bluuebunny/crossref_metadata_2025 huggingface dataset:
Anisotropic homogeneous cosmologies with perfect fluid and electric field
Authors: Cataldo Mauricio , Mitskiévic Nikolai V.
Published: July 2002

A new three-parameter family of cosmological models is found, which are solutions of Einstein–Maxwell equations in a space-time filled with electrically neutral stiff matter. They generalize an anisotropic homogeneous cosmology without electromagnetic field by Vajk and Eltgroth [J. Math. Phys. 11, 2212 (1970)] and support a conjecture about proportionality of electromagnetic four-potential of an Einstein–Maxwell solution and the Killing vector of a corresponding space-time with stiff matter. This conjecture turns out to be the clue to a new solution-generating method of Einstein–Maxwell fields with sources.
Interaction of null dust clouds fronted by impulsive plane gravitational waves
Authors: Taub A. H.
Published: July 2002

This paper discusses the energy-momentum tensor T μν in the region of interaction of a space-time in which two colliding plane impulsive gravitational waves, each followed by a null dust cloud, exist. It is shown that in the interaction region T μν is of three types: (i) that of two noninteracting null dusts; (ii) that of a scalar field (equivalently an irrotational perfect fluid with energy density equal to pressure), and (iii) that of the sum of two independent non-interacting scalar fields [equivalently a complex scalar field or an anisotropic perfect fluid with energy density w and pressures (w,π,π)].
Relativistic theory of spherically symmetric perturbation in dust universe
Authors: Liu Hongya
Published: July 2002

Starting from the initial value formulation of dust solution given in the preceding paper [J. Math. Phys. 31, 2459 (1990)] a relativistic perturbation equation and its general solutions in spherically symmetric universes are derived. It is found that these solutions are in analogy with the Bonnor–Weinberg results of the Newtonian theory and contain three modes: a growing mode, a decaying mode, and a constant mode.
On the asymptotics of finite energy solutions of the Yang–Mills–Higgs equations
Authors: Dostoglou S.
Published: July 2002

Using a priori estimates by Taubes [A. Jaffe and C. H. Taubes, Vortices and Monopoles (Birkhauser, Boston, 1980)] on the rate of decay of finite energy solutions of the Yang–Mills–Higgs equations, the limits of the solution fields are proven to exist at infinity. As expected, the limit of the gauge field is a pure Yang–Mills field on the reduction bundle over S2 defined by the limit of the Higgs field.
A multivectorial Dirac equation
Authors: Keller Jaime , Rodriguez-Romo Suemi
Published: July 2002

The multivectorial generalization of the Cartan map, for 𝒞(1,3) space-time Clifford algebra and an arbitrary gauge group in an isotopic space, is applied to the standard Dirac equation to generate the multivectorial Dirac equation. Using both geometrical and physical reasoning, a particular case discussed by Reifler and Morris [J. Math. Phys. 26, 2059 (1985)] is projected from the general multivectorial Dirac equation, to discuss the properties and limitations associated to their quaternion model. The use of the general multivectorial Dirac equation, which can be defined on any space-time manifold, is also illustrated.
Bounds on the kinetic energy, mean radius, and electric dipole moment of ground states in a class of potential models of quarkonium
Authors: Common A. K.
Published: July 2002

Properties of ‘‘mixed moments,’’ 〈r μ〉l,L≡∫∞0r μuL (r)ul(r)dr of ground state wave functions ul(r) of angular momentum l are investigated. It is shown that for cases where the Laplacian of the central potential V(r) is positive (negative), 〈r μ〉l,L/Γ(μ+L+l+3) is a concave (convex) function of μ. These results are extended to other classes of potentials and inequalities relating ‘‘mixed moments’’ to ‘‘standard moments’’ with L=l, are derived. The latter are used to give a simpler, more comprehensive proof of convexity (concavity) properties of the ground state energies E(0,l) as functions of l. New rigorous bounds on the overlap 〈1〉l,L of two angular momentum ground states, the mean square radius and kinetic energies of ground states and the electric dipole moment for transition rates between two such states are obtained for various classes of potentials using the above ‘‘moment’’ inequalities. These bounds are tested in the framework of a recent potential model description of the upsilon system.
Alternative Lagrangians for central potentials
Authors: Rañada Manuel F.
Published: July 2002

Two methods for the construction of alternative Lagrangians to the standard Lagrangian L=T−V where V is a central potential are presented: the direct integration of the Hessian matrix and the use of the Noether symmetries of L. The Lagrangians obtained are of the so-called nonlinear homogenous (NLH) type and the study is centered in those Lagrangians not preserving the spherical symmetry of L. Finally, the associated Hamiltonian formalism is presented.
Projective reduction in quantum conformal field theory
Authors: Juriev Denis
Published: July 2002

