Index of content:
Volume 45, Issue 7, July 2004
 METHODS OF MATHEMATICAL PHYSICS


3geometries and the Hamilton–Jacobi equation
View Description Hide DescriptionIn the first part of this work we show that on the space of solutions of a certain class of systems of three secondorder PDE’s, and a threedimensional definite or indefinite metric, can be constructed such that the threedimensional Hamilton–Jacobi equation, holds. Furthermore, we remark that this structure is invariant under a subset of contact transformations. In the second part, we obtain analogous results for a certain class of thirdorder ordinary differential equation (ODE’s), In both cases, we apply our general results to the cental force problem.

Distributive and analytic properties of lattice sums
View Description Hide DescriptionWe use sums over Bessel functions of the first kind to derive a convenient form of the Poisson summation identity relating sums over direct lattices in two dimensions to sums over reciprocal lattices. After three simple examples of the use of the identity, we consider sums over complex powers of the radial distance to lattice points, and also sums incorporating factors depending on angles of lattice points. We study the distribution of zeros of lattice sums, and show two which seemingly obey the Riemann hypothesis, and a third which does not. We provide a reflection formula for angular lattice sums, and a Macdonald function sum for the lowest order angular lattice sum.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Existence of the Bogoliubov operator for the quantum field theory
View Description Hide DescriptionWe prove the existence of the Bogoliubov operator for the quantum field theory for coupling functions of compact support in space and time. The construction is nonperturbative and relies on a theorem of Kisyński. It implies almost automatically the properties of unitarity and causality for disjoint supports in the time variable.

 METHODS OF MATHEMATICAL PHYSICS


Killing initial data revisited
View Description Hide DescriptionConsidering the Killing vector fields on a Lorentzian manifold in terms of their lapse and shift along a spacelike hypersurface, we give a new definition of the Killing Initial Data (KID) of Beig and Chruściel [Class. Quantum Grav. 141A, A83 (1996)]. We define, on these new KID’s, the bracket operation induced by the usual Lie bracket of Killing vector fields on the Lorentzian manifold, and study the conditions for our new KID’s to form a Lie algebra. The interesting fact is that these conditions only depend on data along the hypersurface.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On quantum errorcorrection by classical feedback in discrete time
View Description Hide DescriptionWe consider the problem of correcting the errors incurred from sending quantum information through a noisy quantum environment by using classical information obtained from a measurement on the environment. For discrete time Markovian evolutions, in the case of fixed measurement on the environment, we give criteria for quantum information to be perfectly corrigible and characterize the related feedback. Then we analyze the case when perfect correction is not possible and, in the qubit case, we find optimal feedback maximizing the channel fidelity.

Inverse scattering problem in nuclear physics—Optical model
View Description Hide DescriptionWe consider the inverse scattering problem for the Schrödinger operator with optical potential introduced in nuclear physics to study the scattering of nucleons by nuclei. We show that the corresponding spin–orbit interaction and the complex matrix potential can be uniquely reconstructed from the scattering amplitude at fixed energy.

 FLUIDS


On an integrable hierarchy derived from the isentropic gas dynamics
View Description Hide DescriptionIn this paper we study a new hierarchy of equations derived from the system of isentropic gas dynamics equations where the pressure is a nonlocal function of the density. We show that the hierarchy of equations is integrable. We construct the two compatible Hamiltonian structures and show that the first structure has three distinct Casimirs while the second has one. The existence of Casimirs allows us to extend the flows to local ones. We construct an infinite series of commuting local Hamiltonians as well as three infinite series (related to the three Casimirs) of nonlocal charges. We discuss the zero curvature formulation of the system where we obtain a simple expression for the nonlocal conserved charges, which also clarifies the existence of the three series from a Lie algebraic point of view. We point out that the nonlocal hierarchy of Hunter–Zheng equations can be obtained from our nonlocal flows when the dynamical variables are properly constrained.

