparable to or better than coupled-cluster calculations, although we encountered numerical problems associated with non-Hermiticity. We believe that the non-Hermitian DMRG algorithm may further become useful in conjunction with other non-Hermitian Hamiltonians, for example, similarity-transformed coupled-cluster Hamiltonians.
Exceptional groups, symmetric spaces and applications
Energy Technology Data Exchange (ETDEWEB)
Cerchiai, Bianca L.; Cacciatori, Sergio L.
2009-03-31
In this article we provide a detailed description of a technique to obtain a simple parameterization for different exceptional Lie groups, such as G{sub 2}, F{sub 4} and E{sub 6}, based on their fibration structure. For the compact case, we construct a realization which is a generalization of the Euler angles for SU(2), while for the non compact version of G{sub 2(2)}/SO(4) we compute the Iwasawa decomposition. This allows us to obtain not only an explicit expression for the Haar measure on the group manifold, but also for the cosets G{sub 2}/SO(4), G{sub 2}/SU(3), F{sub 4}/Spin(9), E{sub 6}/F{sub 4} and G{sub 2(2)}/SO(4) that we used to find the concrete realization of the general element of the group. Moreover, as a by-product, in the simplest case of G{sub 2}/SO(4), we have been able to compute an Einstein metric and the vielbein. The relevance of these results in physics is discussed.
Non-Hermitian Acoustic Metamaterials: the role of Exceptional Points in sound absorption
Achilleos, V; Richoux, O; Pagneux, V
2016-01-01
Effective non-Hermitian Hamiltonians are obtained to describe coherent perfect absorbing and lasing boundary conditions. PT -symmetry of the Hamiltonians enables to design configurations which perfectly absorb at multiple frequencies. Broadened and flat perfect absorption is predicted at the exceptional point of PT -symmetry breaking while, for a particular case, absorption is enhanced with the use of gain. The aforementioned phenomena are illustrated for acoustic scattering through Helmholtz resonators revealing how tailoring the non-Hermiticity of acoustic metamaterials leads to novel mechanisms for enhanced absorption.
Waghela, Chetan
2015-01-01
Hamiltonian Mechanics works for conserved systems and Quantum Mechanics is given in Hamiltonian language. It is considered that complexifying the quantum Hamiltonian, a balanced loss and gain model can be created. The usual mathematics of density operator formalism and entanglement is extrapolated to such systems and the consequences are studied. Namely, a complete formalism using Density operators is created for real eigenvalue regime of these Non- Hermitian systems and correct forms of Von-Neumann and Entanglement Entropy are created. The consequences are studied in this regime and depicted w.r.t recent papers by [9, 20].
Large N expansions and Painleve hierarchies in the Hermitian matrix model
Energy Technology Data Exchange (ETDEWEB)
Alvarez, Gabriel; Alonso, Luis Martinez [Departamento de Fisica Teorica II, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid (Spain); Medina, Elena, E-mail: galvarez@fis.ucm.es [Departamento de Matematicas, Facultad de Ciencias, Universidad de Cadiz, 11510 Puerto Real, Cadiz (Spain)
2011-07-15
We present a method to characterize and compute the large N formal asymptotics of regular and critical Hermitian matrix models with general even potentials in the one-cut and two-cut cases. Our analysis is based on a method to solve continuum limits of the discrete string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. This method also leads to an explicit formulation, in terms of coupling constants and critical parameters, of the members of the Painleve I and Painleve II hierarchies associated with one-cut and two-cut critical models, respectively.
Dynamic stiffness matrix of thin-walled composite I-beam with symmetric and arbitrary laminations
Kim, Nam-Il; Shin, Dong Ku; Park, Young-Suk
2008-11-01
For the spatially coupled free vibration analysis of thin-walled composite I-beam with symmetric and arbitrary laminations, the exact dynamic stiffness matrix based on the solution of the simultaneous ordinary differential equations is presented. For this, a general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force-displacement relationships are derived from the energy principle and the first order of transformed simultaneous differential equations are constructed by using the displacement state vector consisting of 14 displacement parameters. Then explicit expressions for displacement parameters are derived and the exact dynamic stiffness matrix is determined using force-displacement relationships. In addition, the finite-element (FE) procedure based on Hermitian interpolation polynomials is developed. To verify the validity and the accuracy of this study, the numerical solutions are presented and compared with analytical solutions, the results from available references and the FE analysis using the thin-walled Hermitian beam elements. Particular emphasis is given in showing the phenomenon of vibrational mode change, the effects of increase of the modulus and the bending-twisting coupling stiffness for beams with various boundary conditions.
Irreducible complexity of iterated symmetric bimodal maps
Directory of Open Access Journals (Sweden)
J. P. Lampreia
2005-01-01
Full Text Available We introduce a tree structure for the iterates of symmetric bimodal maps and identify a subset which we prove to be isomorphic to the family of unimodal maps. This subset is used as a second factor for a ∗-product that we define in the space of bimodal kneading sequences. Finally, we give some properties for this product and study the ∗-product induced on the associated Markov shifts.
The Symmetricity of Normal Modes in Symmetric Complexes
Song, Guang
2016-01-01
In this work, we look at the symmetry of normal modes in symmetric structures, particularly structures with cyclic symmetry. We show that normal modes of symmetric structures have different levels of symmetry, or symmetricity. One novel theoretical result of this work is that, for a ring structure with $m$ subunits, the symmetricity of the normal modes falls into $m$ groups of equal size, with normal modes in each group having the same symmetricity. The normal modes in each group can be computed separately, using a much smaller amount of memory and time (up to $m^3$ less), thus making it applicable to larger complexes. We show that normal modes with perfect symmetry or anti-symmetry have no degeneracy while the rest of the modes have a degeneracy of two. We show also how symmetry in normal modes correlates with symmetry in structure. While a broken symmetry in structure generally leads to a loss of symmetricity in symmetric normal modes, the symmetricity of some symmetric normal modes is preserved even when s...
Sphaleron glueballs in NBI theory with symmetrized trace
Dyadichev, V V
2000-01-01
We derive a closed expression for the SU(2) Born-Infeld action with the symmetrized trace for static spherically symmetric purely magnetic configurations. The lagrangian is obtained in terms of elementary functions. Using it, we investigate glueball solutions to the flat space NBI theory and their self-gravitating counterparts. Such solutions, found previously in the NBI model with the 'square root - ordinary trace' lagrangian, are shown to persist in the theory with the symmetrized trace lagrangian as well. Although the symmetrized trace NBI equations differ substantially from those of the theory with the ordinary trace, a qualitative picture of glueballs remains essentially the same. Gravity further reduces the difference between solutions in these two models, and, for sufficiently large values of the effective gravitational coupling, solutions tends to the same limiting form. The black holes in the NBI theory with the symmetrized trace are also discussed.
On Stationary Axially Symmetric Solutions in Brans-Dicke Theory
Kirezli, Pınar
2015-01-01
Stationary axially symmetric Brans-Dicke-Maxwell solutions are re-examined in the framework of the Brans-Dicke theory. We see that, employing a particular parametrization of the standard axially symmetric metric simplifies the procedure of obtaining the Ernst equations for axially symmetric electro-vacuum space-times for this theory. This analysis also permit us to construct a two parameter extension in both Jordan and Einstein frames of an old solution generating technique frequently used to construct axially symmetric solutions for Brans-Dicke theory from a seed solution of General Relativity. As applications of this technique, several known and new solutions are constructed including a general axially symmetric BD-Maxwell solution of Plebanski-Demianski with vanishing cosmological constant, i.e. the Kinnersley solution and general magnetized Kerr-Newman type solutions. Some physical properties and circular motion of test particles for a particular subclass of Kinnersley solution, i.e. Kerr-Newman-NUT type ...
Is space-time symmetry a suitable generalization of parity-time symmetry?
Energy Technology Data Exchange (ETDEWEB)
Amore, Paolo, E-mail: paolo.amore@gmail.com [Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima (Mexico); Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar [INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica, Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina); Garcia, Javier [INIFTA (UNLP, CCT La Plata-CONICET), División Química Teórica, Diag. 113 y 64 (S/N), Sucursal 4, Casilla de Correo 16, 1900 La Plata (Argentina)
2014-11-15
We discuss space-time symmetric Hamiltonian operators of the form H=H{sub 0}+igH{sup ′}, where H{sub 0} is Hermitian and g real. H{sub 0} is invariant under the unitary operations of a point group G while H{sup ′} is invariant under transformation by elements of a subgroup G{sup ′} of G. If G exhibits irreducible representations of dimension greater than unity, then it is possible that H has complex eigenvalues for sufficiently small nonzero values of g. In the particular case that H is parity-time symmetric then it appears to exhibit real eigenvalues for all 0
Plane symmetric cosmological models
Yadav, Anil Kumar; Ray, Saibal; Mallick, A
2016-01-01
In this work, we perform the Lie symmetry analysis on the Einstein-Maxwell field equations in plane symmetric spacetime. Here Lie point symmetries and optimal system of one dimensional subalgebras are determined. The similarity reductions and exact solutions are obtained in connection to the evolution of universe. The present study deals with the electromagnetic energy of inhomogeneous universe where $F_{12}$ is the non-vanishing component of electromagnetic field tensor. To get a deterministic solution, it is assumed that the free gravitational field is Petrov type-II non-degenerate. The electromagnetic field tensor $F_{12}$ is found to be positive and increasing function of time. As a special case, to validate the solution set, we discuss some physical and geometric properties of a specific sub-model.
Institute of Scientific and Technical Information of China (English)
傅育熙
1998-01-01
An alternative presentation of the π－calculus is given.This version of the π-calculus is symmetric in the sense that communications are symmetric and there is no difference between input and output prefixes.The point of the symmetric π-calculus is that it has no abstract names.The set of closed names is therefore homogeneous.The π－calculus can be fully embedded into the symmetric π-calculus.The symmetry changes the emphasis of the communication mechanism of the π-calculus and opens up possibility for further variations.
Biorthonormal transfer-matrix renormalization-group method for non-Hermitian matrices.
Huang, Yu-Kun
2011-03-01
A biorthonormal transfer-matrix renormalization-group (BTMRG) method for non-Hermitian matrices is presented. This BTMRG produces a dual set of biorthonormal bases to construct the renormalized transfer matrix with only half the dimensions of the matrix of a conventional transfer-matrix renormalization group (TMRG). We show that under generic conditions, such biorthonormal bases always exist. Based on a special E·S·E scheme (where S and E represent the system and environment blocks, respectively, and the two dots in between represent two additional physical sites), the BTMRG method can achieve zero truncation of any reduced state in describing both current left and right Perron states so as to reach a high degree of efficiency and accuracy. We believe that the BTMRG constitutes a more powerful and robust tool than conventional TMRG for non-Hermitian matrices and that it would allow us to better understand the collective behaviors and emerging phenomena of strongly correlated many-body systems. We also show that this scheme is particularly adapted to the calculation of the two-site correlation function of a one-dimensional quantum or two-dimensional classical lattice model.
