Volume 43, Issue 3, March 2002
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


The relativistic Jmatrix theory of scattering: An analytic solution
View Description Hide DescriptionThe relativistic matrix is investigated in the case of Coulombfree scattering for a general shortrange spindependent perturbing potential and in two different bases. The resulting recursion relation of the reference problem, in this case, has an analytic solution. The nonrelativistic limit is obtained and shown to be identical to the familiar nonrelativistic matrix.

Stable orbits in a HiggsMonopole field
View Description Hide DescriptionMotion of Yang–Mills particles in Yang–Mills–Higgs fields are studied. By regarding the Higgs field contribution on the particle’s motion, a complete set of equations is worked out for the particle and fields. The planar motions as well as threedimensional bounded motions are studied. Stable orbits are allowed in this scenario.

Quantum mechanics of layers with a finite number of point perturbations
View Description Hide DescriptionWe study spectral and scatteringproperties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Krein’s formula. We prove the existence of bound states, demonstrate their properties, and find the onshell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum.

Semiclassics in the lowest Landau band
View Description Hide DescriptionThis article deals with the comparison between the strong Thomas–Fermi theory and the quantum mechanical ground state energy of a large atom confined to lowest Landau band wave functions. Using the tools of microlocal semiclassical spectral asymptotics we derive precise error estimates. The approach presented in this article suggests the definition of a modified strong Thomas–Fermi functional, where the main modification consists in replacing the integration over the variables perpendicular to the magnetic field by an expansion in angular momentum eigenfunctions. The resulting DSTF theory is studied in detail in the second part of the article.

Coherent states on spheres
View Description Hide DescriptionWe describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the ddimensional sphere The coherent states are labeled by points in the associated phase space These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal–Bargmann representation for the system, the associated unitary Segal–Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of Hall and Stenzel, we give here a substantially different description based on ideas of Thiemann and of Kowalski and Rembieliński. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a selfcontained and explicit way.

Relativistic Nboson systems bound by oscillator pair potentials
View Description Hide DescriptionWe study the lowest energy E of a relativistic system of N identical bosons bound by harmonicoscillator pair potentials in three spatial dimensions. In natural units the system has the semirelativistic (or “spinlessSalpeter”) Hamiltonian, We derive the following energy bounds: where yields a lower bound and yields an upper bound for all A sharper lower bound is given by the function which makes the formula for exact: with this choice of P, the bounds coincide for all in the Schrödinger limit defined by

Maximization of capacity and norms for some product channels
View Description Hide DescriptionIt is conjectured that the Holevo capacity of a product channel is achieved when product states are used as input. Amosov, Holevo, and Werner have also conjectured that the maximal norm of a product channel is achieved with product input states. In this article we establish both of these conjectures in the case that Ω is arbitrary and Φ is a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ω is arbitrary and either Φ is a qubit channel and or Φ is a unital qubit channel and is integer. Our proofs involve a new conjecture for the norm of an output state of the halfnoisy channel when Φ is a qubit channel. We show that this conjecture in some cases also implies additivity of the Holevo capacity.

Pinning phenomena near the lower critical field for superconductor
View Description Hide DescriptionIn this paper, we prove that for fields close to the lower critical field, minimizers of the Ginzburg–Landau functional of an inhomogeneous superconductor have a number of vortices bounded independently from the Ginzburg–Landau parameter. We also locate the vortices.

On asymptotic perturbation theory for quantum mechanics: Almost invariant subspaces and gauge invariant magnetic perturbation theory
View Description Hide DescriptionSingular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory. Under very general conditions, asymptotic estimations on the Rayleigh–Schrödinger expansions of the perturbed spectral projections are obtained. As a consequence almost invariant subspaces of exponential order are constructed. The results cover practically all singular perturbations considered in nonrelativistic quantum mechanics. In the magnetic field case, under the condition that the magnetic field does not increase at infinity, a gauge invariant perturbation theory leading to convergent series with fielddependent coefficients is developed.

On the structure of covariant phase observables
View Description Hide DescriptionWe study the mathematical structure of covariant phase observables. Such observables can alternatively be expressed as phase matrices, as sequences of unit vectors, as sequences of phase states, or as equivalence classes of covariant tracepreserving operations. Covariant generalized operator measures are defined by structure matrices which form a algebra with phase matrices as its subset. The properties of the Radon–Nikodým derivatives of phase probability measures are studied.

Quantum superintegrability and exact solvability in n dimensions
View Description Hide DescriptionA family of maximally superintegrable systems containing the Coulomb atom as a special case is constructed in dimensional Euclidean space. Two different sets of commuting secondorder operators are found, overlapping in the Hamiltonian alone. The system is separable in several coordinate systems and is shown to be exactly solvable. It is solved in terms of classical orthogonal polynomials. The Hamiltonian and further operators are shown to lie in the enveloping algebra of a hidden affine Lie algebra.

Fiber bundles in quantum physics
View Description Hide DescriptionThe theory of fiber bundles provides a natural setting for the description of macroscopic quantum systems, wherein their classical and quantum features are represented by actions on the base manifolds and the fibers, respectively, of the relevant bundles. We provide realizations of this picture in the description of (a) quasiparticleexcitations of manybody systems, especially those in superfluidhelium, (b) the interplay between the microscopic and macroscopic dynamics in certain irreversible processes, such as that of a laser, and (c) local thermodynamic equilibrium. In particular, (b) involves the treatment of a dynamical system which is defined on a vector bundle.