It will be shown in this paper how one can construct the strict functor from the category of quantum conformal field theories (QCFTs) to the category of quantum projective field theories (QPFTs). The conditions that single out the QCFTs from the QPFTs are presented. The local field algebras of QPFT, local projective field algebras (LPFAs), are described as the algebras of matrices with coefficients from the special LPFA—the LPFA linear operators in model (LOEL) of all linear operators in the model of the Verma modules over the Lie algebra sl(2,C).
Differential equation for SU(3) extension of Bessel function with two indices
Authors: Tanimura N. , Tanimura O.
Published: July 2002

For the coefficient function of the character expansion of the single-link action in the SU(3) lattice gauge theory, the linear differential equation of the sixth order, which is specified by two integer indices λ and μ of the SU(3) (λ,μ) representation of the Young tableau, is derived. The asymptotic behavior of the function is derived from the recursion relations. It will be shown that the coefficient function is regarded as the SU(3) extension of the modified Bessel function.
Colliding gravitational waves with Killing–Cauchy horizons
Authors: Li Wei , Hauser Isidore , Ernst Frederick J.
Published: July 2002

A large family of solutions that were constructed earlier [J. Math. Phys. 32, 723 (1991)] is shown to exhibit Killing–Cauchy horizons when one of the parameters, n, is set equal to 3, 1, −1, or −3.
The removal of an energy dependence from the interaction in two-body systems
Authors: Motovilov A. K.
Published: July 2002

Energy-independent potentials that are almost spectrally equivalent to the energy-dependent ones are constructed. A case of polar energy dependence is studied especially. A profound connection between the energy-independent potentials existence and a spectral problem for the initial Schrödinger equation is discovered. An expansion theorem for the eigenfunctions of the resulting non-self-adjoint Hamiltonian is proved.
The brachistochrone problem in a stationary space-time
Authors: Perlick V.
Published: July 2002

In a stationary space-time the brachistochrone problem can be formulated in two different ways, viz., to find the path of shortest travel time with prescribed specific energy from one space-point to another (i) measured in terms of proper time or (ii) measured in terms of the time coordinate distinguished by the stationarity assumption. It is shown that in the static case both brachistochrone problems can be reduced to geodesic problems of appropriate Riemannian three-metrics, in close analogy to the brachistochrone problem in a Newtonian potential. In the stationary but nonstatic case, however, this is true only for the proper time brachistochrones, whereas the coordinate time brachistochrones are influenced by a sort of Coriolis force. These results are illustrated by calculating the brachistochrones in Rindler, Schwarzschild, and Gödel space-times.
Normal forms on partial O*-algebras
Authors: Antoine J.-P. , Inoue A.
Published: July 2002

In the algebraic formulation of quantum theories, a state is often represented by a normal linear functional on some *-algebra 𝔄 of operators on a Hilbert space, i.e., a functional of the form f(X)=Tr TX for some trace class operator X. The question is whether every strongly positive (i.e., positive on positive elements) linear functional is normal. Criteria for this statement to be true are well known in the case where 𝔄 is a *-algebra of bounded operators (C*- or W*-algebra) or a *-algebra of unbounded operators (O*-algebra). Those results are extended to the case where 𝔄 is a (weak) partial *-algebra of closable operators, both for linear functionals and for sesquilinear forms.
Exact solutions of a hierarchy of mixing speeds models
Authors: Cornille H. , Platkowski T.
Published: July 2002

This paper presents several new aspects of discrete kinetic theory (DKT). First a hierarchy of d-dimensional (d=1,2,3) models is proposed with (2d+3) velocities and three moduli speeds: 0, 2, and a third one that can be arbitrary. It is assumed that the particles at rest have an internal energy which, for microscopic collisions, supplies for the loss of the kinetic energy. In a more general way than usual, collisions are allowed that mix particles with different speeds. Second, for the (1+1)-dimensional restriction of the systems of PDE for these models which have two independent quadratic collision terms we construct different exact solutions. The usual types of exact solutions are studied: periodic solutions and shock wave solutions obtained from the standard linearization of the scalar Riccati equations called Riccatian shock waves. Then other types of solutions of the coupled Riccati equations are found called non-Riccatian shock waves and they are compared with the previous ones. The main new result is that, between the upstream and downstream states, these new solutions are not necessarily monotonous. Further, for the shock problem, a two-dimensional dynamical system of ODE is solved numerically with limit values corresponding to the upstream and downstream states. As a by-product of this study two new linearizations for the Riccati coupled equations with two functions are proposed.
The Euclidean root of Snell’s law I. Geometric polarization optics
Authors: Wolf Kurt Bernardo
Published: July 2002

Coset spaces of the Euclidean group of rigid motions in three-space are used as a model of the rays in geometric polarization optics. The Haar measure invariance leads to canonicity of the phase space transformations. The phase space of optical rays undergoes a canonical map also under the effect of smooth refracting or reflecting surfaces between two optical media that is governed by Snell’s law. This is a conservation law between two irreducible representations of the Euclidean group labeled by the refractive indices n, n′ of the media. These surface transformations are canonical and, furthermore, factorize into two root transformations that are also canonical. This factorization applies equally to the transformation of the polarization vector. The root transformation permits the computation of an aberration expansion of the polarization field over the object screen. The generators of infinitesimal surface transformations are found, i.e., those that transform between media n and n+dn. It is shown that, together with displacements, they yield the Hamilton equations of optics for inhomogeneous media.
Qualitative analysis of a class of Bianchi V imperfect fluid cosmologies
Authors: Coley A. A. , Dunn K. A.
Published: July 2002

