Volume 44, Issue 5, May 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


The discretized harmonic oscillator: Mathieu functions and a new class of generalized Hermite polynomials
View Description Hide DescriptionWe present a general, asymptotical solution for the discretized harmonic oscillator. The corresponding Schrödinger equation is canonically conjugate to the Mathieu differential equation, the Schrödinger equation of the quantum pendulum. Thus, in addition to giving an explicit solution for the Hamiltonian of an isolated Josephon junction or a superconductingsingleelectron transistor (SSET), we obtain an asymptotical representation of Mathieu functions. We solve the discretized harmonic oscillator by transforming the infinitedimensional matrixeigenvalue problem into an infinite set of algebraic equations which are later shown to be satisfied by the obtained solution. The proposed ansatz defines a new class of generalized Hermite polynomials which are explicit functions of the coupling parameter and tend to ordinary Hermite polynomials in the limit of vanishing coupling constant. The polynomials become orthogonal as parts of the eigenvectors of a Hermitian matrix and, consequently, the exponential part of the solution can not be excluded. We have conjectured the general structure of the solution, both with respect to the quantum number and the order of the expansion. An explicit proof is given for the three leading orders of the asymptotical solution and we sketch a proof for the asymptotical convergence of eigenvectors with respect to norm. From a more practical point of view, we can estimate the required effort for improving the known solution and the accuracy of the eigenvectors. The applied method can be generalized in order to accommodate several variables.

Perturbative analysis of dynamical localization
View Description Hide DescriptionIn this paper we present a mathematical investigation of the phenomenon of dynamical localization in a class of quasiperiodically and periodically timedependent twolevel systems. Our results are based on a sort of “renormalization” procedure, which is developed here in a systematic way in order to adapt it to the case of dynamical localization. In the quasiperiodic case this procedure leads to a formal perturbative expansion free of secular terms. In the periodic case a convergent perturbative solution is obtained and, in particular, a convergent perturbative expansion for the secular frequency is presented. The case of ac–dc fields is discussed in some detail, leading to the conclusion that the phenomenon of dynamical localization is not exact in that situation.

The increase of binding energy and enhanced binding in nonrelativistic QED
View Description Hide DescriptionWe consider a Pauli–Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of the increase of the binding energy and enhanced binding through coupling to a photon field, and prove that both effects are the results of the existence of the ground state of the selfenergy operator with total momentum

Breit–Wigner formula at barrier tops
View Description Hide DescriptionFor noncritical energies, the asymptotic behavior of the scattering phase and of the timedelay are known to be described by a Weyl type formula and the Breit–Wigner formula, respectively. We consider here the case of critical energy levels in dimension 1. We obtain the semiclassical asymptotics of the scattering phase and of the timedelay, uniformly with respect to the energy in a neighborhood of a critical value.

BiHamiltonian partially integrable systems
View Description Hide DescriptionGiven a first order dynamical system possessing a commutative algebra of dynamical symmetries, we show that, under certain conditions, there exists a Poissonstructure on an open neighborhood of its regular (not necessarily compact) invariant manifold which makes this dynamical system into a partially integrable Hamiltonian system. This Poissonstructure is by no means unique. BiHamiltonian partially integrable systems are described in some detail. As an outcome, we state the conditions of quasiperiodic stability (the KAM theorem) for partially integrable Hamiltonian systems.

The norm1property of a quantum observable
View Description Hide DescriptionA normalized positive operator measure has the norm1property if whenever This property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrarily close to one with suitable state preparations. Some general implications of the norm1property are investigated. As case studies, localization observables, phase observables, and phase space observables are considered.

Rigorous proof of isotope effect by Bardeen–Cooper–Schrieffer theory
View Description Hide DescriptionIn this article, we present a rigorous proof of the isotope effect within the theory of Bardeen–Cooper–Schrieffer for isotropic superconductors. We show that, when the interaction kernel is a positive constant, the isotope effect is exact; when the interaction kernel is a positive function depending on the energies of the pairing electrons, the isotope effect is no longer exact but lies within a sharp range determined by the varying kernel function. Moreover, we show that our method here may be extended to establish an existence and uniqueness theorem for the transition temperature in the Bogoliubov–Tolmachev–Shirkov model which allows separate phonon and Coulomb dominance in their respective energy regimes.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Distributional Borel summability for vacuum polarization by an external electric field
View Description Hide DescriptionIt is proved that the divergent perturbation expansion for the vacuum polarization by an external constant electric field in the pair production sector is Borel summable in the distributional sense.

Modular localization of massive particles with “any” spin in
View Description Hide DescriptionWe discuss a concept of particle localization which is motivated from quantum field theory, and has been proposed by Brunetti, Guido and Longo and by Schroer. It endows the singleparticle Hilbert space with a family of real subspaces indexed by the space–time regions, with certain specific properties reflecting the principles of locality and covariance. We show by construction that such a localization structure exists also in the case of massive anyons in i.e., for particles with positive mass and with arbitrary spin The construction is completely intrinsic to the corresponding ray representation of the (proper orthochronous) Poincaré group. Our result is of particular interest since there are no free fields for anyons, which would fix a localization structure in a straightforward way. We present explicit formulas for the real subspaces, expected to turn out useful for the construction of a quantum field theory for anyons. In accord with wellknown results, only localization in stringlike, instead of pointlike or bounded, regions is achieved. We also prove a singleparticle PCT theorem, exhibiting a PCT operator which acts geometrically correctly on the family of real subspaces.

Towards Euclidean theory of infrared singular quantum fields
View Description Hide DescriptionA new generalized formulation of the spectral condition is proposed for quantum fields with highly singular infrared behavior whose vacuum correlation functions are well defined only under smearing with analytic test functions in momentum space. The Euclidean formulation of QFT developed by Osterwalder and Schrader is extended to theories with infrared singular indefinite metric. The corresponding generalization of the reconstruction theorem is obtained. The fulfilment of the generalized spectral condition is verified for quantum fields representable by infinite series in the Wick powers of indefinite metric free fields.
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 GENERAL RELATIVITY AND GRAVITATION


A symplectic framework for multiplane gravitational lensing
View Description Hide DescriptionWe construct a new framework for the study of multiplane gravitational lensing from the view point of symplectic geometry. Symplectic relations are used to compose the systems and weaker Lagrangian equivalence is applied for classifying the caustics of multiplane gravitational lensing.

Generalized forms and Einstein’s equations
View Description Hide DescriptionGeneralized differential forms of different types are defined and their algebra and calculus are discussed. Complex generalized pforms, a particular class of type two generalized forms, are considered in detail. It is shown that Einstein’s vacuum field equations for Lorentzian fourmetrics are satisfied if and only if a complex generalized oneform on the bundle of two component spinors is closed. A similar result for halfflat and anti selfdual holomorphic fourmetrics is also presented.
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 DYNAMICAL SYSTEMS


Lie point symmetries and first integrals: The Kowalevski top
View Description Hide DescriptionWe show how the Lie group analysis method can be used in order to obtain first integrals of any system of ordinary differential equations. The method of reduction/increase of order developed by Nucci [J. Math. Phys. 37, 1772–1775 (1996)] is essential. Noether’s theorem is neither necessary nor considered. The most striking example we present is the relationship between Lie group analysis and the famous first integral of the Kowalevski top.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Wave scattering in waveguides
View Description Hide DescriptionThe scattering of scalar waves by objects located inside a waveguide or a cavity is discussed using the method of pseudopotentials. Pseudopotentials were introduced to simulate shortrange potentials in quantum mechanics and proved useful in manybody problems and in problems involving multicentered potentials. In this work it is shown that this approach can also be used to describe the scattering of classical scalar waves by objects confined to the interior of a waveguide or a cavity in terms of the scattering amplitudes of those objects in an extended medium.

On harmonic oscillators on the twodimensional sphere and the hyperbolic plane II.
View Description Hide DescriptionThe properties of several noncentral harmonic oscillators are examined on spaces of constant curvature. All the mathematical expressions are presented using the curvature κ as a parameter, in such a way that particularizing for or the corresponding properties are obtained for the system on the sphere the Euclidean plane or the hyperbolic plane respectively. First, the separability in several κdependent systems of coordinates, as well as the existence of four families of κdependent superintegrable potentials related with the harmonic oscillator, are studied. Then three harmonic oscillators (1:1, 2:1 and are studied by using two different methods: superseparability and complex factorization. The second part deals with the problem of the existence of superintegrable but not superseparable systems. Several κdependent superintegrable harmonic oscillators with higherorder constants of motion are studied. The constants of motion are obtained by making use of the method of the complex factorization.

Stability of Beltrami flows
View Description Hide DescriptionStability of a special class of flows (which we call Beltrami flows) can be analyzed by invoking a constant of motion that bounds the energy of perturbations. This stability condition (a sufficient condition) suppresses any instability including nonexponential (secular) growth due to nonHermiticity; it also prohibits nonlinear evolution to a large amplitude. The key to prove is the “coerciveness” of the constant of motion in the topology of the energy norm. The theory has been applied for an ideal (nondissipative) magnetized plasma.
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 STATISTICAL PHYSICS


Anomalous diffusion: Fractional Fokker–Planck equation and its solutions
View Description Hide DescriptionWe analyze a linear fractional Fokker–Planck equation for the case of an external force and diffusion coefficient We also discuss the connection of the solutions found here with the Fox functions and the nonextensive statistics based on the Tsallis entropy.
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 METHODS OF MATHEMATICAL PHYSICS


Wigner’s theorem in a class of Hilbert modules
View Description Hide DescriptionLet be a complex Hilbert space, and let be a algebra such that the ideal of all compact operators on is contained in A. Let be a Hilbert module over A. We prove that any function which preserves the absolute value of the Avalued inner product on is of the form where φ is a phase function and is an Alinear isometry. The result generalizes Wigner’s classical unitaryantiunitary theorem and its extension to Hilbert modules.

Representations of a symplectic type subalgebra of
View Description Hide DescriptionWe classify the irreducible quasifinite highest weight modules over the symplectic Lie subalgebra and realize them in terms of irreducible highest weight representations of classical Lie subalgebras of infinite matrices with finitely many nonzero diagonals.

A bicategorical approach to Morita equivalence for von Neumann algebras
View Description Hide DescriptionWe relate Morita equivalence for von Neumann algebras to the “Connes fusion” tensor product between correspondences. In the purely algebraic setting, it is well known that rings are Morita equivalent iff they are equivalent objects in a bicategory whose 1cells are bimodules. We present a similar result for von Neumann algebras. We show that von Neumann algebras form a bicategory, having Connes’s correspondences as 1morphisms, and (bounded) intertwiners as 2morphisms. Further, we prove that two von Neumann algebras are Morita equivalent iff they are equivalent objects in the bicategory. The proofs make extensive use of the Tomita–Takesaki modular theory.