Large limit of scalar gauge theory
View Description Hide DescriptionIn this paper we study the large limit of gauge theory coupled to a real scalar field following ideas of Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. We will see that the phase space of this resulting classical theory is which is the analog of the Siegel disk in infinite dimensions. The linearized equations of motion give us a version of the wellknown ’t Hooft equation of two dimensional quantum chromodynamics.

Quantum Clifford algebra from classical differential geometry
View Description Hide DescriptionWe show the emergence of Clifford algebras of nonsymmetric bilinear forms as cotangent algebras of Kaluza–Klein (KK) spaces pertaining to teleparallel space–times. These spaces are canonically determined by the horizontal differential invariants of Finsler bundles of the type, where is the set of all the tangent frames to a differentiable manifold and where is the sphere bundle. If is space–time itself, the “geometric phase space,” has dimension seven. This reformulation of the horizontal invariants as pertaining to a KK space removes the mismatch between the dimensionality of the tangent frames to and the dimensionality of In the KK space, a symmetric tangent metric induces a cotangent metric which is not symmetric in general. An interior covariant derivative in the sense of Kaehler is defined. It involves the antisymmetric part of the cotangent metric, which thus enters electrodynamics and the Dirac equation.

On separable Pauli equations
View Description Hide DescriptionWe classify dimensional Pauli equations for a spin particle interacting with the electromagnetic field, that are solvable by the method of separation of variables. As a result, we obtain the 11 classes of vectorpotentials of the electromagnetic field providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into secondorder matrix ordinary differential equations is its equivalence to the system of two uncoupled Schrödinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vectorpotentials of the electromagnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schrödinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vectorpotential of the electromagnetic field is developed. Finally, we describe all vectorpotentials that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe nonzero magnetic field.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


High temperature expansion for a chain model
View Description Hide DescriptionWe consider an arbitrary translationally invariant chain model with nearest neighbors interaction and satisfying periodic boundary condition. The approach developed here allows a thermodynamic description of the chain model directly in terms of grand potential per site. This thermodynamic function is derived from an auxiliary function constructed only from open connected subchains. In order to exemplify its application and how this approach works we consider the Heisenberg XXZ model. We obtain the coefficients of the high temperature expansion of the free energy per site of the model up to third order.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


A biHamiltonian formulation for triangular systems by perturbations
View Description Hide DescriptionA biHamiltonian formulation is proposed for triangular systems resulting from perturbations around solutions, from which infinitely many symmetries and conserved functionals of triangular systems can be explicitly constructed, provided that one operator of the Hamiltonian pair is invertible. Through our formulation, four examples of triangular systems are exhibited, which also show that biHamiltonian systems in both lower dimensions and higher dimensions are many and varied. Two of four examples give local dimensional biHamiltonian systems and illustrate that multiscale perturbations can lead to higherdimensional biHamiltonian systems.

Group invariant classification of separable Hamiltonian systems in the Euclidean plane and the symmetric Yang–Mills theories of Yatsun
View Description Hide DescriptionWe present a new and effective method of determining separable coordinate systems for natural Hamiltonians with two degrees of freedom in flat Riemannian space. The method is based on intrinsic properties of the associated Killing tensors and their invariants under the group of rigid motions Applications to the Hamiltonian systems derived by the late V. A. Yatsun from symmetric Yang–Mills theories are presented. In addition, an equivalence between separability of twodimensional Hamiltonian systems and the existence of Pfaffian quasibiHamiltonian representations is specified.

MultiLagrangians for integrable systems
View Description Hide DescriptionWe propose a general scheme to construct multiple Lagrangians for completely integrable nonlinear evolution equations that admit multiHamiltonian structure. The recursion operator plays a fundamental role in this construction. We use a conserved quantity higher/lower than the Hamiltonian in the potential part of the new Lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map. This leads to some remarkable new developments. We show that nonlinear evolutionary systems that admit Nfold first order local Hamiltonian structure can be cast into variational form with Lagrangians which will be local functionals of Clebsch potentials. This number increases to when the Miura transformation is invertible. Furthermore we construct a new Lagrangian for polytropic gas dynamics in dimensions which is a free, local functional of the physical field variables, namely density and velocity, thus dispensing with the necessity of introducing Clebsch potentials entirely. This is a consequence of biHamiltonian structure with a compatible pair of first and third order Hamiltonian operators derived from Sheftel’s recursion operator.

Perturbation theory for nearly integrable multicomponent nonlinear PDEs
View Description Hide DescriptionThe Riemann–Hilbert problem associated with the integrable PDE is used as a nonlinear transformation of the nearly integrable PDE to the spectral space. The temporal evolution of the spectral data is derived with account for arbitrary perturbations and is given in the form of exact equations, which generate the sequence of approximate ordinary differential equations in successive orders with respect to the perturbation. For vector nearly integrable PDEs, embracing the vector nonlinear Schrödinger and complex modified Korteweg–de Vries equations, the main result is formulated in a theorem. For a single vector soliton the evolution equations for the soliton parameters and firstorder radiation are given in explicit form.