Theory of non-hermitian localization in one dimension: Localization length and eigenenergies
Indian Academy of Sciences (India)
J Heinrichs
2002-02-01
We recall some basic aspects of the pinning of ﬂux lines in a superconducting cylindrical shell subjected to a depinning magnetic ﬁeld, as well as its description by the quantum mechanics of a disordered ring with an imaginary vector potential proportional to the depinning ﬁeld (N Hatano and D R Nelson, Phys. Rev. B56, 8651 (1997)). We then discuss our recent analysis of the pinning-depinning transition in terms of an explicit solution for the inverse localization length of the eigenstates of the non-hermitian quantum system for weak disorder. Our results as to the nature of the non hermitian quantum states, differ qualitatively from earlier studies which did not examine the detailed properties of the localization length. Nevertheless we obtain a well-deﬁned simple picture for the pinning-depinning transition of ﬂux lines. We discuss furthermore a new exact calculation of localized state eigenenergies for weak disorder, which we compare with previous analytic and numerical results.
Zucchini, R
1995-01-01
In this paper, a novel method is presented for the study of the dependence of the functional determinant of the Laplace operator associated to a subbundle F of a hermitian holomorphic vector bundle E over a Riemann surface \\Sigma on the hermitian structure (h,H) of E. The generalized Weyl anomaly of the effective action is computed and found to be expressible in terms of a suitable generalization of the Liouville and Donaldson actions. The general techniques worked out are then applied to the study of a specific model, the Drinfeld--Sokolov (DS) ghost system arising in W--gravity. The expression of generalized Weyl anomaly of the DS ghost effective action is found. It is shown that, by a specific choice of the fiber metric H_h depending on the base metric h, the effective action reduces into that of a conformal field theory. Its central charge is computed and found to agree with that obtained by the methods of hamiltonian reduction and conformal field theory. The DS holomorphic gauge group and the DS moduli s...
Random Matrix Theory for the Hermitian Wilson Dirac Operator and the chGUE-GUE Transition
Akemann, Gernot
2011-01-01
We introduce a random two-matrix model interpolating between a chiral Hermitian (2n+nu)x(2n+nu) matrix and a second Hermitian matrix without symmetries. These are taken from the chiral Gaussian Unitary Ensemble (chGUE) and Gaussian Unitary Ensemble (GUE), respectively. In the microscopic large-n limit in the vicinity of the chGUE (which we denote by weakly non-chiral limit) this theory is in one to one correspondence to the partition function of Wilson chiral perturbation theory in the epsilon regime, such as the related two matrix-model previously introduced in refs. [20,21]. For a generic number of flavours and rectangular block matrices in the chGUE part we derive an eigenvalue representation for the partition function displaying a Pfaffian structure. In the quenched case with nu=0,1 we derive all spectral correlations functions in our model for finite-n, given in terms of skew-orthogonal polynomials. The latter are expressed as Gaussian integrals over standard Laguerre polynomials. In the weakly non-chira...
Symmetric Structure of Induction Motor Systems
Institute of Scientific and Technical Information of China (English)
无
1999-01-01
In this paper, symmetric structure of induction motor system in stationary αβ0 coordinates is studied bythe geometric approach. The results show that the system possesses symmetry (G, θ, Ф) and infinitesimal symme-try. Under certain conditions, the system can be transformed into a form possessing state-space symmetry (G, Ф)and infinitesimal state-space symmetry by means of state feedback and input coordinate base transform. The resultscan be extended to the fifth order induction motor system fed by hysteresis-band current-controlled PWM inverter.
Synthesis of controllers for symmetric systems
Ameur Abid, Chiheb; Zouari, Belhassen
2010-11-01
This article deals with supervisory control problem for coloured Petri (CP) nets. Considering a CP-net, we build a condensed version of the ordinary state-space, namely the symbolic reachability graph (SRG). This latter graph allows to cope with state-space explosion problem for symmetric systems. The control specification can be expressed in terms of either forbidden states or forbidden sequences of transitions. According to these specifications, we derive the controller by applying the theory of regions on the basis of the SRG. Thanks to expressiveness power of CP-nets, the obtained controller to be connected to the plant model is reduced to one single place.
Geometric multiaxial representation of N -qubit mixed symmetric separable states
SP, Suma; Sirsi, Swarnamala; Hegde, Subramanya; Bharath, Karthik
2017-08-01
The study of N -qubit mixed symmetric separable states is a longstanding challenging problem as no unique separability criterion exists. In this regard, we take up the N -qubit mixed symmetric separable states for a detailed study as these states are of experimental importance and offer an elegant mathematical analysis since the dimension of the Hilbert space is reduced from 2N to N +1 . Since there exists a one-to-one correspondence between the spin-j system and an N -qubit symmetric state, we employ Fano statistical tensor parameters for the parametrization of the spin-density matrix. Further, we use a geometric multiaxial representation (MAR) of the density matrix to characterize the mixed symmetric separable states. Since the separability problem is NP-hard, we choose to study it in the continuum limit where mixed symmetric separable states are characterized by the P -distribution function λ (θ ,ϕ ) . We show that the N -qubit mixed symmetric separable states can be visualized as a uniaxial system if the distribution function is independent of θ and ϕ . We further choose a distribution function to be the most general positive function on a sphere and observe that the statistical tensor parameters characterizing the N -qubit symmetric system are the expansion coefficients of the distribution function. As an example for the discrete case, we investigate the MAR of a uniformly weighted two-qubit mixed symmetric separable state. We also observe that there exists a correspondence between the separability and classicality of states.
Afzal, Muhammad Imran; Lee, Yong Tak
2016-01-01
Von Neumann and Wigner theorized bounding of asymmetric eigenstates and anti-crossing of symmetric eigenstates. Experiments have shown that owing to anti-crossing and similar radiation rates, graphene-like resonance of inhomogeneously strained photonic eigenstates can generate pseudomagnetic field, bandgaps and Landau levels, while dissimilar rates induce non-Hermicity. Here, we showed experimentally higher-order supersymmetry and quantum phase transitions by resonance between similar one dimensional lattices. The lattices consisted of inhomgeneously strain-like phases of triangular solitons. The resonance created two dimensional inhomogeneously deformed photonic graphene. All parent eigenstates are annihilated. Where eigenstates of mildly strained solitons are annihilated with similar (power law) rates through one tail only and generated Hermitianally bounded eigenstates. The strongly strained solitons, positive defects are annihilated exponentially through both tails with dissimilar rates. Which bounded eig...
Complete classification of spherically symmetric static spacetimes via Noether symmetries
Ali, Farhad; Ali, Sajid
2013-01-01
In this paper we give a complete classification of spherically symmetric static space-times by their Noether symmetries. The determining equations for Noether symmetries are obtained by using the usual Lagrangian of a general spherically symmetric static spacetime which are integrated for each case. In particular we observe that spherically symmetric static spacetimes are categorized into six distinct classes corresponding to Noether algebra of dimensions 5, 6, 7, 9, 11 and 17. Using Noether`s theorem we also write down the first integrals for each class of such spacetimes corresponding to their Noether symmetries.
A de Finetti representation for finite symmetric quantum states
König, R; Koenig, Robert; Renner, Renato
2004-01-01
Consider a symmetric quantum state on an n-fold product space, that is, the state is invariant under permutations of the n subsystems. We show that, conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems, the state in the remaining subsystems is close to having product form. This immediately generalizes the so-called de Finetti representation to the case of finite symmetric quantum states.
Representation of Fuzzy Symmetric Relations
1986-03-19
Std Z39-18 REPRESENTATION OF FUZZY SYMMETRIC RELATIONS L. Valverde Dept. de Matematiques i Estadistica Universitat Politecnica de Catalunya Avda...REPRESENTATION OF FUZZY SYMMETRIC RELATIONS L. "Valverde* Dept. de Matematiques i Estadistica Universitat Politecnica de Catalunya Avda. Diagonal, 649
Parallel Symmetric Eigenvalue Problem Solvers
2015-05-01
Plemmons G. Golub and A. Sameh. High-speed computing : scientific appli- cations and algorithm design. University of Illinois Press, Champaign, Illinois , 1988...16. SECURITY CLASSIFICATION OF: Sparse symmetric eigenvalue problems arise in many computational science and engineering applications such as...Eigenvalue Problem Solvers Report Title Sparse symmetric eigenvalue problems arise in many computational science and engineering applications such as
INVERSE EIGENVALUE PROBLEM OF HERMITIAN GENERALIZED ANTI-HAMILTONIAN MATRICES%HGAH矩阵的逆特征值问题
Institute of Scientific and Technical Information of China (English)
张忠志; Liu Changrong
2004-01-01
In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamiltonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.
Mandal, B P
2004-01-01
We consider a spin half particle in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction. We find that the energy eigenvalues for this system are real even though the interaction is not PT invariant.
Maximally Symmetric Spacetimes emerging from thermodynamic fluctuations
Bravetti, A; Quevedo, H
2015-01-01
In this work we prove that the maximally symmetric vacuum solutions of General Relativity emerge from the geometric structure of statistical mechanics and thermodynamic fluctuation theory. To present our argument, we begin by showing that the pseudo-Riemannian structure of the Thermodynamic Phase Space is a solution to the vacuum Einstein-Gauss-Bonnet theory of gravity with a cosmological constant. Then, we use the geometry of equilibrium thermodynamics to demonstrate that the maximally symmetric vacuum solutions of Einstein's Field Equations -- Minkowski, de-Sitter and Anti-de-Sitter spacetimes -- correspond to thermodynamic fluctuations. Moreover, we argue that these might be the only possible solutions that can be derived in this manner. Thus, the results presented here are the first concrete examples of spacetimes effectively emerging from the thermodynamic limit over an unspecified microscopic theory without any further assumptions.
Nonreciprocal light transmission in parity-time-symmetric whispering-gallery microcavities
Peng, Bo; Lei, Fuchuan; Monifi, Faraz; Gianfreda, Mariagiovanna; Long, Gui Lu; Fan, Shanhui; Nori, Franco; Bender, Carl M; Yang, Lan
2013-01-01
Optical systems combining balanced loss and gain profiles provide a unique platform to implement classical analogues of quantum systems described by non-Hermitian parity-time- (PT-) symmetric Hamiltonians and to originate new synthetic materials with novel properties. To date, experimental works on PT-symmetric optical systems have been limited to waveguides in which resonances do not play a role. Here we report the first demonstration of PT-symmetry breaking in optical resonator systems by using two directly coupled on-chip optical whispering-gallery-mode (WGM) microtoroid silica resonators. Gain in one of the resonators is provided by optically pumping Erbium (Er3+) ions embedded in the silica matrix; the other resonator exhibits passive loss. The coupling strength between the resonators is adjusted by using nanopositioning stages to tune their distance. We have observed reciprocal behavior of the PT-symmetric system in the linear regime, as well as a transition to nonreciprocity in the PT symmetry-breaking...