It has been shown that by utilizing a set of ‘‘dimensionless’’ equations of state, the Einstein field equations governing Bianchi V imperfect fluid cosmologies reduce to a plane-autonomous system of equations. This plane-autonomous system shall be investigated here, and the qualitative behavior of the underlying cosmological models shall thereby be obtained.
Boundary terms in Lovelock gravity from dimensionally continued Chern–Simons forms
Authors: Verwimp Theo
Published: July 2002

For a space-time of dimension D with nonempty boundary, it was shown by Müller-Hoissen [Nucl. Phys. B 337, 709 (1990)] that every term in the boundary-terms-modified Lovelock action results from the dimensional continuation of the index theorem for the de Rham complex from an even dimension lower than D. Here, the surface corrections are expressed in terms of dimensionally continued Chern–Simons forms and the cancellation of boundary integrals is checked explicitly by independent variations with respect to the D-bein form and the connection one-form.
Concomitants theory and renormalization in even dimensions
Authors: Castagnino M. , Domenech G. , Frolov V. , Noriega R. J. , Schifini G.
Published: July 2002

The complete set of concomitants is given of the metric, a scalar function, and their derivatives in six dimensions without imposing conditions on the order of the derivatives and their properties are studied under conformal transformations. These results are useful in several physical problems, like finding 〈φ2〉ren and 〈Tμν〉ren in any geometry.
Space-times with plane-symmetric scalar waves
Authors: Li Jianzeng
Published: July 2002

The general metric yielded by the plane-symmetric scalar wave has been found. It is either static or spatially homogenous. Its symmetries and singularities are also investigated.
A note on Borel summability
Authors: Rezende Jorge
Published: July 2002

In this article Nevanlinna’s theorem about Borel summability is improved. Moreover, the Borel summability of the asymptotic expansion in the method of stationary phase in infinitely many dimensions is discussed.
Group interpretation of the spectral parameter in the case of nonhomogeneous, nonlinear Schrödinger system
Authors: Cieśliński Jan
Published: July 2002

A one-parameter group inserting the spectral parameter into the nonparametric Lax pair of the nonhomogeneous, nonlinear Schrödinger system is presented and discussed. Being a nonlocal extension of a Lie point transformation this group is probably not contained in any known class of symmetries.
Lattice soliton: Solutions to the Kadomtsev–Petviashvili equation with positive dispersion
Authors: Tajiri Masayoshi , Murakami Youichi
Published: July 2002

Lattice soliton solutions that have doubly periodic array of the localized structures in the plane are presented to the Kadomtsev–Petviashvili equation with the positive dispersion by using the superposition of the rational functions. Each structure is similar to the rational soliton. The existence condition and some properties are also presented.
Space-time extensions I
Authors: Rácz István
Published: July 2002

The local extendibility of causal geodesically incomplete space-times is examined. It is shown that for a space-time including an incomplete inextendible causal geodesic curve γ there exists a particular Ck (resp. Ck−) local extension provided that the curvature and its covariant derivatives are uniformly bounded up to order k+1 (resp. k) in a synchronized frame field along a family of causal geodesics in some neighborhood of a final segment of γ and the strong causality condition is satisfied locally.
First integrals and symmetries of time-dependent Hamiltonian systems
Authors: Dewisme Alain , Bouquet Serge
Published: July 2002

In this paper, first integrals of one-dimensional time-dependent Hamiltonians, H(q,p,t), are studied. First of all, it is shown that the second first integral, J(q,H,t)=t−∫dq/Hp, valid for autonomous Hamiltonians H(q,p), can be extended to the case of nonautonomous Hamiltonians H(q,p,t), provided Hp (resp. t) is replaced with Ip [where I(q,p,t) is a first integral of H(q,p,t)] (resp. a new time T): J(q,I,T)=T−∫dq/Ip. In a second part, we derive a technique leading to a first integral, I(q,p,t), from a Lie symmetry. As an example, this technique, together with the expression of the second first integral, J(q,I,t), is used to completely integrate a special case of the linear harmonic oscillator with a time-dependent frequency. As a result, Lewis’ first integral is recovered from a Lie symmetry in a way that generalizes the recent work of Shivamoggi and Muilenberg [Phys. Lett. A 154, 1 (1991)].
Möbius transform pairs
Authors: Millane R. P.
Published: July 2002

A continuous version of a theorem from classical number theory due to Möbius has recently been derived and used to solve some inverse problems in physics that are otherwise difficult to solve explicitly. Using standard results, a table of Möbius transform pairs is presented that may be useful in identifying other applications of the transform to physical problems.
Free braided differential calculus, braided binomial theorem, and the braided exponential map
Authors: Majid S.
Published: July 2002