 DYNAMICAL SYSTEMS


Deformed Harry Dym and Hunter–Zheng equations
View Description Hide DescriptionWe study the deformed Harry Dym and Hunter–Zheng equations with two arbitrary deformation parameters. These reduce to various other known models in appropriate limits. We show that both these systems are biHamiltonian with the same Hamiltonian structures. They are integrable and belong to the same hierarchy corresponding to positive and negative flows. We present the Lax pair description for both the systems and construct the conserved charges of negative order from the Lax operator. For the deformed Harry Dym equation, we construct the nonstandard Lax representation for two special classes of values of the deformation parameters. In general, we argue that a nonstandard description will involve a pseudodifferential operator of infinite order.

 METHODS OF MATHEMATICAL PHYSICS


Expansion around halfinteger values, binomial sums, and inverse binomial sums
View Description Hide DescriptionI consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around halfinteger values. I present algorithms which are suitable for an implementation within a symbolic computer algebra system. The method is an extension of the technique of nested sums. The algorithms allow in addition the evaluation of binomial sums, inverse binomial sums and generalizations thereof.

The use of so(2,1) algebra for the evaluation of atomic integrals: The study of twoelectron atoms
View Description Hide DescriptionThe use of the so(2,1) algebra for the study of the twoelectron atoms is suggested. The radial part of the twoelectron function is expanded into the products of the oneelectron functions. These oneelectron functions form complete, entirely discrete set and are identified as the eigenfunctions of one of the generators of the so(2,1) algebra. By applying this algebra we are able to express all the matrix elements in analytic and numericaly stable form. For matrix elements of the twoelectron interaction this is done in three steps, all of them completely novel from the methodological point of view. First, repulsion integrals over four radial functions are written as a linear combination of the integrals over two radial functions and the coefficients of the linear combination are given in terms of hypergeometric functions. Second, combining algebraic technique with the integration by parts we derive recurrence relations for the repulsion integrals over two radial functions. Third, the derived recurrence relations are solved analytically in terms of the hypergeometric functions. Thus we succeed in expressing the repulsion integrals as rational functions of the hypergeometric functions. In this way we resolve the problem of the numerical stability of calculation of the repulsion integrals. Finally, as an illustration, the configuration interaction calculation of the lowest lying states of the He atom is discussed.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Temporally stable coherent states for a free magnetic Schrödinger operator
View Description Hide DescriptionEigenfunctions and eigenvalues of the free magnetic Schrödinger operator, describing a spinless particle confined to an infinite layer of fixed width, are discussed in detail. The eigenfunctions are realized as an orthonormal basis of a suitable Hilbert space. Four different classes of temporally stable coherent states associated with the operator are presented. The first two classes are derived as coherent states with one degree of freedom and the last two classes are derived with two degrees of freedom. The dynamical algebra of each class is found. Statistical quantities associated to each class of coherent states are calculated explicitly.

 FLUIDS


On limits to convective heat transport at infinite Prandtl number with or without rotation
View Description Hide DescriptionWe prove rigorous upper bounds for the bulk heat transport in infinite Prandtl number Rayleigh–Bénard convection with or without rotation. For the rotation free case, our estimate shows that the Nusselt number is bounded by Rayleigh number according to with constant In the presence of rotation, we prove with constant Moreover, for weak rotating constraint the Nusselt number is uniformly bounded above by

 GENERAL RELATIVITY AND GRAVITATION


A diffusion process in curved space–time
View Description Hide DescriptionWe construct a curved space–time generalization of the special relativistic Ornstein–Uhlenbeck Process. This is done by deriving a manifestly covariant Kolmogorov equation that describes diffusion in curved space–times. The simple case of diffusion in a spatially flat Friedmann–Robertson–Walker universe is then considered. It is proven that, at least in these space–times, Kolmogorov equation admits as possible solution a natural generalization of the flat space–time Jüttner equilibrium solution. The first correction to Jüttner’s distribution in a slowly expanding universe is also obtained explicitly.