Energy Technology Data Exchange (ETDEWEB)
Clemens, M.; Weiland, T. [Technische Hochschule Darmstadt (Germany)
1996-12-31
In the field of computational electrodynamics the discretization of Maxwell`s equations using the Finite Integration Theory (FIT) yields very large, sparse, complex symmetric linear systems of equations. For this class of complex non-Hermitian systems a number of conjugate gradient-type algorithms is considered. The complex version of the biconjugate gradient (BiCG) method by Jacobs can be extended to a whole class of methods for complex-symmetric algorithms SCBiCG(T, n), which only require one matrix vector multiplication per iteration step. In this class the well-known conjugate orthogonal conjugate gradient (COCG) method for complex-symmetric systems corresponds to the case n = 0. The case n = 1 yields the BiCGCR method which corresponds to the conjugate residual algorithm for the real-valued case. These methods in combination with a minimal residual smoothing process are applied separately to practical 3D electro-quasistatical and eddy-current problems in electrodynamics. The practical performance of the SCBiCG methods is compared with other methods such as QMR and TFQMR.
Product numerical range in a space with tensor product structure
Puchała, Zbigniew; Miszczak, Jarosław Adam; Skowronek, Łukasz; Choi, Man-Duen; Zyczkowski, Karol \\
2010-01-01
We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.
Eigenvalue Outliers of Non-Hermitian Random Matrices with a Local Tree Structure.
Neri, Izaak; Metz, Fernando Lucas
2016-11-25
Spectra of sparse non-Hermitian random matrices determine the dynamics of complex processes on graphs. Eigenvalue outliers in the spectrum are of particular interest, since they determine the stationary state and the stability of dynamical processes. We present a general and exact theory for the eigenvalue outliers of random matrices with a local tree structure. For adjacency and Laplacian matrices of oriented random graphs, we derive analytical expressions for the eigenvalue outliers, the first moments of the distribution of eigenvector elements associated with an outlier, the support of the spectral density, and the spectral gap. We show that these spectral observables obey universal expressions, which hold for a broad class of oriented random matrices.
Non-Hermitian Hamiltonian and Lamb shift in circular dielectric microcavity
Park, Kyu-Won; Kim, Jaewan; Jeong, Kabgyun
2016-06-01
We study the normal modes and quasi-normal modes (QNMs) in circular dielectric microcavities through non-Hermitian Hamiltonian, which come from the modifications due to system-environment coupling. Differences between the two types of modes are studied in detail, including the existence of resonances tails. Numerical calculations of the eigenvalues reveal the Lamb shift in the microcavity due to its interaction with the environment. We also investigate relations between the Lamb shift and quantized angular momentum of the whispering gallery mode as well as the refractive index of the microcavity. For the latter, we make use of the similarity between the Helmholtz equation and the Schrödinger equation, in which the refractive index can be treated as a control parameter of effective potential. Our result can be generalized to other open quantum systems with a potential term.
A New Class of non-Hermitian Quantum Hamiltonians with PT Symmetry
Jones-Smith, Katherine
2009-01-01
In a remarkable development Bender and coworkers have shown that it is possible to formulate quantum mechanics consistently even if the Hamiltonian and other observables are not Hermitian. Their formulation, dubbed PT quantum mechanics, replaces hermiticity by another set of requirements, notably that the Hamiltonian should be invariant under the discrete symmetry PT, where P denotes parity and T denotes time reversal. All prior work has focused on the case that time reversal is even (T^2 = 1). We generalize the formalism to the case of odd time reversal (T^2 = -1). We discover an analogue of Kramer's theorem for PT quantum mechanics, present a prototypical example of a PT quantum system with odd time reversal, and discuss potential applications of the formalism. Odd time reversal symmetry applies to fermionic systems including quarks and leptons and a plethora of models in nuclear, atomic and condensed matter physics. PT quantum mechanics makes it possible to enlarge the set of possible Hamiltonians that phy...
Energy Technology Data Exchange (ETDEWEB)
Alvarez, Gabriel, E-mail: galvarez@fis.ucm.e [Departamento de Fisica Teorica II, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid (Spain); Martinez Alonso, Luis, E-mail: luism@fis.ucm.e [Departamento de Fisica Teorica II, Facultad de Ciencias Fisicas, Universidad Complutense, 28040 Madrid (Spain); Medina, Elena, E-mail: elena.medina@uca.e [Departamento de Matematicas, Facultad de Ciencias, Universidad de Cadiz, 11510 Puerto Real, Cadiz (Spain)
2011-07-11
We present a method to compute the genus expansion of the free energy of Hermitian matrix models from the large N expansion of the recurrence coefficients of the associated family of orthogonal polynomials. The method is based on the Bleher-Its deformation of the model, on its associated integral representation of the free energy, and on a method for solving the string equation which uses the resolvent of the Lax operator of the underlying Toda hierarchy. As a byproduct we obtain an efficient algorithm to compute generating functions for the enumeration of labeled k-maps which does not require the explicit expressions of the coefficients of the topological expansion. Finally we discuss the regularization of singular one-cut models within this approach.
Microscopic correlations for non-Hermitian Dirac operators in three-dimensional QCD
Akemann, G.
2001-12-01
In the presence of a nonvanishing chemical potential the eigenvalues of the Dirac operator become complex. We calculate spectral correlation functions of complex eigenvalues using a random matrix model approach. Our results apply to non-Hermitian Dirac operators in three-dimensional QCD with broken flavor symmetry and in four-dimensional QCD in the bulk of the spectrum. The derivation follows earlier results of Fyodorov, Khoruzhenko, and Sommers for complex spectra exploiting the existence of orthogonal polynomials in the complex plane. Explicit analytic expressions are given for all microscopic k-point correlation functions in the presence of an arbitrary even number of massive quarks, both in the limit of strong and weak non-Hermiticity. In the latter case the parameter governing the non-Hermiticity of the Dirac matrices is identified with the influence of the chemical potential.
Microscopic correlations of non-Hermitian Dirac operators in three-dimensional QCD
Akemann, G
2001-01-01
In the presence of a non-vanishing chemical potential the eigenvalues of the Dirac operator become complex. We calculate spectral correlation functions of complex eigenvalues using a random matrix model approach. Our results apply to non-Hermitian Dirac operators in three-dimensional QCD with broken flavor symmetry and in four-dimensional QCD in the bulk of the spectrum. The derivation follows earlier results of Fyodorov, Khoruzhenko and Sommers for complex spectra exploiting the existence of orthogonal polynomials in the complex plane. Explicit analytic expressions are given for all microscopic k-point correlation functions in the presence of an arbitrary even number of massive quarks, both in the limit of strong and weak non-Hermiticity. In the latter case the parameter governing the non-Hermiticity of the Dirac matrices is identified with the influence of the chemical potential.
Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation
Institute of Scientific and Technical Information of China (English)
Zhen Yun PENG; Yuan Bei DENG; Jin Wang LIU
2006-01-01
In this paper, we first consider the least-squares solution of the matrix inverse problem as follows: Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA‖AX - B‖. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is: Given an arbitrary A*, find a matrix A ∈ SE which is nearest to A* in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.
MINIMIZATION PROBLEM FOR SYMMETRIC ORTHOGONAL ANTI-SYMMETRIC MATRICES
Institute of Scientific and Technical Information of China (English)
Yuan Lei; Anping Liao; Lei Zhang
2007-01-01
By applying the generalized singular value decomposition and the canonical correlation decomposition simultaneously, we derive an analytical expression of the optimal approximate solution (X), which is both a least-squares symmetric orthogonal anti-symmetric solution of the matrix equation ATXA ＝ B and a best approximation to a given matrix X*.Moreover, a numerical algorithm for finding this optimal approximate solution is described in detail, and a numerical example is presented to show the validity of our algorithm.
Molecular response properties from a Hermitian eigenvalue equation for a time-periodic Hamiltonian.
Pawłowski, Filip; Olsen, Jeppe; Jørgensen, Poul
2015-03-21
The time-dependent Schrödinger equation for a time-periodic perturbation is recasted into a Hermitian eigenvalue equation, where the quasi-energy is an eigenvalue and the time-periodic regular wave function an eigenstate. From this Hermitian eigenvalue equation, a rigorous and transparent formulation of response function theory is developed where (i) molecular properties are defined as derivatives of the quasi-energy with respect to perturbation strengths, (ii) the quasi-energy can be determined from the time-periodic regular wave function using a variational principle or via projection, and (iii) the parametrization of the unperturbed state can differ from the parametrization of the time evolution of this state. This development brings the definition of molecular properties and their determination on par for static and time-periodic perturbations and removes inaccuracies and inconsistencies of previous response function theory formulations. The development where the parametrization of the unperturbed state and its time evolution may differ also extends the range of the wave function models for which response functions can be determined. The simplicity and universality of the presented formulation is illustrated by applying it to the configuration interaction (CI) and the coupled cluster (CC) wave function models and by introducing a new model-the coupled cluster configuration interaction (CC-CI) model-where a coupled cluster exponential parametrization is used for the unperturbed state and a linear parametrization for its time evolution. For static perturbations, the CC-CI response functions are shown to be the analytical analogues of the static molecular properties obtained from finite field equation-of-motion coupled cluster (EOMCC) energy calculations. The structural similarities and differences between the CI, CC, and CC-CI response functions are also discussed with emphasis on linear versus non-linear parametrizations and the size-extensivity of the obtained
A state-space algorithm for the spectral factorization
Kraffer, F.; Kwakernaak, H.
1997-01-01
This paper presents an algorithm for the spectral factorization of a para-Hermitian polynomial matrix. The algorithm is based on polynomial matrix to state space and vice versa conversions, and avoids elementary polynomial operations in computations; It relies on well-proven methods of numerical lin
Weierstrass representations for harmonic morphisms on Euclidean spaces and spheres
Baird, P
1995-01-01
We construct large families of harmonic morphisms which are holomorphic with respect to Hermitian structures by finding heierarchies of Weierstrass-type representations. This enables us to find new examples of complex-valued harmonic morphisms from Euclidean spaces and spheres.
Riemann-Roch Spaces and Linear Network Codes
DEFF Research Database (Denmark)
Hansen, Johan P.
2015-01-01
We construct linear network codes utilizing algebraic curves over finite fields and certain associated Riemann-Roch spaces and present methods to obtain their parameters. In particular we treat the Hermitian curve and the curves associated with the Suzuki and Ree groups all having the maximal...