Braided differential operators ∂i are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang–Baxter matrix R. The quantum eigenfunctions expR(x‖v) of the ∂i (braided-plane waves) are introduced in the free case where the position components xi are totally noncommuting. A braided R-binomial theorem and a braided Taylor theorem expR(a‖∂)f(x)=f(a+x) are proven. These various results precisely generalize to a generic R-matrix (and hence to n dimensions) the well-known properties of the usual one-dimensional q-differential and q-exponential. As a related application, it is shown that the q-Heisenberg algebra px−qxp=1 is a braided semidirect product C[x]×C[ p] of the braided line acting on itself (a braided Weyl algebra) and similarly for its generalization to an arbitrary R- matrix.
Fluctuations of a spherical gravitational impulsive wave
Authors: Hortaçsu M.
Published: July 2002

It is shown that quantum fluctuations, in particular vacuum polarization, vanish in the background of a spherical impulsive wave solution of the Einstein field equations, recently found by Nutku and Penrose. The calculation is done in first-order perturbation theory but arguments are given why it should persist to all orders.
A new method to introduce the Möbius function and the Möbius transform
Authors: Chen Nan-xian
Published: July 2002

The Möbius function and Möbius transform are introduced and derived in a manner that parallels a typical inverse problem of physics, rather than in the usual number-theoretic manner.
Group-theoretic solution of the scalar wave equation in a multiply connected domain
Authors: Singh S. N. , Singh G. S.
Published: July 2002

The solution of the two-dimensional scalar Helmholtz wave equation is discussed in a domain consisting of N+2 boundaries wherein there is one central boundary and N circles are located symmetrically within a closed space such that the system as a whole has N-fold rotational symmetry. The general solution for this complicated domain has been found to reproduce the results corresponding to simpler structures with lesser number of boundaries. Furthermore, the rotational symmetry of the system is exploited to obtain the group-theoretic solution of the composite problem leading to symmetry-induced simplifications. The numerical results pertaining to the lowest and some of the higher modes are presented for N=1 and 2. The plot of the lowest eigenvalue as a function of eccentricity exhibits maximum for symmetric as well as antisymmetric modes whenever there are at least two inner boundaries. This behavior has universal character in the sense that the maximum shows up irrespective of the values of the geometrical parameters other than the eccentricity. The computations involving the group-theoretic aspects have been found to have fast convergence highlighting the significance of our approach to deal with the solution for a class of multidisciplinary problems coming within the purview of the Helmholtz equation.
Taylor series expansions for eigenvalues and eigenfunctions of parametrized composition operators
Authors: Thron Chris
Published: July 2002

Operators of the form Tβf(x)=∑Jj=1 φj(βx)f(σj +ρjx) are considered herein where β is a real parameter; σj and ρj are real numbers, and {φj} are holomorphic functions which satisfy certain additional conditions. Transfer operators of this form appear in statistical physics: the leading eigenvalue may be interpreted as the free energy of a lattice of interacting particles, as a function of inverse temperature β. A method for obtaining Taylor series expansions in β of the leading and second leading eigenvalues, and the leading eigenfunction are given. The method may be extended to find series expansions for other eigenvalues and eigenfunctions. The radii of convergence of these series have not been found. These results generalize the author’s findings in another article [J. Math. Phys. 32, 2718 (1991)].
Energy-momentum conservation in gauge theories
Authors: van den Heuvel B. M.
Published: July 2002

Using the formulation in terms of principal fiber bundles, it is shown that the external symmetry implies the conservation of the energy-momentum tensor for a general gauge theory. When applying this theory to the case of Dirac fields, one is naturally forced to replace the dependence on the metric by the dependence on the underlying spin structure in the variational problems.
Noncommutative differential geometry on deformed quantum mechanical phase spaces
Authors: Parashar Preeti , Bhasin V. S. , Soni S. K.
Published: July 2002

The problem of formulating noncommutative differential geometry on multiparametric deformations of arbitrary quantum mechanical phase spaces involving bosonic or (even or odd-dimensional) fermionic variables is investigated. A suitable enlargement of the basic quadratic algebras enables one to solve all the consistency conditions on the calculus with the help of plausible ansatzes.
The relativistic Hermite polynomial is a Gegenbauer polynomial
Authors: Nagel Bengt
Published: July 2002

It is shown that the polynomials introduced recently by Aldaya, Bisquert, and Navarro-Salas [Phys. Lett. A 156, 381 (1991)] in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of Gegenbauer polynomials. This fact is useful in the investigation of the properties of the corresponding wave function. Some examples are given, in particular, related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers.
On convergence to equilibrium for Kac’s caricature of a Maxwell gas
Authors: Gabetta E. , Toscani G.
Published: July 2002

