Volume 44, Issue 4, April 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


An investigation of entanglement and quasiprobability distribution in a generalized JaynesCummings model
View Description Hide DescriptionIn this paper, we consider a unified approach to study the degree of entanglement of twolevel systems interacting with a quantized electromagnetic field. We investigate a range of parameters in a generalized Jaynes–Cummings model (JCmodel) with intensitydependent, field nonlinearity and nonresonant coupling. In terms of the density matrix and without the diagonal approximation we derive an explicit expression for the entanglement degree using a function analogous to mutual entropy. This notion is inspired by the fact that the quantum state may be interpreted as a measure of information. With the aid of the quasiprobability distribution function, the statistical properties of the field are analyzed. It is shown that when the atom is initially in its upperlevel and the onephoton at resonance, the function splits into two peaks and counter rotate in phase space.

Wigner functions for curved spaces. II. On spheres
View Description Hide DescriptionThe form of the Wigner distribution function for Hamiltonian systems in spaces of constant negative curvature (i.e., hyperboloids) proposed in M. A. Alonso, G. S. Pogosyan, and K. B. Wolf, “Wigner functions for curved spaces. I. On hyperboloids” [J. Math. Phys. 43, 5857 (2002)], is extended here to spaces whose curvature is constant and positive, i.e., spheres. An essential part of this construction is the use of the functions of Sherman and Volobuyev, which are an overcomplete set of planewavelike solutions of the Laplace–Beltrami equation for this space. Rotations that displace the poles transform these functions with a multiplier factor, and their momentum direction becomes formally complex; the covariance properties of the proposed Wigner function are understood in these terms. As an example for the onedimensional case, we consider the energy eigenstates of the oscillator on the circle in a Pöschl–Teller potential. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.

On the boson spectrum of the N particle Schrödinger operator with periodic binary potential (dense matter)
View Description Hide DescriptionA new asymptotic method is proposed for calculation of spectral series and corresponding asymptotic eigenfunctions for the spinless particle Schrödinger equation when tends to infinitum but the physical parameters and the volume are fixed (dense matter case).

Duality versus dual flatness in quantum information geometry
View Description Hide DescriptionWe investigate questions in quantum informationgeometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finitedimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove, moreover, that there is never a dually flat structure on the set of pure states. Thus any general theory of quantum informationgeometry that involves duality concepts must inevitably be based on dual structures which are nonflat.

Multiresolution analysis generated by a seed function
View Description Hide DescriptionIn this paper we use the equivalence result originally proved by the author, which relates a multiresolution analysis (MRA) of and an orthonormal set of single electron wave functions in the lowest Landau level, to build up a procedure which produces, starting with a certain squareintegrable function, a MRA of

Continuum quantum systems as limits of discrete quantum systems. IV. Affine canonical transforms
View Description Hide DescriptionAffine canonical transforms, complexorder Fourier transforms, and their associated coherent states appear in two scenarios: finitediscrete and continuum. We examine the relationship between the two scenarios, making systematic use of inductive limits, which were developed in the preceding articles in this series.

Upper and lower limits for the number of Swave bound states in an attractive potential
View Description Hide DescriptionNew upper and lower limits are given for the number of Swave bound states yielded by an attractive (monotonic) potential in the context of the Schrödinger or Klein–Gordon equation.

Multiple algebraisations of an elliptic Calogero–Sutherland model
View Description Hide DescriptionRecently, GómezUllate et al. [Phys. Lett. B 511, 112 (2001)] have studied an particle quantum problem with ellipticfunction potentials. They have shown that the Hamiltonian operator preserves a finite dimensional space of functions and as such is quasiexactly solvable (QES). In this article we show that other types of invariant function spaces exist, which are in close relation to the algebraic properties of the elliptic functions. Accordingly, series of new algebraic eigenfunctions can be constructed.

Positive measure spectrum for Schrödinger operators with periodic magnetic fields
View Description Hide DescriptionWe study Schrödinger operators with periodic magnetic field in in the case of irrational magnetic flux. Positive measure Cantor spectrum is generically expected in the presence of an electric potential. We show that, even without electric potential, the spectrum has positive measure if the magnetic field is a perturbation of a constant one.

Wave function confinement via transfer matrix methods
View Description Hide DescriptionThe exact transfer matrix approach used in studying sectionally constant potentials in one dimension is generalized to cylindrical and spherical geometries, where the potential depends only on radius. In each geometry two transfer matrices suffice to completely describe the wave function: one for handling a discontinuity in potential and one for handling a deltafunction potential barrier. This method is then applied to the problem of confining a wave function in a cylindrical configuration using only a series of carefully placed delta function potential barriers. It is found that confinement can be made to increase nearly exponentially with the number of barriers if placed correctly, but that this arrangement has an exponentially sharp dependence on both barrier position and energy.