Dirichlet spectra of the paradigm model of complex PT-symmetric potential: V(x) = -(ix) N
Ahmed, Zafar; Kumar, Sachin; Sharma, Dhruv
2017-08-01
So far the spectra En(N) of the paradigm model of complex PT(Parity-Time)-symmetric potential VBB(x , N) = -(ix) N is known to be analytically continued for N > 4. Consequently, the well known eigenvalues of the Hermitian cases (N = 6 , 10) cannot be recovered. Here, we illustrate Kato's theorem that even if a Hamiltonian H(λ) is an analytic function of a real parameter λ, its eigenvalues En(λ) may not be analytic at finite number of Isolated Points (IPs). In this light, we present the Dirichlet spectra En(N) of VBB(x , N) for 2 ≤ N < 12 using the numerical integration of Schrödinger equation with ψ(x = ± ∞) = 0 and the diagonalization of H =p2 / 2 μ +VBB(x , N) in the harmonic oscillator basis. We show that these real discrete spectra are consistent with the most simple two-turning point CWKB (C refers to complex turning points) method provided we choose the maximal turning points (MxTP) [ - a + ib , a + ib , a , b ∈ R] such that | a | is the largest for a given energy among all (multiple) turning points. We find that En(N) are continuous function of N but non-analytic (their first derivative is discontinuous) at IPs N = 4 , 8; where the Dirichlet spectrum is null (as VBB becomes a Hermitian flat-top potential barrier). At N = 6 and 10, VBB(x , N) becomes a Hermitian well and we recover its well known eigenvalues.
Spherically symmetric brane spacetime with bulk f(R) gravity
Energy Technology Data Exchange (ETDEWEB)
Chakraborty, Sumanta [IUCAA, Ganeshkhind, Pune University Campus, Post Bag 4, Pune (India); SenGupta, Soumitra [Indian Association for the Cultivation of Science, Department of Theoretical Physics, Kolkata (India)
2015-01-01
Introducing f(R) term in the five-dimensional bulk action we derive effective Einstein's equation on the brane using Gauss-Codazzi equation. This effective equation is then solved for different conditions on dark radiation and dark pressure to obtain various spherically symmetric solutions. Some of these static spherically symmetric solutions correspond to black hole solutions, with parameters induced from the bulk. Specially, the dark pressure and dark radiation terms (electric part of Weyl curvature) affect the brane spherically symmetric solutions significantly. We have solved for one parameter group of conformal motions where the dark radiation and dark pressure terms are exactly obtained exploiting the corresponding Lie symmetry. Various thermodynamic features of these spherically symmetric space-times are studied, showing existence of second order phase transition. This phenomenon has its origin in the higher curvature term with f(R) gravity in the bulk. (orig.)
EQUIVARIANT COHOMOLOGY AND REPRESENTATIONS OF THE SYMMETRIC GROUP
Institute of Scientific and Technical Information of China (English)
M.ATIYAH
2001-01-01
In a recent paper the author constructed a continuous map from the configuration space of n distinct ordered points in 3-space to the flag manifold of the unitary group U(n), which is compatible with the action of the symmetric group. This map is also compatible with appropriate actions of the rotation group SO(3). In this paper the author studies the induced homomorphism in SO(3)-equivariant cohomology and shows that this contains much interesting information involving representations of the symmetric group.
Positive projections of symmetric matrices and Jordan algebras
DEFF Research Database (Denmark)
Fuglede, Bent; Jensen, Søren Tolver
2013-01-01
An elementary proof is given that the projection from the space of all symmetric p×p matrices onto a linear subspace is positive if and only if the subspace is a Jordan algebra. This solves a problem in a statistical model.......An elementary proof is given that the projection from the space of all symmetric p×p matrices onto a linear subspace is positive if and only if the subspace is a Jordan algebra. This solves a problem in a statistical model....
On the Hermitian Positive Definite Solutions of Nonlinear Matrix Equation Xs+A∗X−t1A+B∗X−t2B=Q
Directory of Open Access Journals (Sweden)
Aijing Liu
2011-01-01
Full Text Available Nonlinear matrix equation Xs+A∗X−t1A+B∗X−t2B=Q has many applications in engineering; control theory; dynamic programming; ladder networks; stochastic filtering; statistics and so forth. In this paper, the Hermitian positive definite solutions of nonlinear matrix equation Xs+A∗X−t1A+B∗X−t2B=Q are considered, where Q is a Hermitian positive definite matrix, A, B are nonsingular complex matrices, s is a positive number, and 0
On the dual code of points and generators on the Hermitian variety $\\mathcal{H}(2n+1,q^2)$
De Boeck, Maarten; Vandendriessche, Peter
2016-01-01
We study the dual linear code of points and generators on a non-singular Hermitian variety $\\mathcal{H}(2n+1,q^2)$. We improve the earlier results for $n=2$, we solve the minimum distance problem for general $n$, we classify the $n$ smallest types of code words and we characterize the small weight code words as being a linear combination of these $n$ types.
Spectral singularity in confined PT symmetric optical potential
Energy Technology Data Exchange (ETDEWEB)
Sinha, Anjana [Department of Instrumentation Science, Jadavpur University, Kolkata - 700 032 (India); Roychoudhury, R. [Department of Mathematics, Bethune College, Kolkata - 700 006, India and Advanced Centre for Nonlinear and Complex Phenomena, 1175 Survey Park, Kolkata - 700075 (India)
2013-11-15
We present an analytical study for the scattering amplitudes (Reflection ‖R‖ and Transmission ‖T‖), of the periodic PT symmetric optical potential V(x)=W{sub 0}cos{sup 2}x+iV{sub 0}sin2x confined within the region 0 ⩽x⩽L, embedded in a homogeneous medium having uniform potential W{sub 0}. The confining length L is considered to be some integral multiple of the period π. We give some new and interesting results. Scattering is observed to be normal (‖T‖{sup 2}⩽ 1, ‖R‖{sup 2}⩽ 1) for V{sub 0}⩽ 0.5, when the above potential can be mapped to a Hermitian potential by a similarity transformation. Beyond this point (V{sub 0} > 0.5) scattering is found to be anomalous (‖T‖{sup 2}, ‖R‖{sup 2} not necessarily ⩽1). Additionally, in this parameter regime of V{sub 0}, one observes infinite number of spectral singularities E{sub SS} at different values of V{sub 0}. Furthermore, for L= 2nπ, the transition point V{sub 0}= 0.5 shows unidirectional invisibility with zero reflection when the beam is incident from the absorptive side (Im[V(x)] < 0) but with finite reflection when the beam is incident from the emissive side (Im[V(x)] > 0), transmission being identically unity in both cases. Finally, the scattering coefficients ‖R‖{sup 2} and ‖T‖{sup 2} always obey the generalized unitarity relation : ‖T|{sup 2}−1|=√(|R{sub R}|{sup 2}|R{sub L}|{sup 2}), where subscripts R and L stand for right and left incidence, respectively.
A symmetric scalar constraint for loop quantum gravity
Lewandowski, Jerzy
2014-01-01
In the framework of loop quantum gravity, we define a new Hilbert space of states which are solutions of a large number of components of the diffeomorphism constraint. On this Hilbert space, using the methods of Thiemann, we obtain a family of gravitational scalar constraints. They preserve the Hilbert space for every choice of lapse function. Thus adjointness and commutator properties of the constraint can be investigated in a straightforward manner. We show how the space of solutions of the symmetrized constraint can be defined by spectral decomposition, and the Hilbert space of physical states by subsequently fully implementing the diffeomorphism constraint.
Circularly symmetric light scattering from nanoplasmonic spirals.
Trevino, Jacob; Cao, Hui; Dal Negro, Luca
2011-05-11
In this paper, we combine experimental dark-field imaging, scattering, and fluorescence spectroscopy with rigorous electrodynamics calculations in order to investigate light scattering from planar arrays of Au nanoparticles arranged in aperiodic spirals with diffuse, circularly symmetric Fourier space. In particular, by studying the three main types of Vogel's spirals fabricated by electron-beam lithography on quartz substrates, we demonstrate polarization-insensitive planar light diffraction in the visible spectral range. Moreover, by combining dark-field imaging with analytical multiparticle calculations in the framework of the generalized Mie theory, we show that plasmonic spirals support distinctive structural resonances with circular symmetry carrying orbital angular momentum. The engineering of light scattering phenomena in deterministic structures with circular Fourier space provides a novel strategy for the realization of optical devices that fully leverage on enhanced, polarization-insensitive light-matter coupling over planar surfaces, such as thin-film plasmonic solar cells, plasmonic polarization devices, and optical biosensors.
Particle-vortex symmetric liquid
Mulligan, Michael
2016-01-01
We introduce an effective theory with manifest particle-vortex symmetry for disordered thin films undergoing a magnetic field-tuned superconductor-insulator transition. The theory may enable one to access both the critical properties of the strong-disorder limit, which has recently been confirmed [Breznay et al., PNAS 113, 280 (2016)] to exhibit particle-vortex symmetric electrical response, and the metallic phase discovered earlier [Mason and Kapitulnik, Phys. Rev. Lett. 82, 5341 (1999)] in less disordered samples. Within the effective theory, the Cooper-pair and field-induced vortex degrees of freedom are simultaneously incorporated into an electrically-neutral Dirac fermion minimally coupled to an (emergent) Chern-Simons gauge field. A derivation of the theory follows upon mapping the superconductor-insulator transition to the integer quantum Hall plateau transition and the subsequent use of Son's particle-hole symmetric composite Fermi liquid. Remarkably, particle-vortex symmetric response does not requir...
Symmetric autocompensating quantum key distribution
Walton, Zachary D.; Sergienko, Alexander V.; Levitin, Lev B.; Saleh, Bahaa E. A.; Teich, Malvin C.
2004-08-01
We present quantum key distribution schemes which are autocompensating (require no alignment) and symmetric (Alice and Bob receive photons from a central source) for both polarization and time-bin qubits. The primary benefit of the symmetric configuration is that both Alice and Bob may have passive setups (neither Alice nor Bob is required to make active changes for each run of the protocol). We show that both the polarization and the time-bin schemes may be implemented with existing technology. The new schemes are related to previously described schemes by the concept of advanced waves.