The trend towards equilibrium of the solution to the one-dimensional Kac’s caricature of a Maxwell gas is investigated. When the initial value has finite energy and finite Linnik’s entropy, it is proven that the solution converges in L1∩L∞ to the equilibrium Maxwellian distribution.
On the number of states bound by one-dimensional finite periodic potentials
Authors: Sassoli de Bianchi M. , Di Ventra M.
Published: July 2002

Bound states and zero-energy resonances of one-dimensional finite periodic potentials are investigated, by means of Levinson’s theorem. For finite range potentials supporting no bound states, a lower bound for the (reduced) time delay at threshold is derived.
A phase-space technique for the perturbation expansion of Schrödinger propagators
Authors: Barvinsky A. O. , Osborn T. A. , Gusev Yu. V.
Published: July 2002

A perturbation theory for Schrödinger and heat equations that is based on phase-space variables is developed. The Dyson series representing the evolution kernel is described in terms of two basic classical quantities: the free classical motion along flat space geodesics and the Green function for the Jacobi operator in phase space. Further, for problems with Abelian interactions it is demonstrated that the perturbation theory may be summed to all orders yielding an exponentiated connected graph description for the evolution kernel. Connected graph representations provide an efficient method of constructing various semiclassical approximations wherein expansion coefficients are directly determined by explicit cluster integrals. This type of application is discussed for the case of Schrödinger and heat equations with external electromagnetic fields. Detailed expressions for coefficients are obtained for both the gauge invariant large mass expansion as well as the short time Schwinger–DeWitt expansion. Finally it is shown how to apply this phase-space method so that it incorporates a recently proposed covariant perturbation theory.
Tau functions, Korteweg–de Vries flows, and supersymmetric quantum mechanics
Authors: Sukumar C. V. , Guha P.
Published: July 2002

In this article the connection between the τ functions of the Korteweg–de Vries (KdV) hierarchy and reflectionless potentials with an arbitrary number of bound states constructed using supersymmetric quantum mechanics is studied. It is shown how the underlying KdV flows can be extracted from the construction based on supersymmetric quantum mechanics.
Knots and classical 3-geometries
Authors: Toh Tze-Chuen , Anderson Malcolm R.
Published: July 2002

It has been conjectured by Rovelli that there is a correspondence between the space of link classes of a Riemannian 3-manifold and the space of 3-geometries (on the same manifold). An exact statement of his conjecture will be established and then verified for the case when the 3-manifold is compact, orientable, and closed.
The Hamiltonian structures of the super-KP hierarchy associated with an even parity super-Lax operator
Authors: Barcelos-Neto J. , Ghosh Sasanka , Roy Shibaji
Published: July 2002

The even parity super-Lax operator is considered for the supersymmetric KP hierarchy of the form L=D2+∑∞i=0 ui−2D−i+1 and two Hamiltonian structures are obtained following the standard method of Gel’fand and Dikii. It is observed that the first Hamiltonian structure is local and linear whereas the second Hamiltonian structure is nonlocal and nonlinear among the superfields appearing in the Lax operator. Their connections with the super-w∞ algebra are discussed briefly.
Singular and unstable solutions of the Korteweg–de Vries hierarchy
Authors: Dorren H. J. S.
Published: July 2002

The Korteweg–de Vries (KdV) equation is solved using an inverse scattering transform approach using the continuous part of the spectrum of the Schrödinger equation. It is assumed that the reflection coefficient that corresponds to the initial condition of the KdV equation is a rational function of the wave number. It is shown that the Lyapunov exponent of the associated nonlinear evolution equation can be larger than zero if the reflection coefficient has poles that are close to zero. A positive Lyapunov exponent suggests that corresponding solutions of the KdV equation are unstable. This approach is generalized to the KdV hierarchy.
The contraction of the SU(1,1) discrete series of representations by means of coherent states
Authors: Renaud Jacques
Published: July 2002

The 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.
On the asymptotic integrability of a higher-order evolution equation describing internal waves in a deep fluid
Authors: Fokas A. S. , Grimshaw R. H. J. , Pelinovsky D. E.
Published: July 2002

A 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.
Variational method: How it can generate false instabilities
Authors: Kaup D. J. , Lakoba T. I.
Published: July 2002

When 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.
Perennials and the group-theoretical quantization of a parametrized scalar field on a curved background
Authors: Hájíček P. , Isham C. J.
Published: July 2002