 METHODS OF MATHEMATICAL PHYSICS


Spherical harmonics and basic coupling coefficients for the group SO(5) in an SO(3) basis
View Description Hide DescriptionAn easily programmable algorithm is given for the computation of SO(5) spherical harmonics needed to complement the radial (beta) wave functions to form an orthonormal basis of wave functions for the fivedimensional harmonic oscillator. It is shown how these functions can be used to compute the (Clebsch–Gordan a.k.a. Wigner) coupling coefficients for combining pairs of irreps in this space to other irreps. This is of particular value for the construction of the matrices of Hamiltonians and transition operators that arise in applications of nuclear collective models. Tables of the most useful coupling coefficients are given in the Appendix.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


A generalization of Chetaev’s principle for a class of higher order nonholonomic constraints
View Description Hide DescriptionThe constraint distribution in nonholonomic mechanics has a double role. On the one hand, it is a kinematic constraint, that is, it is a restriction on the motion itself. On the other hand, it is also a restriction on the allowed variations when using D’Alembert’s principle to derive the equations of motion. We will show that many systems of physical interest where D’Alembert’s principle does not apply can be conveniently modeled within the general idea of the principle of virtual work by the introduction of both kinematic constraints and variational constraints as being independent entities. This includes, for example, elastic rolling bodies and pneumatic tires. Also, D’Alembert’s principle and Chetaev’s principle fall into this scheme. We emphasize the geometric point of view, avoiding the use of local coordinates, which is the appropriate setting for dealing with questions of global nature, like reduction.

 GENERAL RELATIVITY AND GRAVITATION


Separable Hilbert space in loop quantum gravity
View Description Hide DescriptionWe study the separability of the state space of loop quantum gravity. In the standard construction, the kinematical Hilbert space of the diffeomorphisminvariant states is nonseparable. This is a consequence of the fact that the knot space of the equivalence classes of graphs under diffeomorphisms is noncountable. However, the continuous moduli labeling these classes do not appear to affect the physics of the theory. We investigate the possibility that these moduli could be only the consequence of a poor choice in the finetuning of the mathematical setting. We show that by simply choosing a minor extension of the functional class of the classical fields and coordinates, the moduli disappear, the knot classes become countable, and the kinematical Hilbert space of loop quantum gravity becomes separable.

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


The loss of stability of surface superconductivity
View Description Hide DescriptionThe Ginzburg–Landau equations in a halfplane are considered in the large κ limit. We look at the reduced set of equations obtained in that limit. It is proved that the onedimensional solution presented by Pan [Commun. Math. Phys. 228, 327 (2002)] undergoes a bifurcation for an infinite number of applied magnetic field values which are lower than We also prove that each bifurcating mode is energetically preferable to the onedimensional surfacesuperconductivity solution, and thus, prove that the surfacesuperconductivity becomes unstable for applied fields which are lower than

 METHODS OF MATHEMATICAL PHYSICS


Existence and construction of the transmutation operator
View Description Hide DescriptionWe prove the existence of a transmutation operator between two weighted Sturm–Liouville operators. We also provide an explicit formula for the transmutation operator and a construction algorithm. An example and an application to an inverse spectral problem are also considered.

Covariant Poisson equation with compact Lie algebras
View Description Hide DescriptionThe covariant Poisson equation for Lie algebravalued mappings defined on is studied using functional analytic methods. Weighted covariant Sobolev spaces are defined and used to derive sufficient conditions for the existence and smoothness of solutions to the covariant Poisson equation. These conditions require, apart from suitable continuity, appropriate local integrability of the gauge potential and global weighted integrability of the curvature form and the source. The possibility of nontrivial asymptotic behavior of a solution is also considered. As a byproduct, weighted covariant generalizations of Sobolev embeddings are established.

 DYNAMICAL SYSTEMS


AKS hierarchy and biHamiltonian geometry of Gelfand–Zakharevich type
View Description Hide DescriptionA biHamiltonian system is a system of differential equations which can be written in Hamiltonian form in two distinct ways. The applications of Gelfand–Zakharevich biHamiltonian structure, which is an extension of a Poisson–Nijenhuis structure on phase space, has been extensively explored by Falqui, Magri, and Pedroni in the context of separation of variables. It is well known that the integrable Hamiltonian systems defined by the Adler–Kostant–Symes (AKS) scheme contains biHamiltonian structure. In this paper we unveil the connection between Adler–Kostant–Symes formalism applied to loop algebra and the Gelfand–Zakharevich biHamiltonian structure by superposition the results of Fordy and Kulish in the AKS scheme. We also study the commuting flows of the AKS hierachy and its connection to the Zakharov–Shabat hierarchy.