Ge, Li
2016-01-01
The scattering matrix $S$ obeys the unitary relation $S^\\dagger S=1$ in a Hermitian system and the symmetry property ${\\cal PT}S{\\cal PT}=S^{-1}$ in a Parity-Time (${\\cal PT}$) symmetric system. Here we report a different symmetry relation of the $S$ matrix in a one-dimensional heterostructure, which is given by the amplitude ratio of the incident waves in the scattering eigenstates. It originates from the optical reciprocity and holds independent of the Hermiticity or $\\cal PT$ symmetry of the system. Using this symmetry relation, we probe a quasi-transition that is reminiscent of the spontaneous symmetry breaking of a $\\cal PT$-symmetric $S$ matrix, now with unbalanced gain and loss and even in the absence of gain. We show that the additional symmetry relation provides a clear evidence of an exceptional point, even when all other signatures of the $\\cal PT$ symmetry breaking are completely erased. We also discuss the existence of a final exceptional point in this correspondence, which is attributed to asymm...
Observation of Bloch oscillations in complex PT-symmetric photonic lattices.
Wimmer, Martin; Miri, Mohammed-Ali; Christodoulides, Demetrios; Peschel, Ulf
2015-12-07
Light propagation in periodic environments is often associated with a number of interesting and potentially useful processes. If a crystalline optical potential is also linearly ramped, light can undergo periodic Bloch oscillations, a direct outcome of localized Wannier-Stark states and their equidistant eigenvalue spectrum. Even though these effects have been extensively explored in conservative settings, this is by no means the case in non-Hermitian photonic lattices encompassing both amplification and attenuation. Quite recently, Bloch oscillations have been predicted in parity-time-symmetric structures involving gain and loss in a balanced fashion. While in a complex bulk medium, one intuitively expects that light will typically follow the path of highest amplification, in a periodic system this behavior can be substantially altered by the underlying band structure. Here, we report the first experimental observation of Bloch oscillations in parity-time-symmetric mesh lattices. We show that these revivals exhibit unusual properties like secondary emissions and resonant restoration of PT symmetry. In addition, we present a versatile method for reconstructing the real and imaginary components of the band structure by directly monitoring the light evolution during a cycle of these oscillations.
Symmetric Periodic Solutions of the Anisotropic Manev Problem
Santoprete, Manuele
2002-01-01
We consider the Manev Potential in an anisotropic space, i.e., such that the force acts differently in each direction. Using a generalization of the Poincare' continuation method we study the existence of periodic solutions for weak anisotropy. In particular we find that the symmetric periodic orbits of the Manev system are perturbed to periodic orbits in the anisotropic problem.
Composite spherically symmetric configurations in Jordan-Brans-Dicke theory
Kozyrev, S
2010-01-01
In this article, a study of the scalar field shells in relativistic spherically symmetric configurations has been performed. We construct the composite solution of Jordan-Brans-Dicke field equation by matching the conformal Brans solutions at each junction surfaces. This approach allows us to associate rigorously with all solutions as a single glued "space", which is a unique differentiable manifold M^4.
On the generation techniques of axially symmetric stationary metrics
Indian Academy of Sciences (India)
S Chaudhuri
2002-03-01
In the present paper, a relationship between the method of Gutsunaev–Manko and the soliton technique (for two-soliton solutions) of Belinskii–Zakharov, for generating solutions of axially symmetric stationary space-times in general relativity is discussed.
Structures of generalized 3-circular projections for symmetric norms
Indian Academy of Sciences (India)
A B Abubaker; S Dutta
2016-05-01
Recently several authors investigated structures of generalized bi-circular projections in spaces where the descriptions of the group of surjective isometries are known. Following the same idea in this paper we give complete descriptions of generalized 3-circular projections for symmetric norms on ${\\mathbb C}^n$ and ${\\mathbb M}_{m \\times n}({\\mathbb C})$.
Physical unitarity in the lagrangian Sp(2)-symmetric formalism
Lavrov, P M
1996-01-01
The structure of state vector space for a general (non-anomalous) gauge theory is studied within the Lagrangian version of the Sp(2)-symmetric quantization method. The physical {\\it S}-matrix unitarity conditions are formulated. The general results are illustrated on the basis of simple gauge theory models.
Whittaker Vector of Deformed Virasoro Algebra and Macdonald Symmetric Functions
Yanagida, Shintarou
2016-03-01
We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with factored coefficients. In the proof, we fully use currents appearing in the Fock representation of Ding-Iohara-Miki quantum algebra.
Bertola, Marco; Lee, Seung-Yeop; Pierce, Virgil U
2010-01-01
Random Hermitian matrices with a source term arise, for instance, in the study of non-intersecting Brownian walkers \\cite{Adler:2009a, Daems:2007} and sample covariance matrices \\cite{Baik:2005}. We consider the case when the $n\\times n$ external source matrix has two distinct real eigenvalues: $a$ with multiplicity $r$ and zero with multiplicity $n-r$. The source is small in the sense that $r$ is finite or $r=\\mathcal O(n^\\gamma)$, for $0< \\gamma<1$. For a Gaussian potential, P\\'ech\\'e \\cite{Peche:2006} showed that for $|a|$ sufficiently small (the subcritical regime) the external source has no leading-order effect on the eigenvalues, while for $|a|$ sufficiently large (the supercritical regime) $r$ eigenvalues exit the bulk of the spectrum and behave as the eigenvalues of $r\\times r$ Gaussian unitary ensemble (GUE). We establish the universality of these results for a general class of analytic potentials in the supercritical and subcritical regimes.
A remark on band-Toeplitz preconditions for Hermitian Toeplitz systems
Energy Technology Data Exchange (ETDEWEB)
Parter, S.V.; You, J. [Univ. of Wisconsin, Madison, WI (United States)
1994-12-31
This note presents a modification of an idea of R.H. Chan and P.T.P. Tang. Let f(0) {>=} 0 be a real valued, bounded, continuous function defined on ({minus}{pi}, {pi}). Let T{sub n}[f] be the Toeplitz matrix of order n + 1 generated by f(0). Chan and Tang suggest that for given {ell} {>=} 1 one chose g{sub {ell}}(0) {>=} 0 as a real valued function of fixed degree {ell} which minimizes a particular matrix. They construct g(0) via the Remez algorithm. Then T{sub n}[g{sub {ell}}]{sup {minus}1} is used as the precondition for T{sub n}[f]. The author suggests that g(0) be chosen as the even trigonometric polynomial of minimal degree which {open_quotes}matches{close_quotes} f(0) at all points {theta}{sub j} at which f({theta}) assumes its minimum. Clearly this g({theta}) is much easier to determine that the g{sub {ell}}({theta}). This choice is based on earlier work on the extreme eigenvalues of Hermitian Toeplitz matrices and the more recent work of Manteuffel and Parter on Preconditioning finite element discretizations of elliptic operators. It is shown that the condition number of T{sub n}[g]{sup {minus}1} T{sub N}[F] is uniformly bounded for all n. Experimental results demonstrate the efficacy of these preconditioners.
Adiabatic theorem for non-hermitian time-dependent open systems
Fleischer, A; Fleischer, Avner; Moiseyev, Nimrod
2005-01-01
In the conventional quantum mechanics (i.e., hermitian QM) the adia- batic theorem for systems subjected to time periodic fields holds only for bound systems and not for open ones (where ionization and dissociation take place) [D. W. Hone, R. Ketzmerik, and W. Kohn, Phys. Rev. A 56, 4045 (1997)]. Here with the help of the (t,t') formalism combined with the complex scaling method we derive an adiabatic theorem for open systems and provide an analytical criteria for the validity of the adiabatic limit. The use of the complex scaling transformation plays a key role in our derivation. As a numerical example we apply the adiabatic theorem we derived to a 1D model Hamiltonian of Xe atom which interacts with strong, monochromatic sine-square laser pulses. We show that the gener- ation of odd-order harmonics and the absence of hyper-Raman lines, even when the pulses are extremely short, can be explained with the help of the adiabatic theorem we derived.
Nesterov, Alexander I; Bishop, Alan R
2012-01-01
We model the quantum electron transfer (ET) in the photosynthetic reaction center (RC), using a non-Hermitian Hamiltonian approach. Our model includes (i) two protein cofactors, donor and acceptor, with discrete energy levels and (ii) a third protein pigment (sink) which has a continuous energy spectrum. Interactions are introduced between the donor and acceptor, and between the acceptor and the sink, with noise acting between the donor and acceptor. The noise is considered classically (as an external random force), and it is described by an ensemble of two-level systems (random fluctuators). Each fluctuator has two independent parameters, an amplitude and a switching rate. We represent the noise by a set of fluctuators with fitting parameters (boundaries of switching rates), which allows us to build a desired spectral density of noise in a wide range of frequencies. We analyze the quantum dynamics and the efficiency of the ET as a function of (i) the energy gap between the donor and acceptor, (ii) the streng...
Axially Symmetric, Spatially Homothetic Spacetimes
Wagh, S M; Wagh, Sanjay M.; Govinder, Keshlan S.
2002-01-01
We show that the existence of appropriate spatial homothetic Killing vectors is directly related to the separability of the metric functions for axially symmetric spacetimes. The density profile for such spacetimes is (spatially) arbitrary and admits any equation of state for the matter in the spacetime. When used for studying axisymmetric gravitational collapse, such solutions do not result in a locally naked singularity.
Shearfree Spherically Symmetric Fluid Models
Sharif, M
2013-01-01
We try to find some exact analytical models of spherically symmetric spacetime of collapsing fluid under shearfree condition. We consider two types of solutions: one is to impose a condition on the mass function while the other is to restrict the pressure. We obtain totally of five exact models, and some of them satisfy the Darmois conditions.
Particle-vortex symmetric liquid
Mulligan, Michael
2017-01-01
We introduce an effective theory with manifest particle-vortex symmetry for disordered thin films undergoing a magnetic field-tuned superconductor-insulator transition. The theory may enable one to access both the critical properties of the strong-disorder limit, which has recently been confirmed by Breznay et al. [Proc. Natl. Acad. Sci. USA 113, 280 (2016), 10.1073/pnas.1522435113] to exhibit particle-vortex symmetric electrical response, and the nearby metallic phase discovered earlier by Mason and Kapitulnik [Phys. Rev. Lett. 82, 5341 (1999), 10.1103/PhysRevLett.82.5341] in less disordered samples. Within the effective theory, the Cooper-pair and field-induced vortex degrees of freedom are simultaneously incorporated into an electrically neutral Dirac fermion minimally coupled to a (emergent) Chern-Simons gauge field. A derivation of the theory follows upon mapping the superconductor-insulator transition to the integer quantum Hall plateau transition and the subsequent use of Son's particle-hole symmetric composite Fermi liquid. Remarkably, particle-vortex symmetric response does not require the introduction of disorder; rather, it results when the Dirac fermions exhibit vanishing Hall effect. The theory predicts approximately equal (diagonal) thermopower and Nernst signal with a deviation parameterized by the measured electrical Hall response at the symmetric point.
Thermophoresis of Axially Symmetric Bodies
2007-11-02
Sweden Abstract. Thermophoresis of axially symmetric bodies is investigated to first order in the Knudsen-mimber, Kn. The study is made in the limit...derived. Asymptotic solutions are studied. INTRODUCTION Thermophoresis as a phenomenon has been known for a long time, and several authors have approached
Axiomatizations of symmetrically weighted solutions
Kleppe, John; Reijnierse, Hans; Sudhölter, P.
2013-01-01
If the excesses of the coalitions in a transferable utility game are weighted, then we show that the arising weighted modifications of the well-known (pre)nucleolus and (pre)kernel satisfy the equal treatment property if and only if the weight system is symmetric in the sense that the weight of a su
Computationally Efficient Searchable Symmetric Encryption
Liesdonk, van Peter; Sedghi, Saeed; Doumen, Jeroen; Hartel, Pieter; Jonker, Willem; Jonker, Willem; Petkovic, Milan
2010-01-01
Searchable encryption is a technique that allows a client to store documents on a server in encrypted form. Stored documents can be retrieved selectively while revealing as little information as possible to the server. In the symmetric searchable encryption domain, the storage and the retrieval are
Symmetrical progressive erythro-keratoderma
Directory of Open Access Journals (Sweden)
Sunil Gupta
1999-01-01
Full Text Available A 13-year-old male child had gradually progressive, bilaterall, symmetrical, erythematous hyperkeratotic plaques over knees, elbows, natal cleft, dorsa of hands and feet with palmoplantar keratoderma. High arched palate, fissured tongue and sternal depression (pectus-excavatum were unusual associations.
DEFF Research Database (Denmark)
Gustavson, Fred G.; Quintania-Orti, Enrique S.; Quintana-Orti, Gregorio
We describe a minor format change for representing a symmetric band matrix AB using the same array space specified by LAPACK. In LAPACK, band codes operating on the lower part of a symmetric matrix reference matrix element (i, j) as AB1+i−j,j . The format change we propose allows LAPACK band code...
Representation of spectra of algebras of block-symmetric analytic functions of bounded type
Directory of Open Access Journals (Sweden)
V. V. Kravtsiv
2016-12-01
Full Text Available The paper contains a description of symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on $\\ell_{1}$-sum of the space $\\mathbb{C}^{2}.$ We show that the specrum of such algebra does not coincide of point evaluation functionals and described characters of the algebra as functions of exponential type with plane zeros.
Electroweak Phase Transitions in left-right symmetric models
Barenboim, G; Barenboim, Gabriela; Rius, Nuria
1998-01-01
We study the finite-temperature effective potential of minimal left-right symmetric models containing a bidoublet and two triplets in the scalar sector. We perform a numerical analysis of the parameter space compatible We perform a numerical analysis of the parameter space compatible with the requirement that baryon asymmetry is not washed out by sphaleron processes after the electroweak phase transition. We find that the spectrum of scalar particles for these acceptable cases is consistent with present experimental bounds.
Symmetric Euler orientation representations for orientational averaging.
Mayerhöfer, Thomas G
2005-09-01
A new kind of orientation representation called symmetric Euler orientation representation (SEOR) is presented. It is based on a combination of the conventional Euler orientation representations (Euler angles) and Hamilton's quaternions. The properties of the SEORs concerning orientational averaging are explored and compared to those of averaging schemes that are based on conventional Euler orientation representations. To that aim, the reflectance of a hypothetical polycrystalline material with orthorhombic crystal symmetry was calculated. The calculation was carried out according to the average refractive index theory (ARIT [T.G. Mayerhöfer, Appl. Spectrosc. 56 (2002) 1194]). It is shown that the use of averaging schemes based on conventional Euler orientation representations leads to a dependence of the result from the specific Euler orientation representation that was utilized and from the initial position of the crystal. The latter problem can be overcome partly by the introduction of a weighing factor, but only for two-axes-type Euler orientation representations. In case of a numerical evaluation of the average, a residual difference remains also if a two-axes type Euler orientation representation is used despite of the utilization of a weighing factor. In contrast, this problem does not occur if a symmetric Euler orientation representation is used as a matter of principle, while the result of the averaging for both types of orientation representations converges with increasing number of orientations considered in the numerical evaluation. Additionally, the use of a weighing factor and/or non-equally spaced steps in the numerical evaluation of the average is not necessary. The symmetrical Euler orientation representations are therefore ideally suited for the use in orientational averaging procedures.
Accretion processes for general spherically symmetric compact objects
Energy Technology Data Exchange (ETDEWEB)
Bahamonde, Sebastian [University College London, Department of Mathematics, London (United Kingdom); Jamil, Mubasher [National University of Sciences and Technology (NUST), H-12, Department of Mathematics, School of Natural Sciences (SNS), Islamabad (Pakistan)
2015-10-15
We investigate the accretion process for different spherically symmetric space-time geometries for a static fluid. We analyze this procedure using the most general black hole metric ansatz. After that, we examine the accretion process for specific spherically symmetric metrics obtaining the velocity of the sound during the process and the critical speed of the flow of the fluid around the black hole. In addition, we study the behavior of the rate of change of the mass for each chosen metric for a barotropic fluid. (orig.)
Accretion Processes for General Spherically Symmetric Compact Objects
Bahamonde, Sebastian
2015-01-01
We investigate the accretion process for different spherically symmetric space-time geometries for a static fluid. We analyse this procedure using the most general black hole metric ansatz. After that, we examine the accretion process for specific spherically symmetric metrics obtaining the velocity of the sound during the process and the critical speed of the flow of the fluid around the black hole. In addition, we study the behaviour of the rate of change of the mass for each chosen metric for a barotropic fluid.
Understanding symmetrical components for power system modeling
Das, J C
2017-01-01
This book utilizes symmetrical components for analyzing unbalanced three-phase electrical systems, by applying single-phase analysis tools. The author covers two approaches for studying symmetrical components; the physical approach, avoiding many mathematical matrix algebra equations, and a mathematical approach, using matrix theory. Divided into seven sections, topics include: symmetrical components using matrix methods, fundamental concepts of symmetrical components, symmetrical components –transmission lines and cables, sequence components of rotating equipment and static load, three-phase models of transformers and conductors, unsymmetrical fault calculations, and some limitations of symmetrical components.
Nam-Il, Kim; Moon-Young, Kim
2005-06-01
An improved numerical method to exactly evaluate the dynamic element stiffness matrix is proposed for the spatially coupled free vibration analysis of non-symmetric thin-walled curved beams subjected to uniform axial force. For this purpose, firstly equations of motion, boundary conditions and force-deformation relations are rigorously derived from the total potential energy for a curved beam element. Next systems of linear algebraic equations with non-symmetric matrices are constructed by introducing 14 displacement parameters and transforming the fourth-order simultaneous differential equations into the first-order simultaneous equations. And then explicit expressions for displacement parameters are numerically evaluated via eigensolutions and the exact 14×14 element stiffness matrix is determined using force-deformation relations. In order to demonstrate the validity and the accuracy of this study, the spatially coupled natural frequencies of non-symmetric thin-walled curved beams subjected to uniform compressive and tensile forces are evaluated and compared with analytical and finite element solutions using Hermitian curved beam elements or ABAQUS's shell element. In addition, some results by the parametric study are reported.
The Einstein field equations for cylindrically symmetric elastic configurations
Energy Technology Data Exchange (ETDEWEB)
Brito, I; Vaz, E G L R [Departamento de Matematica e Aplicacoes, Universidade do Minho, 4800-058 Guimaraes (Portugal); Carot, J, E-mail: ireneb@math.uminho.pt, E-mail: jcarot@uib.cat, E-mail: evaz@math.uminho.pt [Departament de Fisica, Universitat de les Illes Balears, Cra Valdemossa pk 7.5, E-07122 Palma (Spain)
2011-09-22
In the context of relativistic elasticity it is interesting to study axially symmetric space-times due to their significance in modeling neutron stars and other astrophysical systems of interest. To approach this problem, here, a particular class of these space-times is considered. A cylindrically symmetric elastic space-time configuration is studied, where the material metric is taken to be flat. The components of the energy-momentum tensor for elastic matter are written in terms of the invariants of the strain tensor, here chosen to be the eigenvalues of the pulled-back material metric. The Einstein field equations are presented and a condition confirming the existence of a constitutive function is obtained. This condition leads to special cases, in one of which a new system for the metric functions and an expression for the constitutive function are deduced. The new system depends on a particular function, which builds up the constitutive equation.
Symmetric normalisation for intuitionistic logic
DEFF Research Database (Denmark)
Guenot, Nicolas; Straßburger, Lutz
2014-01-01
, but using a non-local rewriting. The second system is the symmetric completion of the first, as normally given in deep inference for logics with a DeMorgan duality: all inference rules have duals, as cut is dual to the identity axiom. We prove a generalisation of cut elimination, that we call symmetric...... normalisation, where all rules dual to standard ones are permuted up in the derivation. The result is a decomposition theorem having cut elimination and interpolation as corollaries.......We present two proof systems for implication-only intuitionistic logic in the calculus of structures. The first is a direct adaptation of the standard sequent calculus to the deep inference setting, and we describe a procedure for cut elimination, similar to the one from the sequent calculus...
Block quasi-minimal residual iterations for non-Hermitian linear systems
Energy Technology Data Exchange (ETDEWEB)
Freund, R.W. [AT& T Bell Labs., Murray Hill, NJ (United States)
1994-12-31
Many applications require the solution of multiple linear systems that have the same coefficient matrix, but differ only in their right-hand sides. Instead of applying an iterative method to each of these systems individually, it is usually more efficient to employ a block version of the method that generates blocks of iterates for all the systems simultaneously. An example of such an iteration is the block conjugate gradient algorithm, which was first studied by Underwood and O`Leary. On parallel architectures, block versions of conjugate gradient-type methods are attractive even for the solution of single linear systems, since they have fewer synchronization points than the standard versions of these algorithms. In this talk, the author presents a block version of Freund and Nachtigal`s quasi-minimal residual (QMR) method for the iterative solution of non-Hermitian linear systems. He describes two different implementations of the block-QMR method, one based on a block version of the three-term Lanczos algorithm and one based on coupled two-term block recurrences. In both cases, the underlying block-Lanczos process still allows arbitrary normalizations of the vectors within each block, and the author discusses different normalization strategies. To maintain linear independence within each block, it is usually necessary to reduce the block size in the course of the iteration, and the author describes a deflation technique for performing this reduction. He also present some convergence results, and reports results of numerical experiments with the block-QMR method. Finally, the author discusses possible block versions of transpose-free Lanczos-based iterations such as the TFQMR method.
Symmetric two-coordinate photodiode
Directory of Open Access Journals (Sweden)
Dobrovolskiy Yu. G.