The perennial formalism is applied to the real, massive Klein–Gordon field on a globally-hyperbolic background space–time with compact Cauchy hypersurfaces. The parametrized form of this system is taken over from the accompanying paper. Two different algebras 𝒮can and 𝒮loc of elementary perennials are constructed. The elements of 𝒮can correspond to the usual creation and annihilation operators for particle modes of the quantum field theory, whereas those of 𝒮loc are the smeared fields. Both are shown to have the structure of a Heisenberg algebra, and the corresponding Heisenberg groups are described. Time evolution is constructed using transversal surfaces and time shifts in the phase space. Important roles are played by the transversal surfaces associated with embeddings of the Cauchy hypersurface in the space–time, and by the time shifts that are generated by space–time isometries. The automorphisms of the algebras generated by this particular type of time shift are calculated explicitly. The construction of the quantum theory using the perennial formalism is shown to be equivalent to the Segal quantization of a Weyl system if the time shift automorphisms of the algebra 𝒮can are used. In this way, the absence of any timelike Killing vector field in the background space–time leads naturally to the ‘‘problem of time’’ for quantum field theory on a background space–time. Within the perennial formalism, this problem is formally identical to the problem of time for any parametrized system, including general relativity itself. Two existing strategies—the ‘‘scattering’’ approach, and the ‘‘algebraic’’ approach—for dealing with this problem in quantum field theory on a background space–time are translated into the language of the perennial formalism in the hope that this may give some insight into how the general problem can be solved. The non-unitary time evolution typical of the Hawking effect is shown to be due to global properties of the corresponding phase space: specifically, the time shifts map a global transversal surface to a non-global one. Thus, the existence of this effect is closely related to the global time problem.
Quasi-exactly solvable potentials on the line and orthogonal polynomials
Authors: Finkel Federico , González-López Artemio , Rodríguez Miguel A.
Published: July 2002

In this paper we show that a quasi-exactly solvable (normalizable or periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three-term recursion relation. In particular, we prove that (normalizable) exactly solvable one-dimensional systems are characterized by the fact that their associated polynomials satisfy a two-term recursion relation. We study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that its associated Stieltjes measure is supported on a finite set. From this we deduce that the corresponding moment problem is determined, and that the kth moment grows like the kth power of a constant as k tends to infinity. We also show that the moments satisfy a constant coefficient linear difference equation, and that this property actually characterizes weakly orthogonal polynomial systems.
An invariant imbedding analysis of general wave scattering problems
Authors: Haines A. John , de Hoop Maarten V.
Published: July 2002

The invariant imbedding technique, via the solution of a Riccati-type equation, is modified to calculate the wave fields inside and scattered from a strongly (laterally and vertically) heterogenous, anisotropic inclusion, which may be large but remains compact. The factorization underlying this approach is carried out with respect to direction of average power flow rather than the more conventional factorization with respect to local direction of propagation. The solution of the operator Riccati equation is related to the Dirichlet-to-Neumann map. The formulation is robust in the sense that it can handle a rather extreme range of modal wave speeds, and allows continuous as well as discontinuous medium variations on different (wave) length scales. It also, inherently, takes care of critical-angle phenomena. The algorithm, based on the invariant imbedding approach, yields the internal fields for a full survey of sources and receivers simultaneously. The wave field solution in the inclusion is coupled to the external field via a boundary element approach.
The behavior of nearby trajectories in magnetic billiards
Authors: Tasnádi Tamás
Published: July 2002

In chaos theory the separation of infinitesimally close trajectories has great importance. In present paper this behavior is investigated for classical magnetic billiard systems on Riemannian manifolds. The separation of the trajectories during the bounceless segments as well as at the reflections is studied generally, with a method similar to that of Jacobi fields for geodesic flows. For two-dimensional manifolds the results are also given in a natural coordinate frame, and they are illustrated in special (homogeneous) cases. We relate our issues to the known properties of the curvature of the horocycles, too.
Painlevé analysis of new higher-dimensional soliton equation
Authors: Porsezian K.
Published: July 2002

In this note, we prove that the recently proposed new higher-dimensional nonlinear partial differential equation admits the Painlevé property. We briefly discuss the integrability properties of the equation.
Transformation brackets between U(N)⊃SO(N)⊃SO(Na)⊕SO(Nb) and U(N)⊃U(Na)⊕U(Nb)⊃SO(Na)⊕SO(Nb)
Authors: Kota V. K. B.
Published: July 2002

Transformation brackets between U(N)⊃SO(N)⊃SO(Na)⊕SO(Nb) and U(N)⊃U(Na)⊕U(Nb)⊃SO(Na)⊕SO(Nb) for symmetric U(N) irreducible representations are derived for Na, Nb⩾2 by using bispherical coordinates in N=Na+Nb dimensions and a convolution identity for generalized Laguerre polynomials. The formula derived for the transformation brackets also applies to the special cases with Na⩾2, Nb=1 and Na=1, Nb⩾2.
Discrete time adiabatic theorems for quantum mechanical systems
Authors: Dranov A. , Kellendonk J. , Seiler R.
Published: July 2002

The theory of adiabatic asymptotics is adapted to systems with discrete time evolution. The corresponding theorems about the approximation of physical time evolution by the adiabatic time evolution are shown to hold true in a discrete setting.
Lattice approximation of quantum statistical traces at a complex temperature
Authors: Lukkarinen J.
Published: July 2002

