Volume 37, Issue 7, July 1996
Index of content:

Antisymmetric tensor fields on spheres: Functional determinants and non‐local counterterms
View Description Hide DescriptionThe Hodge–de Rham Laplacian on spheres acting on antisymmetric tensor fields is considered. Explicit expressions for the spectrum are derived in a quite direct way, confirming previous results. Associated functional determinants and the heat kernel expansion are evaluated. Using this method, new non‐local counterterms in the quantum effective action are obtained, which can be expressed in terms of Betti numbers.

Proper time and path integral representations for the commutation function
View Description Hide DescriptionOn the example of the quantized spinor field, interacting with arbitrary external electromagnetic field, the commutation function is studied. It is shown that a proper time representation is available in any dimensions. Using it, all the light cone singularities of the function are found explicitly, generalizing the Fock formula in four dimensions, and a path integral representation is constructed.

Quantum Lobachevsky planes
View Description Hide DescriptionWe classify all SL(2,R)‐covariant Poisson structures on the Lobachevsky plane with respect to all multiplicative Poisson structures on SL(2,R) and describe quantizations for all these Poisson structures.

On localization and regularization
View Description Hide DescriptionDifferent regularizations are studied in localization of path integrals. We discuss the effect of the choice of regularization by evaluating the partition functions for the harmonic oscillator and the Weyl character for SU(2). In particular, we solve the Weyl shift problem that arises in path integral evaluation of the Weyl character by using the Atiyah–Patodi–Singer η‐invariant and the Borel–Weil theory.

One dimensional periodic Dirac Hamiltonians: Semiclassical and high‐energy asymptotics for gaps
View Description Hide DescriptionIn this article we consider a one‐dimensional Dirac operator with a potential of Gevrey class α and study the semiclassical and high‐energy asymptotics of the spectral gaps for a region of energies that in the Schrödinger case corresponds to unbounded motion. An exponential upper bound for the gap’s widths as well as the asymptotic expansion of their positions are derived for both cases; the first two terms in the asymptotic expansions are explicitly written down for the scalar potential case.

The contraction of the SU(1,1) discrete series of representations by means of coherent states
View Description Hide DescriptionThe group SU(1,1) is a deformation of the Poincaré group. This relationship is studied both at the classical level (coadjoint orbits) and at the quantum level (unitary representations). The contraction of the Lie algebras is written in such a way that the limit of coadjoint orbits, and hence of the classical mechanics, appears clearly. At the quantum level the representations are written on holomorphic functions Hilbert spaces and the contraction is realized by restricting these functions. It is shown that this restriction is a continuous operator. Moreover, using suitable coherent states, it is proved that the contraction extends to the representation of the whole enveloping algebras of the groups, hence it allows us to define the contraction of the quantum mechanics observables.

Quantum mechanics of charged particles in random electromagnetic fields
View Description Hide DescriptionIn this paper we consider the ℏ→0 asymptotics for the solutions to stochastic Schrödinger equations for quantum mechanical particles in random electromagnetic fields in R ^{ n }. We obtain semi‐classical expansions for their solutions up to any order in L ^{2}(R ^{ n }) a.s. by using a stochastic Hamilton Jacobi equation and a stochastic continuity equation. We conclude that as ℏ→0 the stochastic quantum mechanics with random electromagnetic fields tends to stochastic classical mechanics.

Conditional symmetry and spectrum of the one‐dimensional Schrödinger equation
View Description Hide DescriptionWe develop an algebraic approach to studying the spectral properties of the stationary Schrödinger equation in one dimension based on its high‐order conditional symmetries. This approach makes it possible to obtain in explicit form representations of the Schrödinger operator by n×n matrices for any n∈N and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n×n matrices. The connection to so‐called quasiexactly solvable models is discussed. It is established, in particular, that the case, when conditional symmetries reduce to high‐order Lie symmetries, corresponds to exactly solvable Schrödinger equations. A symmetry classification of Schrödinger equation admitting nontrivial high‐order Lie symmetries is carried out, which yields a hierarchy of exactly solvable Schrödinger equations. Exact solutions of these are constructed in explicit form. Possible applications of the technique developed to multidimensional linear and one‐dimensional nonlinear Schrödinger equations are briefly discussed.