2008-12-01
Full Text Available The two-coordinate photodiode is developed and explored on the longitudinal photoeffect, which allows to get the coordinate descriptions symmetric on the steepness and longitudinal resistance great exactness. It was shown, that the best type of the coordinate description is observed in the case of scanning by the optical probe on the central part of the photosensitive element. The ways of improvement of steepness and linear of its coordinate description were analyzed.
Rotationally symmetric viscous gas flows
Weigant, W.; Plotnikov, P. I.
2017-03-01
The Dirichlet boundary value problem for the Navier-Stokes equations of a barotropic viscous compressible fluid is considered. The flow region and the data of the problem are assumed to be invariant under rotations about a fixed axis. The existence of rotationally symmetric weak solutions for all adiabatic exponents from the interval (γ*,∞) with a critical exponent γ* < 4/3 is proved.
Live-Axis Turning for the Fabrication of Non-Rotationally Symmetric Optics Project
National Aeronautics and Space Administration — The goal of this proposal is to develop a new method to create Non-Rotationally Symmetric (NRS) surfaces that overcomes the limitations of the current techniques and...
Exact multiplicity of solutions to perturbed logistic type equations on a symmetric domain
Institute of Scientific and Technical Information of China (English)
LIU Ping; SHI JunPing; WANG YuWen
2008-01-01
We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain.We obtain the precise bifurcation diagrams.
Exact multiplicity of solutions to perturbed logistic type equations on a symmetric domain
Institute of Scientific and Technical Information of China (English)
2008-01-01
We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain. We obtain the precise bifurcation diagrams.
Non-Hermitian wave packet approximation for coupled two-level systems in weak and intense fields
Puthumpally-joseph, Raiju; Charron, Eric
2016-01-01
We introduce an accurate non-Hermitian Schr\\"odinger-type approximation of Bloch optical equations for two-level systems. This approximation provides a complete description of the excitation, relaxation and decoherence dynamics in both weak and strong laser fields. In this approach, it is sufficient to propagate the wave function of the quantum system instead of the density matrix, providing that relaxation and dephasing are taken into account via automatically-adjusted time-dependent gain and decay rates. The developed formalism is applied to the problem of scattering and absorption of electromagnetic radiation by a thin layer comprised of interacting two-level emitters.
Puente, Elsa
2011-01-01
For $n \\geq 1$, the twistor space $\\mathfrak{Z}(\\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\\mathfrak{Z}(\\mathbb{S}^{2n})$ with $n \\geq 3$, in terms of what we have called {\\it generalised complex orthogonal Stiefel manifolds} of $\\mathbb{C}^{n+1}$. In particular, the twistor space $\\mathfrak{Z}(\\mathbb{S}^{6})$ is biholomorphic to a non-singular complex quadric hypersurface in $\\mathbb{P}^{7}$. We explicitly construct a real-analytic foliation, by linear 3-folds, of this quadric hypersurface such that the quotient space is isomorphic to the 6-sphere with its standard metric. This foliation is Riemannian with respect to the Fubini-Study metric and isometrically equivalent to the twistor fibration over the 6-sphere.
Symmetric products of mixed Hodge modules
Maxim, Laurentiu; Schuermann, Joerg
2010-01-01
Generalizing a theorem of Macdonald, we show a formula for the mixed Hodge structure on the cohomology of the symmetric products of bounded complexes of mixed Hodge modules by showing the existence of the canonical action of the symmetric group on the multiple external self-products of complexes of mixed Hodge modules. We also generalize a theorem of Hirzebruch and Zagier on the signature of the symmetric products of manifolds to the case of the symmetric products of symmetric parings on bounded complexes with constructible cohomology sheaves where the pairing is not assumed to be non-degenerate.
Singular Value Decomposition for Unitary Symmetric Matrix
Institute of Scientific and Technical Information of China (English)
ZOUHongxing; WANGDianjun; DAIQionghai; LIYanda
2003-01-01
A special architecture called unitary sym-metric matrix which embodies orthogonal, Givens, House-holder, permutation, and row (or column) symmetric ma-trices as its special cases, is proposed, and a precise corre-spondence of singular values and singular vectors between the unitary symmetric matrix and its mother matrix is de-rived. As an illustration of potential, it is shown that, for a class of unitary symmetric matrices, the singular value decomposition (SVD) using the mother matrix rather than the unitary symmetric matrix per se can save dramatically the CPU time and memory without loss of any numerical precision.
Discrete Torsion and Symmetric Products
Dijkgraaf, R
1999-01-01
In this note we point out that a symmetric product orbifold CFT can be twisted by a unique nontrivial two-cocycle of the permutation group. This discrete torsion changes the spins and statistics of corresponding second-quantized string theory making it essentially ``supersymmetric.'' The long strings of even length become fermionic (or ghosts), those of odd length bosonic. The partition function and elliptic genus can be described by a sum over stringy spin structures. The usual cubic interaction vertex is odd and nilpotent, so this construction gives rise to a DLCQ string theory with a leading quartic interaction.
A charged spherically symmetric solution
Indian Academy of Sciences (India)
K Moodley; S D Maharaj; K S Govinder
2003-09-01
We ﬁnd a solution of the Einstein–Maxwell system of ﬁeld equations for a class of accelerating, expanding and shearing spherically symmetric metrics. This solution depends on a particular ansatz for the line element. The radial behaviour of the solution is fully speciﬁed while the temporal behaviour is given in terms of a quadrature. By setting the charge contribution to zero we regain an (uncharged) perfect ﬂuid solution found previously with the equation of state =+ constant, which is a generalisation of a stiff equation of state. Our class of charged shearing solutions is characterised geometrically by a conformal Killing vector.
Spherically symmetric scalar field collapse
Indian Academy of Sciences (India)
Koyel Ganguly; Narayan Banerjee
2013-03-01
It is shown that a scalar field, minimally coupled to gravity, may have collapsing modes even when the energy condition is violated, that is, for ( + 3) < 0. This result may be useful in the investigation of the possible clustering of dark energy. All the examples dealt with have apparent horizons formed before the formation of singularity. The singularities formed are shell focussing in nature. The density of the scalar field distribution is seen to diverge at singularity. The Ricci scalar also diverges at the singularity. The interior spherically symmetric metric is matched with exterior Vaidya metric at the hypersurface and the appropriate junction conditions are obtained.
Tucholska, Aleksandra M; Lesiuk, Michał; Moszynski, Robert
2017-01-21
We introduce a new method for the computation of the transition moments between the excited electronic states based on the expectation value formalism of the coupled cluster theory [B. Jeziorski and R. Moszynski, Int. J. Quantum Chem. 48, 161 (1993)]. The working expressions of the new method solely employ the coupled cluster operator T and an auxiliary operator S that is expressed as a finite commutator expansion in terms of T and T(†). In the approximation adopted in the present paper, the cluster expansion is limited to single, double, and linear triple excitations. The computed dipole transition probabilities for the singlet-singlet and triplet-triplet transitions in alkali earth atoms agree well with the available theoretical and experimental data. In contrast to the existing coupled cluster response theory, the matrix elements obtained by using our approach satisfy the Hermitian symmetry even if the excitations in the cluster operator are truncated, but the operator S is exact. The Hermitian symmetry is slightly broken if the commutator series for the operator S are truncated. As a part of the numerical evidence for the new method, we report calculations of the transition moments between the excited triplet states which have not yet been reported in the literature within the coupled cluster theory. Slater-type basis sets constructed according to the correlation-consistency principle are used in our calculations.
Tucholska, Aleksandra M.; Lesiuk, Michał; Moszynski, Robert
2017-01-01
We introduce a new method for the computation of the transition moments between the excited electronic states based on the expectation value formalism of the coupled cluster theory [B. Jeziorski and R. Moszynski, Int. J. Quantum Chem. 48, 161 (1993)]. The working expressions of the new method solely employ the coupled cluster operator T and an auxiliary operator S that is expressed as a finite commutator expansion in terms of T and T†. In the approximation adopted in the present paper, the cluster expansion is limited to single, double, and linear triple excitations. The computed dipole transition probabilities for the singlet-singlet and triplet-triplet transitions in alkali earth atoms agree well with the available theoretical and experimental data. In contrast to the existing coupled cluster response theory, the matrix elements obtained by using our approach satisfy the Hermitian symmetry even if the excitations in the cluster operator are truncated, but the operator S is exact. The Hermitian symmetry is slightly broken if the commutator series for the operator S are truncated. As a part of the numerical evidence for the new method, we report calculations of the transition moments between the excited triplet states which have not yet been reported in the literature within the coupled cluster theory. Slater-type basis sets constructed according to the correlation-consistency principle are used in our calculations.
Directory of Open Access Journals (Sweden)
Maziar Nekovee
2010-01-01
Full Text Available Cognitive radio is being intensively researched as the enabling technology for license-exempt access to the so-called TV White Spaces (TVWS, large portions of spectrum in the UHF/VHF bands which become available on a geographical basis after digital switchover. Both in the US, and more recently, in the UK the regulators have given conditional endorsement to this new mode of access. This paper reviews the state-of-the-art in technology, regulation, and standardisation of cognitive access to TVWS. It examines the spectrum opportunity and commercial use cases associated with this form of secondary access.
Mazziotti, David A
2007-05-14
Two-electron reduced density matrices (2-RDMs) have recently been directly determined from the solution of the anti-Hermitian contracted Schrodinger equation (ACSE) to obtain 95%-100% of the ground-state correlation energy of atoms and molecules, which significantly improves upon the accuracy of the contracted Schrodinger equation (CSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. Two subsets of the CSE, the ACSE and the contraction of the CSE onto the one-particle space, known as the 1,3-CSE, have two important properties: (i) dependence upon only the 3-RDM and (ii) inclusion of all second-order terms when the 3-RDM is reconstructed as only a first-order functional of the 2-RDM. The error in the 1,3-CSE has an important role as a stopping criterion in solving the ACSE for the 2-RDM. Using a computationally more efficient implementation of the ACSE, the author treats a variety of molecules, including H2O, NH3, HCN, and HO3-, in larger basis sets such as correlation-consistent polarized double- and triple-zeta. The ground-state energy of neon is also calculated in a polarized quadruple-zeta basis set with extrapolation to the complete basis-set limit, and the equilibrium bond length and harmonic frequency of N2 are computed with comparison to experimental values. The author observes that increasing the basis set enhances the ability of the ACSE to capture correlation effects in ground-state energies and properties. In the triple-zeta basis set, for example, the ACSE yields energies and properties that are closer in accuracy to coupled cluster with single, double, and triple excitations than to coupled cluster with single and double excitations. In all basis sets, the computed 2-RDMs very closely satisfy known N-representability conditions.
Duality symmetric string and M-theory
Berman, David S.; Thompson, Daniel C.
2015-03-01
We review recent developments in duality symmetric string theory. We begin with the world-sheet doubled formalism which describes strings in an extended spacetime with extra coordinates conjugate to winding modes. This formalism is T-duality symmetric and can accommodate non-geometric T-fold backgrounds which are beyond the scope of Riemannian geometry. Vanishing of the conformal anomaly of this theory can be interpreted as a set of spacetime equations for the background fields. These equations follow from an action principle that has been dubbed Double Field Theory (DFT). We review the aspects of generalised geometry relevant for DFT. We outline recent extensions of DFT and explain how, by relaxing the so-called strong constraint with a Scherk-Schwarz ansatz, one can obtain backgrounds that simultaneously depend on both the regular and T-dual coordinates. This provides a purely geometric higher dimensional origin to gauged supergravities that arise from non-geometric compactification. We then turn to M-theory and describe recent progress in formulating an En(n) U-duality covariant description of the dynamics. We describe how spacetime may be extended to accommodate coordinates conjugate to brane wrapping modes and the construction of generalised metrics in this extended space that unite the bosonic fields of supergravity into a single object. We review the action principles for these theories and their novel gauge symmetries. We also describe how a Scherk-Schwarz reduction can be applied in the M-theory context and the resulting relationship to the embedding tensor formulation of maximal gauged supergravities.
Energy Technology Data Exchange (ETDEWEB)
Sand, Andrew M.; Mazziotti, David A., E-mail: damazz@uchicago.edu [Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637 (United States)
2015-10-07
Determination of the two-electron reduced density matrix (2-RDM) from the solution of the anti-Hermitian contracted Schrödinger equation (ACSE) yields accurate energies and properties for both ground and excited states. Here, we develop a more efficient method to solving the ACSE that uses second-order information to select a more optimal step towards the solution. Calculations on the ground and excited states of water, hydrogen fluoride, and conjugated π systems show that the improved ACSE algorithm is 10-20 times faster than the previous ACSE algorithm. The ACSE can treat both single- and multi-reference electron correlation with the initial 2-RDM from a complete-active-space self-consistent-field (CASSCF) calculation. Using the improved algorithm, we explore the relationship between truncation of the active space in the CASSCF calculation and the accuracy of the energy and 2-RDM from the ACSE calculation. The accuracy of the ACSE, we find, is less sensitive to the size of the active space than the accuracy of other wavefunction methods, which is useful when large active space calculations are computationally infeasible.
Sand, Andrew M; Mazziotti, David A
2015-10-01
Determination of the two-electron reduced density matrix (2-RDM) from the solution of the anti-Hermitian contracted Schrödinger equation (ACSE) yields accurate energies and properties for both ground and excited states. Here, we develop a more efficient method to solving the ACSE that uses second-order information to select a more optimal step towards the solution. Calculations on the ground and excited states of water, hydrogen fluoride, and conjugated π systems show that the improved ACSE algorithm is 10-20 times faster than the previous ACSE algorithm. The ACSE can treat both single- and multi-reference electron correlation with the initial 2-RDM from a complete-active-space self-consistent-field (CASSCF) calculation. Using the improved algorithm, we explore the relationship between truncation of the active space in the CASSCF calculation and the accuracy of the energy and 2-RDM from the ACSE calculation. The accuracy of the ACSE, we find, is less sensitive to the size of the active space than the accuracy of other wavefunction methods, which is useful when large active space calculations are computationally infeasible.
DERIVATIVES OF EIGENPAIRS OF SYMMETRIC QUADRATIC EIGENVALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system.
Whittaker vector of deformed Virasoro algebra and Macdonald symmetric functions
Yanagida, Shintarou
2014-01-01
We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with factored coefficients. In the proof we fully use currents appearing in the Fock representation of Ding-Iohara-Miki quantum algebra. We also mention an interpretation of Whittaker vector in terms of the geometry of the Hilbert schemes of points on the affine plane.
Schwarz Methods: To Symmetrize or Not to Symmetrize
Holst, Michael
2010-01-01
A preconditioning theory is presented which establishes sufficient conditions for multiplicative and additive Schwarz algorithms to yield self-adjoint positive definite preconditioners. It allows for the analysis and use of non-variational and non-convergent linear methods as preconditioners for conjugate gradient methods, and it is applied to domain decomposition and multigrid. It is illustrated why symmetrizing may be a bad idea for linear methods. It is conjectured that enforcing minimal symmetry achieves the best results when combined with conjugate gradient acceleration. Also, it is shown that absence of symmetry in the linear preconditioner is advantageous when the linear method is accelerated by using the Bi-CGstab method. Numerical examples are presented for two test problems which illustrate the theory and conjectures.
域上2×2对称矩阵空间的加法秩保持%Additive preservers of rank on the spaces of 2 × 2 symmetric matrices over fields
Institute of Scientific and Technical Information of China (English)
张显
2004-01-01
Let F be a field and n be a positive integer. Denote the set of all n × n symmetric matrices over F by Sn(F).An operator f: Sn(F) → Sn(F) is said to be additive if f(A + B) = f(A) + f(B) for any A, B ∈ Sn(F), and is called a preserver of rank on Sn(F) if rankf(X) =rankX for every X ∈ Sn(F). When n ≥ 3, for any F the forms of the additive preservers of rank on Sn(F) have been determined by the author in [4]. Here, when F is arbitrary, the general form of all additive operators from S2(F) to itself satisfying rankf(X) =rankX for every X ∈ S2(F) \\ {xD12| x ∈ F \\ {0}} is determined. Thereby, the additive preservers of rank on S2(F) for any F are characterized.%令F是一个域,n是一个正整数.Sn(F)记F上所有n×n对称矩阵的集合.若一个算子f:Sn(F)→Sn(F)满足对任意的A,B∈Sn(F)都有f(A+B)=f(A)+f(B),则称之为加法的;若对任意的X∈Sn(F)都有rankf(X)=rankX,则称f为Sn(F)上的秩保持.当n≥3及F为任意域时,Sn(F)上的所有加法秩保持已被作者在[4]中确定.这里,对于任意的F,S2(F)上所有的满足对每个X∈S2(F)\\{xD12|x∈F\\{0}}都有rankf(X)=rankX的加法算子的一般形式被确定,由此S2(F)上的所有加法秩保持被刻划.
Duality, Phase Structures and Dilemmas in Symmetric Quantum Games
Ichikawa, T; Ichikawa, Tsubasa; Tsutsui, Izumi
2006-01-01
Symmetric quantum games for 2-player, 2-qubit strategies are analyzed in detail by using a scheme in which all pure states in the 2-qubit Hilbert space are utilized for strategies. We consider two different types of symmetric games exemplified by the familiar games, the Battle of the Sexes (BoS) and the Prisoners' Dilemma (PD). These two types of symmetric games are shown to be related by a duality map, which ensures that they share common phase structures with respect to the equilibria of the strategies. We find eight distinct phase structures possible for the symmetric games, which are determined by the classical payoff matrices from which the quantum games are defined. We also discuss the possibility of resolving the dilemmas in the classical BoS, PD and the Stag Hunt (SH) game based on the phase structures obtained in the quantum games. It is observed that quantization cannot resolve the dilemma fully for the BoS, while it generically can for the PD and SH if appropriate correlations for the strategies of...
Modulational instability in a PT-symmetric vector nonlinear Schrödinger system
Cole, J. T.; Makris, K. G.; Musslimani, Z. H.; Christodoulides, D. N.; Rotter, S.
2016-12-01
A class of exact multi-component constant intensity solutions to a vector nonlinear Schrödinger (NLS) system in the presence of an external PT-symmetric complex potential is constructed. This type of uniform wave pattern displays a non-trivial phase whose spatial dependence is induced by the lattice structure. In this regard, light can propagate without scattering while retaining its original form despite the presence of inhomogeneous gain and loss. These constant-intensity continuous waves are then used to perform a modulational instability analysis in the presence of both non-hermitian media and cubic nonlinearity. A linear stability eigenvalue problem is formulated that governs the dynamical evolution of the periodic perturbation and its spectrum is numerically determined using Fourier-Floquet-Bloch theory. In the self-focusing case, we identify an intensity threshold above which the constant-intensity modes are modulationally unstable for any Floquet-Bloch momentum belonging to the first Brillouin zone. The picture in the self-defocusing case is different. Contrary to the bulk vector case, where instability develops only when the waves are strongly coupled, here an instability occurs in the strong and weak coupling regimes. The linear stability results are supplemented with direct (nonlinear) numerical simulations.
Poisson kernel and Cauchy formula of a non-symmetric transitive domain
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In 1965, Lu Yu-Qian discovered that the Poisson kernel of the homogenous domain S m,p,q={Z∈Cm×m, Z1∈Cm×p,Z2 ∈Cq×m|2i1( Z-Z+)-Z1Z1′-Z2′Z2>0} does not satisfy the Laplace-Beltrami equation associated with the Bergman metric when S m,p,q is not symmetric. However the map T0:Z→Z, Z1→Z1 , Z2→Z2 transforms S m,p,q into a domain S I (m, m + p + q) which can be mapped by the Cayley transformation into the classical domains R I (m, m + p + q). The pull back of the Bergman metric of R I (m, m + p + q) to S m,p,q is a Riemann metric ds 2 which is not a Khler metric and even not a Hermitian metric in general. It is proved that the Laplace-Beltrami operator associated with the metric ds 2 when it acts on the Poisson kernel of S m,p,q equals 0. Consequently, the Cauchy formula of S m,p,q can be obtained from the Poisson formula.
Fine Spectra of Symmetric Toeplitz Operators
Directory of Open Access Journals (Sweden)
Muhammed Altun
2012-01-01
Full Text Available The fine spectra of 2-banded and 3-banded infinite Toeplitz matrices were examined by several authors. The fine spectra of n-banded triangular Toeplitz matrices and tridiagonal symmetric matrices were computed in the following papers: Altun, “On the fine spectra of triangular toeplitz operators” (2011 and Altun, “Fine spectra of tridiagonal symmetric matrices” (2011. Here, we generalize those results to the (2+1-banded symmetric Toeplitz matrix operators for arbitrary positive integer .
Half-spectral unidirectional invisibility in non-Hermitian periodic optical structures
Longhi, Stefano
2015-01-01
The phenomenon of half-spectral unidirectional invisibility is introduced for one-dimensional periodic optical structures with tailored real and imaginary refractive index distributions in a non-$\\mathcal{PT}$-symmetric configuration. The effect refers to the property that the optical medium appears to be invisible, both in reflection and transmission, below the Bragg frequency when probed from one side, and above the Bragg frequency when probed from the opposite side. Half-spectral invisibility is obtained by a combination of in-phase index and gain gratings whose spatial amplitudes are related each other by a Hilbert transform.