We show that the addition of an imaginary part to the temperature in the Gibbs ensemble will not destroy the convergence of the standard lattice approximation scheme for quantum mechanical systems with Hamiltonians of the type Ĥ=p̂2/2m+V(x̂), provided the potential function V is real and bounded from below and it satisfies the condition ∫dnx exp(−tV(x))<∞ for all t>0. As a by-product we obtain an explicit bound for the real-temperature lattice kernels and a simple condition for the convergence of the real-temperature lattice expectation values of observables given by polynomially bounded functions.
The secular solutions of the linearized Korteweg–de Vries equation
Authors: Leblond H.
Published: July 2002

We study the inhomogeneous linearized Korteweg–de Vries (KdV) equation. It is solved by the inverse scattering transform method. The secular-producing terms on the right-hand side (rhs) are characterized in several ways: first we give a mathematical characterization as resonant terms. Second, the secular-producing terms are interpreted as conserved densities of the KdV equation. Third, it is checked that the removal of all linear terms from the rhs, polynomial in the solution of KdV, ensures the boundness of the solution of the linearized equation. Fourth, considering this solution itself as the rhs, we determine which part of it is secular producing, and which part is not.
Solutions of (2+1)-dimensional spin systems
Authors: Thomova Z. , Winternitz P. , Zakrzewski W. J.
Published: July 2002

We use the methods of group theory to reduce the equations of motion of two spin systems in (2+1) dimensions to sets of coupled ordinary differential equations. We present solutions of some classes of these sets and discuss their physical significance.
A proof of the dynamical version of the Bombieri–Taylor conjecture
Authors: de Oliveira César R.
Published: July 2002

An adaptation of the Bombieri–Taylor conjecture on diffraction has been applied to some time-dependent quantum systems. Here the problem is mathematically stated and a proof is presented. Simple examples give an idea as to what extent the conjecture is valid.
JWKB connection-formula problem revisited again via Borel summation
Authors: Fröman Nanny , Fröman Per Olof
Published: July 2002

Silverstone [Phys. Rev. Lett. 55, 2523–2526 (1985)] has expressed the opinion that the traditional version of the JWKB connection formulas associated with a classical turning point is incorrect and that the correct version follows from the Borel-based summability of the asymptotic expansions for the Airy functions. In the present paper we show that this assertion is incorrect. After the Borel summation of the asymptotic expressions for the Airy functions has actually been performed explicitly, there are no connection problems, and thus no connection formulas, but exact relations expressing the functions Ai(z) and Bi(z) in terms of the Whittaker function, although with two defects: Due to the derivation there appear false restrictions on arg z, and there is an artificial cut that coincides with the Stokes line along the real z-axis. We also show that these exact relations can be obtained more appropriately in a direct way with the aid of standard formulas available in handbooks, whereby the defects mentioned do not appear. On the other hand, when one uses truncated asymptotic series, the Stokes phenomenon gives rise to connection problems, and the connection formulas, which one uses for handling these connection problems, are inherently one-directional. This property of the connection formulas is demonstrated and discussed anew, in general terms as well as with special regard to the Airy functions.
Mathematical derivation of chiral anomaly in lattice gauge theory with Wilson’s action
Authors: Hattori Tetsuya , Watanabe Hiroshi
Published: July 2002

Chiral U(1) anomaly is derived with mathematical rigor for a Euclidean fermion coupled to a smooth external U(1) gauge field on an even dimensional torus as a continuum limit of lattice regularized fermion field theory with the Wilson term in the action. The present work rigorously proves for the first time that the Wilson term correctly reproduces the chiral anomaly.
Two-component formulation of the Wheeler–DeWitt equation
Authors: Mostafazadeh Ali
Published: July 2002

The Wheeler–DeWitt equation for the minimally coupled Friedman–Robertson–Walker-massive-scalar-field minisuperspace is written as a two-component Schrödinger equation with an explicitly “time”-dependent Hamiltonian. This reduces the solution of the Wheeler–DeWitt equation to the eigenvalue problem for a nonrelativistic one-dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler–DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler–DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definite-parity solutions. Moreover, it is shown that the mode expansion of both even- and odd-parity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt’s boundary condition at initial singularity.
Quantum Anosov flows: A new family of examples
Authors: Peter Ingo J. , Emch Gérard G.
Published: July 2002

A quantum version is presented for the Anosov system defined by the time evolution implemented by the geodesic coflow on the cotangent bundle of any compact quotient manifold obtained from the Poincaré half-plane. While the canonical Weyl algebra does not close under time evolution, the symplectic structure of these classical systems can be exploited to produce objects akin to the CCR algebras encountered in quantum field theory. This construction allows one to lift both the geodesic and the horocyclic flows to a Weyl algebra describing the quantum dynamics corresponding to the systems under consideration. The Anosov relations as proposed in Ref. Reference 1 are found to be valid for these models. A quantum version of the classical ergodicity of these systems is discussed in the last section.
Thermodynamic fermion loop in a constant magnetic field
Authors: Sato Haru-Tada
Published: July 2002