Integral equation methods for the inverse problem with discontinuous wave speed
View Description Hide DescriptionThe recovery of the coefficient H(x) in the one‐dimensional generalized Schrödinger equationd ^{2}ψ/dx ^{2}+k ^{2} H(x)^{2}ψ=Q(x)ψ, where H(x) is a positive, piecewise continuous function with positive limits H _{±} as x→±∞, is studied. The large‐k asymptotics of the wave functions and the scattering coefficients are analyzed. A factorization formula is given expressing the total scattering matrix as a product of simpler scattering matrices. Using this factorization an algorithm is presented to obtain the discontinuities in H(x) and H′(x)/H(x) in terms of the large‐k asymptotics of the reflection coefficient. When there are no bound states, it is shown that H(x) is recovered from an appropriate set of scattering data by using the solution of a singular integral equation, and the unique solvability of this integral equation is established. An equivalent Marchenko integral equation is derived and is shown to be uniquely solvable; the unique recovery of H(x) from the solution of this Marchenko equation is presented. Some explicit examples are given, illustrating the recovery of H(x) from the solution of the singular integral equation and from that of the Marchenko equation.

Generalization of the Bremmer coupling series
View Description Hide DescriptionAn operator formalism is developed to expand the acoustic wave field in a multi‐dimensionally smoothly varying medium, generated by a source localized in space and time, into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times. This expansion is a generalization of the Bremmer coupling series. The condition of smoothness of the medium relates to the width of the signature of the source in the configuration. Both the existence and the convergence (in the weak sense) of the expansion are discussed. The operator calculus involved leads to a natural generalization of the concept of slowness surface to multi‐dimensionally smoothly varying media. The operator associated with the corresponding generalized vertical slowness induces the full one‐way wave operator in the type of media under consideration. In addition, a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents, are derived.

Diffuse tomography modulo Grassmann and Laplace
View Description Hide DescriptionComplicated and unphysical families of modified transition probabilities for the 4×4 diffuse tomographic problem are presented. Grassmann–Plücker identities and Laplace expansions of determinants are used to simplify the initial transition probabilities. Besides restoring self‐consistency to the system, enforcing range conditions eliminates one‐half of the parameters.

On a hierarchy of macroscopic models for semiconductors
View Description Hide DescriptionThis paper shows that various models of electron transport in semiconductors that have been previously proposed in the literature can be connected one with each other by the diffusion approximation methodology. We first investigate the diffusion limit of the semiconductorBoltzmann equation towards the so‐called ‘‘spherical harmonic expansion model,’’ under the assumption of dominant elastic scattering. Then, this model is again connected, either to the energy‐transport model or to a ‘‘periodic spherical harmonic expansion model’’ through a diffusion approximation, respectively making electron–electron or phonon scattering large. We provide the mathematical background which makes the Hilbert expansions associated with these various diffusion limits rigorous.

Energy extremes and spin configurations for the one‐dimensional antiferromagnetic Ising model with arbitrary‐range interaction
View Description Hide DescriptionThe one‐dimensional antiferromagnetic spin‐1/2 Ising model is investigated using the formalism of Maximally/Minimally Even sets. The salient features of Maximally/Minimally Even set theory are introduced. Energy and spin content vectors are defined to facilitate the use of interval spectra used in Maximally/Minimally Even set theory. It is shown that Maximally Even sets of up‐ and down‐spins minimize the configurational energy per spin and that Minimally Even sets maximize configurational energy per spin. An exponentially decreasing antiferromagnetic pairwise interaction of arbitrary range is used as an example interaction. The asymptotic (N→∞) configurational energy per spin and the energy per spin calculated for seven‐near neighbors are compared.