Boundary behavior of quantum Green’s functions
View Description Hide DescriptionWe consider the timeindependent Green’s function for the Schrödinger operator with a oneparticle potential, defined in a dimensional domain. Recently, in one dimension (1D), the Green’s function problem was solved explicitly in inverse form, with diagonal elements of the Green’s function as prescribed variables. In this article, the 1D inverse solution is used to derive leading behavior of the Green’s function close to the domain boundary. The emphasis is put onto “universal” expansion terms which are dominated by the boundary and do not depend on the particular shape of the applied potential. The inverse formalism is extended to higher dimensions, especially to 3D, and subsequently the boundary form of the Green’s function is predicted for an arbitrarily shaped domain boundary.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Renormalization of Coulomb interactions for the 1D Dirac equation
View Description Hide DescriptionThe Coulomb interaction for the Dirac equation in one space dimension is singular in the sense that there exists a fourparameter family of selfadjoint extensions of the associated Hamiltonian operator. The purpose of this paper is to represent the dynamical group generated by some of the selfadjoint extensions as a path integral. The Feynman–Kac functional we use is constructed by a renormalization process that subtracts divergences as paths cross the isolated singularity of the interaction.

General relations between radial integrals in nonrelativistic and relativistic calculation schemes
View Description Hide DescriptionUsing the equivalent relativistic operator and the correspondence of its terms to the operators in the Breit–Pauli approximation the relativistic analogs for the integrals of Coulomb, spin–contact, spin–orbit, spin–spin and other interactions are obtained. They give the possibility to take into account not only direct but also indirect relativistic effects by performing the calculations of atomic structure with existing general programs in a nonrelativistic scheme with relativistic Breit–Pauli corrections.
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 GENERAL RELATIVITY AND GRAVITATION


Moment problems and the causal set approach to quantum gravity
View Description Hide DescriptionWe study a collection of discrete Markov chains related to the causal set approach to modeling discrete theories of quantum gravity. The transition probabilities of these chains satisfy a general covariance principle, a causality principle, and a renormalizability condition. The corresponding dynamics are completely determined by a sequence of nonnegative real coupling constants. Using techniques related to the classical moment problem, we give a complete description of any such sequence of coupling constants. We prove a representation theorem: every discrete theory of quantum gravity arising from causal set dynamics satisfying covariance, causality, and renormalizability corresponds to a unique probability distributionfunction on the nonnegative real numbers, with the coupling constants defining the theory given by the moments of the distribution.

A new approach of the stationary axisymmetric vacuum S(A) solutions
View Description Hide DescriptionWe revisit axisymmetric stationary vacuum solutions of the Einstein equations, like we did for the cylindrical case [J. Math. Phys. 41, 7535 (2000)]. We explicitly formulate the simplest hypothesis under which the S(A) solutions, or axisymmetric Lewis solutions can be found and demonstrate that this hypothesis leads to a linear relation between the potentials. We show that the field equations still can be associated to the motion of a classical particle in a central field, where an arbitrary harmonic χ function plays the role of time. Three classes of solutions are obtained without the need of invoking the Papapetrou class. They depend on two real parameters, and the potentials are functions of χ only. The new approach exempts the need of complex parameters. We interpret one of the parameters as related to the vorticity of the source.
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 DYNAMICAL SYSTEMS


Total variation in Hamiltonian formalism and symplecticenergy integrators
View Description Hide DescriptionWe present a discrete total variation calculus in Hamiltonian formalism in this paper. Using this discrete variation calculus and generating functions for the flows of Hamiltonian systems, we derive symplecticenergy integrators of any finite order for Hamiltonian systems from a variational perspective. The relationship between the symplectic integrators derived directly from the Hamiltonian systems and the variationally derived symplecticenergy integrators is explored.

Integrability of Kersten–Krasil’shchik coupled KdV–mKdV equations: singularity analysis and Lax pair
View Description Hide DescriptionThe integrability of a coupled KdV–mKdV system is tested by means of singularity analysis. The true Lax pair associated with this system is obtained by the use of prolongation technique.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Legendre transformation and analytical mechanics: A geometric approach
View Description Hide DescriptionA revisitation of the Legendre transformation in the context of affine principal bundles is presented. The argument, merged with the gaugetheoretical considerations developed by Massa et al., provides a unified representation of Lagrangian and Hamiltonian mechanics, extending to arbitrary nonautonomous systems the symplectic approach of Tulczyjew.
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 METHODS OF MATHEMATICAL PHYSICS


Asymptotic expansion of the quasiconfluent hypergeometric function
View Description Hide DescriptionThe asymptotic expansion of the hypergeometric function in the case of quasiconfluence, i.e., for is revised. A very simple expansion, in terms of a semiasymptotic sequence of polynomials, is presented. Some properties of those polynomials are discussed.

Noncommutative phase and unitarization of
View Description Hide DescriptionIn this article, imposing Hermitian conjugate relations on the twoparameter deformed quantum group is studied. This results in a noncommutative phase associated with the unitarization of the quantum group. After the achievement of the quantum group with real via a noncommutative phase, the representation of the algebra is built by means of the action of the operators constituting the matrix on states.