The one-loop effective potential of a thermodynamic fermion loop in a constant magnetic field is studied. As expected, it can be interpreted literally as a discretized sum of (D−2)-dimensional energy density above the Dirac sea. Large/small mass expansions of the potential are also examined.
A one-dimensional many-body integrable model from Zn Belavin model with open boundary conditions
Authors: Fan Heng , Hou Bo-Yu , Li Guang-Liang , Shi Kang-Jie , Wang Yan-Shen
Published: July 2002

We use a factorized L operator to construct an integrable model with open boundary conditions. By taking the trigonometric (τ→−1∞) and scaling limits (w→0), we get a Hamiltonian of a classical integrable system. It shows that this integrable system is similar to those found by Calogero et al. [J. Math. Phys. 10, 2191 (1969); 12, 419 (1971)].
Binary Darboux–Bäcklund transformations for the Manin–Radul super KdV hierarchy
Authors: Shaw Jiin-Chang , Tu Ming-Hsien
Published: July 2002

We construct the supersymmetric extensions of the Darboux–Bäcklund transformations (DBTs) for the Manin–Radul super Korteweg–de Vries (KdV) hierarchy using the super-pseudo-differential operators. The elementary DBTs are triggered by the gauge operators constructed from the wave functions and adjoint wave functions of the hierarchy. Iterating these elementary DBTs, we obtain not only Wronskian-type but also binary-type superdeterminant representations of the solutions.
The elliptic quantum algebra Aq,p(sln∧) and its bosonization at level one
Authors: Fan Heng , Hou Bo-yu , Shi Kang-jie , Yang Wen-li
Published: July 2002

We extend the work of Foda et al. and propose an elliptic quantum algebra Aq,p(sln∧). Similar to the case of Aq,p(sl2∧), our presentation of the algebra is based on the relation RLL=LLR*, where R and R* are Zn symmetric R matrices with the elliptic moduli chosen differently, and a scalar factor is also involved. With the help of the results obtained by Asai et al., we realize type I and type II vertex operators in terms of bosonic free fields for the Zn symmetric Belavin model. We also give a bosonization for the elliptic quantum algebra Aq,p(sln∧) at level one.
The Lanczos potential for vacuum space–times with an Ernst potential
Authors: Dolan P. , Muratori B. D.
Published: July 2002

By a particular choice of tetrad and coordinate systems, we find the Lanczos potential for general vacuum stationary axisymmetric space–times and for general vacuum cylindrically symmetric solutions. This result is particularly elegant and unifies the treatment of the Lanczos potential and the Ernst potential for vacuum stationary axially symmetric and cylindrically symmetric space–times. The Lanczos potential for such large classes of vacuum space–times is expressed very simply in terms of spin-coefficients for a null tetrad.
Symmetries of discrete dynamical systems involving two species
Authors: Gómez-Ullate D. , Lafortune S. , Winternitz P.
Published: July 2002

The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in other cases up to ten dimensional. The equations can describe the interaction of two long molecular chains, each involving one type of atoms.
Metric symmetries and spin asymmetries of Ricci-flat Riemannian manifolds
Authors: McInnes Brett
Published: July 2002

The Calabi–Yau and Joyce manifolds used in string and M-theory compactifications have no continuous groups of isometries, but they often have nontrivial discrete (actually finite) isometry groups. Discrete isometries of nonsimply connected Riemannian manifolds do not necessarily map spin structures into themselves, however; thus, inconsistencies are possible if a spin connection is used to construct the gauge vacuum. We consider this problem in detail and show how it may be avoided.
Generalized Lamé functions. I. The elliptic case
Authors: Ruijsenaars S. N. M.
Published: July 2002

We present and study a class of functions associated with the two-particle quantum relativistic Calogero–Moser system with elliptic interactions. The functions may be viewed as joint eigenfunctions of two independent commuting analytic difference operators, one of which is the defining quantum dynamics; The second one is obtained by interchanging the step size and the imaginary period. The functions depend on parameters that are dense in the natural parameter domain. In essence, they consist of products of Weierstrass σ-functions and plane waves. The zeros of the σ-functions satisfy a constraint system encoding both Schrödinger equations at once.
Frequency analysis based on general periodic functions
Authors: Wei Yuchuan
Published: July 2002

Following the sine–cosine function, the sawtooth wave, square wave, triangular wave, trapezoidal wave, and so on become new easily generated periodic functions in modern electronics. Similar to Fourier’s idea, a natural question is whether a signal can be considered as a superposition of easily generated functions with different frequencies. Therefore it is necessary to generalize Fourier analysis based on sine–cosine functions into frequency analysis based on general periodic functions. In this paper, we introduce the frequency series and frequency transformation based on general periodic functions. We discuss when a frequency system is a complete system or an unconditional basis in L2[−π,π], and when a frequency transformation can be carried out in L2(−∞,+∞). For practical convenience almost all easily generated functions in electronics are considered carefully as examples. As a new and practical generalization of classical Fourier analysis, these results will become a theoretical foundation for the technique of easily generated function analysis in signal processing.