On the Poisson integrals representation in the classical statistical mechanics of continuous systems
View Description Hide DescriptionDescription of the grand canonical Gibbs ensemble for classical continuous systems in terms of the nonlocally perturbed infinite‐divisible generalized random fields is presented. The equivalence of the traditional description with the ones presented here on the level of DLR equations is established. The antiferromagnetism for the purely repulsive interactions has been observed. Finally, the usefulness of our description for an analysis of the high‐temperature cluster expansion has been demonstrated.

Lagrangian dynamics for classical, Brownian, and quantum mechanical particles
View Description Hide DescriptionIn the framework of Nelson’s stochastic mechanics [E. Nelson, Dynamical Theories of Brownian Motion (Princeton University, Princeton, 1967); F. Guerra, Phys. Rep. 77, 263 (1981); E. Nelson, Quantum Fluctuations (Princeton University, Princeton, 1985)] we seek to develop the particle counterpart of the hydrodynamic results of M. Pavon [J. Math. Phys. 36, 6774 (1995); Phys. Lett. A 209, 143 (1995)]. In particular, a first form of Hamilton’s principle is established. We show that this variational principle leads to the correct equations of motion for the classical particle, the Brownian particle in thermodynamical equilibrium, and the quantum particle. In the latter case, the critical process q satisfies a stochastic Newton law. We then introduce the momentum process p, and show that the pair (q,p) satisfies canonical‐like equations.

On the geometry of non‐holonomic Lagrangian systems
View Description Hide DescriptionWe present a geometric framework for non‐holonomic Lagrangian systems in terms of distributions on the configuration manifold. If the constrained system is regular, an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics. If the constrained system is singular, we develop a constraint algorithm which is very similar to that developed by Dirac and Bergmann, and later globalized by Gotay and Nester. Special attention to the case of constrained systems given by connections is paid. In particular, we extend the results of Koiller for Čaplygin systems. An application to the so‐called non‐holonomic geometry is given.

On the asymptotic integrability of a higher‐order evolution equation describing internal waves in a deep fluid
View Description Hide DescriptionA higher‐order nonlocal evolution equation describing internal waves in a deep fluid is shown to be asymptotically integrable only if the coefficients of the higher‐order terms satisfy certain constraints. In this case, the nonlocal equation can be transformed to the integrable Benjamin–Ono equation. The asymptotic integrability of the reductions of the higher‐order evolution equation to a complex Burgers equation, to an envelope‐wave equation, and to a finite‐dimensional dynamical system is also considered.

Soliton solutions and nontrivial scattering in an integrable chiral model in (2+1) dimensions
View Description Hide DescriptionThe behavior of solitons in integrable theories is strongly constrained by the integrability of the theory; i.e., by the existence of an infinite number of conserved quantities that these theories are known to possess. One usually expects the scattering of solitons in such theories to be rather simple, i.e., trivial. By contrast, in this paper we generate new soliton solutions for the planar integrable chiralmodel whose scattering properties are highly nontrivial; more precisely, in head‐on collisions of N indistinguishable solitons the scattering angle (of the emerging structures relative to the incoming ones) is π/N. We also generate soliton–antisoliton solutions with elastic scattering; in particular, a head‐on collision of a soliton and an antisoliton resulting in 90° scattering.

Variational method: How it can generate false instabilities
View Description Hide DescriptionWhen the variational method is applied to nonlinear evolution equations for determining solitary wave dynamics, it is possible for the method to predict the pulse to be unstable when in fact it is stable. We determine the necessary conditions for this to occur as well as give sufficient conditions for avoiding such false instabilities. We also discuss the general problem of applying the method to a general evolution equation.

Stationary problems for equation of the KdV type and dynamical r‐matrices
View Description Hide DescriptionWe study a new quite general family of dynamical r‐matrices for an auxiliary loop algebraL( su(2)) related to restricted flows for equations of the KdV type. This underlying r‐matrix structure allows us to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems.