bluuebunny/crossref_metadata_2025 huggingface dataset:
Application of nonstandard analysis to quantum mechanics
Authors: Farrukh M. O.
Published: March 2003

Quantum mechanics is formulated using a nonstandard Hilbert space. The concept of an eigen vector of a linear operator, which applies to standard as well as nonstandard Hilbert spaces, is replaced by the more general concept of an ultra eigen vector, which applies to nonstandard Hilbert spaces alone. Ultra eigen vectors corresponding to all spectral points of internal self−adjoint operators are proved to exist. This result enables us to set up a formalism which is equally valid for the discrete, as well as the continuous spectrum. Finally, Dirac’s formalism is reproduced, in a rigorous form within the nonstandard Hilbert space structure.
An algebraic representation of continuous superselection rules
Authors: Cirelli Renzo , Gallone Franco , Gubbay Bella
Published: March 2003

From the logic approach to quantum and classical mechanics, the W*−algebraic approach is deduced in dependence of a suitable ’’prestate.’’ An algebraic representation of the logic description is in fact constructed in a framework in which continuous superselection rules can be present. Logic propositions, observables, and states are represented by decomposable projections, decomposable self−adjoint operators, and normal states in a direct integral of Hilbert spaces. In this representation each algebraic term becomes the representative of a homologous logic one and the expectation values as well as the superselection rules are conserved. When a principle of ’’undistinguishability’’ is taken into account, the representation is faithful. In the classical case, the representation results in Koopman’s formalism.
Scattering of a finite beam in a random medium with a nonhomogeneous background
Authors: Beran M. J. , Whitman A. M.
Published: March 2003

We consider the scattering of a finite beam of radiation in a random medium with a nonhomogeneous background that contains linear and quadratic variation. We assume that both the fluctuations in the random medium and the background nonhomogeneity are weak. We obtain general expressions for the coherence function and intensity distribution. We present explicit solutions in the multiple scattering region. Results are compared to the case in which the background is homogeneous.
Clebsch−Gordan coefficients for crystal space groups
Authors: Berenson Rhoda , Birman Joseph L.
Published: March 2003

A practical method for calculating Clebsch−Gordan coefficients for crystal space groups is presented. It is based on properties of the space group irreducible representations as induced from ray representations of subgroups. Using this method, we obtain all Clebsch−Gordan coefficients for a family of representations in a single calculation: For space groups, for a given triangle of stars *k, *k′, *k″, where *k ⊗ *k′ - *k″, the coefficients for all allowable little group representations l, l′, l″ are obtained. In the following paper this is applied to rocksalt O5h−Fm3m and diamond O7h−Fd3m space groups.
Clebsch−Gordan coefficients for *X ⊗ *X in diamond O7h−Fd3m and rocksalt O5n−Fm3m
Authors: Berenson Rhoda , Itzkan Irving , Birman Joseph L.
Published: March 2003

By using the method described in the previous paper, based on properties of space group irreducible representations as induced from ray representations of subgroups, Clebsch−Gordan coefficients are calculated for *X ⊗ *X in diamond O7h−Fd3m and rocksalt O5h−Fm3m structures. Tables of coefficients for these stars are presented. An example of explicit calculation of the coefficients is given for these symmorphic and nonsymmorphic groups with multiplicity included in the former.
Variational methods and nonlinear forced oscillations
Authors: Hsieh D. Y.
Published: March 2003

An approximate, direct variational method, simple in concept, and straightforward in application is presented to deal with the problem of forced oscillation of nonlinear systems. The general procedure is illustrated in detail by treating a particular example, i.e., the Duffing’s equation. The same procedure is also applied to some other examples in mathematical physics.
Electrostatic screening
Authors: Rauch Jeffrey , Taylor Michael
Published: March 2003

Using the methods of partial differential equations and functional analysis, we investigate the electric field in the presence of a screen composed of wires of radius r spaced at distance R spread over a surface S. In the limit as r and R converge to zero if [R lnr]−1 → −∞, the field in the presence of the screen converges to the field with a conducting sheet spread over S. If [R lnr]−1 → 0, the field
A correction to the Sen and Dunn gravitational field equations
Authors: Jeavons J. S. , McIntosh C. B. G. , Sen D. K.
Published: March 2003

It is shown that the gravitational field equations proposed by Sen and Dunn do not follow from their variational principle and that in the vacuum case the correct field equations are related to Einstein’s field equations by a conformal mapping.
Some properties of asymptotically flat, static, Einstein−Maxwell space−times
Authors: Hallidy William
Published: March 2003

We show that the generators of the Weyl tensor for asymptotically flat, static, Einstein−Maxwell space−times are electric type to asymptotic order r−7 and that for space−times which are stable under static perturbations the generators of the electromagnetic field tensor are electric type to order r−5. We also argue that the Weyl tensor, the Ricci tensor, and the electromagnetic field tensor may fail to be electric type in general for asymptotically flat, static, Einstein−Maxwell space−times but that the generators of the Weyl tensor and the electromagnetic field tensor may still be electric type in general. The variable r is an affine parameter along null geodesics.
Explicit representations of a single parabose operator
Authors: Gray D. A. , Hurst C. A.
Published: March 2003

Representations for a single parabose operator are found as operators over a Bargmann and over a Schrödinger or harmonic oscillator space. The equivalence of the two representations is proved by the construction of a unitary integral transform connecting them.
Formula for the computation of the representation matrix elements of the group S
O (n)
Authors: Maekawa Takayoshi
Published: March 2003

The formulas of expressing the infinitesimal operators of the parameter groups of SO (n) in terms of the Euler angles are given. By using these, recursion relations which are useful for calculations of the representation matrix elements of SO (n) are obtained. Expressions for the d functions with the highest and some weights are obtained explicitly, and it is shown that the d functions of SO (2j+1) or SO (2j) with special weights agree with those of SO (3) or SO (4) respectively. By using the results, a formula of computing the D−matrix elements of SO (n) are given in terms of lowering operators corresponding to those of Pang and Hecht.
A new approach to the stability of matter problem. I
Authors: Federbush Paul
Published: March 2003

The stability of matter problem solved by Dyson and Lenard is studied by more field theoretic techniques. Stability is proven for matter in a periodic cube.
Wave propagation with stochastically coupled propagating and evanescent modes
Authors: Kohler Werner E.
Published: March 2003

The problem of electromagnetic wave propagation in a randomly perturbed waveguide is analyzed in the forward scattering approximation. Both propagating and evanescent modes are taken into account. Coupled power equations are derived in the asymptotic limit of long guide length and small perturbations. More generally, a diffusion equation is derived which governs the evolution of functions of the process in this same asymptotic limit. The resulting dynamical equations characterize the evolution of the propagating modes; the evanescent modes affect this propagation through their modification of the parameters in these equations. However, in the presence of evanescent modes, the forward scattering approximation leads to nonconservative coupled power equations; energy is apparently exchanged with the neglected backward waves through their coupling to the evanescent modes.
On Chapman−Kolmogorov equations for neutron population in a multiplying assembly
Authors: Belleni−Morante A.
Published: March 2003

We prove that the Chapman−Kolmogorov system for neutron population in a multiplying assembly admits a unique positive and norm invariant solution belonging to the Banach space of summable sequences. We then show that the standard kinetic equation can be deduced in a rigorous way from such a system of a countably infinite number of partial differential equations. Finally, we indicate how the Chapman−Kolmogorov initial−value problem can be approximated by a problem in a finite−dimensional space.
Tensor operators and twisted group algebras
Authors: Backhouse N. B.
Published: March 2003

Two generalizations of Frame’s theory of the conjugating representation of a finite group G are explored and applied to the problem of forming tensor operators out of a group algebra. In each case the group G acts on the group algebra A (Ḡ), where Ḡ either contains G or covers G as a central extension. By construction, an affirmative answer is given to the question, raised by de Vries and van Zanten, as to whether or not Ḡ may be found such that A (Ḡ) carries all the irreducible representations of G. Several examples are given, with the groups chosen from among the crystallographic point groups.
Useful extremum principle for the variational calculation of matrix elements. II
Authors: Gerjuoy Edward , Rosenberg L. , Spruch Larry
Published: March 2003

Recent work [Phys. Rev. A 9, 108 (1974)] on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Phys. Rev. A 9, 108 (1974) originally were based.
Lie theory and separation of variables. 6. The equation i
U
t + Δ2U = 0
Authors: Boyer C. P. , Kalnins E. G. , Miller W.
Published: March 2003

This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrödinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrödinger equation, many of which are new.
Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
Authors: Boyer C. P. , Kalnins E. G. , Miller W.
Published: March 2003

As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.
Low frequency coupling in the planar rectangular lattice
Authors: Twersky Victor
Published: March 2003

The solution for the three−dimensional problem of multiple scattering by a doubly periodic planar array of bounded obstacles is specialized to consider low frequency coupling effects for spheres. For all of the usual boundary conditions, we derive closed form approximations for the transmission and reflection amplitudes explicitly in terms of the physical parameters of the isolated scatterers, lattice spacings, and direction of incidence. The results are compared with those obtained earlier for the grating of parallel cylinders and for the periodic line (and planar random distribution) of spheres. We show, for example, that the pseudo−Brewster effects are anisotropic for the rectangular lattice cell and that the leading effects of close packing for a square array of rigid spheres may be approximated by single scattering by an equivalent spheroid representing elongation in the plane of the array and contraction along the normal. For dense packing at very low frequencies, we obtain explicit closed form approximations including up to octupole−octupole coupling effects.
Reversible dynamics in a proposition−state structure
Authors: Gorini Vittorio , Zecca Antonio
Published: March 2003

We propose an intrinsic definition of reversible dynamical evolution of a physical system based on a unified formulation of the principle of superposition within an axiomatic approach to quantum mechanics.
The Cauchy problem in general relativity. II. Main and constraint equations in the characteristic initial−value problem
Authors: d’Inverno R. A.
Published: March 2003

This paper applies the ’’yA−formalism’’ of a previous paper (Paper I) to a Lagrangian formulation of the characteristic initial−value problem in general relativity. The essential content of the paper is the respective identifications of the ’’field equations’’ and ’’Bianchi identities’’ of the yA−formalism with the ’’main equations’’ and ’’constraint equations’’ of a metric in a Bondi coordinate system. The identifications are developed in detail in the case of Bondi’s (axially symmetric) radiating metric.
The Weyl and Dirac equations in terms of functions over the group S
U2
Authors: Malin S.
Published: March 2003

Carmeli’s approach to Maxwell’s equations, in which the field variables are considered as functions of the group SU2, is extended to two−component Weyl spinors and four−component Dirac spinors. The Weyl and Dirac equations are formulated over the group SU2, the relevent functions are shown to be ’’quantities of spin−weight s=±1/2’’; they are expanded in a generalized Fourier series and the equations for the coefficients are derived. The ’’quantities of spin−weight s’’ are shown to be closely related to eigenvectors and eigenspinors of the radial spin operator ?⋅s.
Percolation on a Cayley tree with second nearest neighbor bonds
Authors: Eggarter T. P.
Published: March 2003

We derive the critical concentration for site percolation on a Cayley tree ’’decorated’’ with second nearest neighbor bonds. For a tree with connectivity K the result is ccrit =〈1+K−[(1+K)2−4K]1/2〉/2K.
A definition of quantum detection point processes
Authors: Bénard C.
Published: March 2003

A definition of quantum particle detection point processes is given for normal and locally normal systems. This is realized by building coincidence measurement operators, describing in a thoroughly quantum formalism the coincidence experiments that have been actually performed (photon) or that might be performed (electron, neutron). The quantum mean values of these coincidence operators are shown to be the coincidence probabilities of a mathematically well−defined regular process. Some applications of the latter result are given, in particular in the case of fermion point processes.
A model of quantum field theory treated in the Fock−Cook formalism. II
Authors: Schroeck Franklin E.
Published: March 2003

The Fock−Cook formalism for quantum field theory is further generalized, and the methods developed are used to remove all cutoffs from a variety of Yukawa−type interactions in one and two space dimensions. We conclude that the usual operator−valued distribution approach to quantum field theory is not always necessary.
Change of Petrov type under generation of solutions of Einstein’s equations
Authors: Sneddon G. E.
Published: March 2003

A method given by Geroch [J. Math. Phys. 12, 918 (1971)] for generating new solutions of the vacuum gravitational field equations from known solutions is investigated in order to find out how the Petrov type of the new metric will be related to that of the old metric. Conditions are found under which algebraically special metrics will generate algebraically special metrics. It is shown how the Petrov type of these metrics will change under the generation method of Geroch.
Adiabatic expansions of solutions of coupled second−order linear differential equations. I
Authors: Fulling S. A.
Published: March 2003

A generalized higher−order WKB approximation is found for the set of equations h″j(t) + u2 𝒥Nk=1 Mjk(t)hk(t) = 0 (u → ∞), when the coefficients Mjk form a positive definite Hermitian matrix M satisfying a smoothness condition as a function of t. In the construction, essential use is made of a transformation introduced by Kato to connect smoothly the eigenvectors of M(t) at different values of t. Eigenvalue degeneracies which exist for all t are covered by the method. The expansion breaks down at points t where the multiplicities of the eigenvalues of M(t) change; this phenomenon, analogous to the ’’turning point’’ problem of the ordinary WKB method, will be studied in a second paper. The asymptotic nature of the expansion is proved; error bounds can be extracted from the proof but are not studied here.
Vector and tensor fields on conformal space
Authors: Mayer D. H.
Published: March 2003

We consider in this paper vector and tensor fields on the compactified Minkowski space M4c and investigate their transformation properties under the group SO (2,4) of conformal transformations which are well defined on the manifold M4c contrary to their singular behavior on the pseudo−Euclidean noncompact Minkowski space M4. In writing down field equations on the manifold M4c, we get immediately the well−known conformal invariance of Maxwell’s equations. Also the problem of gauge transformations of the vector potential is treated both from the global point of view on the manifold M4c and from the local point of view on the Minkowski space M4. There we show which gauge instead of the so−called Lorentz gauge one has to choose to get a conformal covariant formulation of Maxwell’s equations on the pseudo−Euclidean noncompact Minkowski space M4.
Generalized free fields and the representations of Weyl group. II. Reducible representations
Authors: Lukierski J. , Sienkiewicz W.
Published: March 2003

The multiplets of generalized free fields, transforming according to triangle indecomposable representation of dilatation group are studied. Further, using the technique of Mellin transform, we represent an arbitrary generalized free field as a continuous superposition of fields with definite
Electrostatic or gravitational interaction energy of coaxial ellipsoids
Authors: Schultheis H. , Schultheis R.
Published: March 2003

An expression, which is appropriate for high−precision calculations, is derived for the interaction energy of two coaxial ellipsoidal distributions of charge or mass. By starting from Dirichlet’s expression for the potential it is shown that the (in general) sixfold integration can be considerably reduced by integrating over equipotential surfaces and introducing elliptic integrals. The evaluation of the final expression requires only a double numerical integration.
From a ’’laboratory’’ Galilei−Hilbert bundle to an algebra of observables
Authors: Abbati Maria C. , Cirelli Renzo , Gallone Franco
Published: March 2003

After a schematic examination of the ’’physical’’ representations of the Galilei group, a quantum mechanical description is drawn from a ’’laboratory’’ one in the case of a free elementary system with spin, assuming that the kinematical group is the Galilei group. In fact, the ’’laboratory states’’ of the system naturally fit in a Galilei−Hilbert bundle. Then, according to the general theory of the unitary representations of groups in the framework of Hilbert bundles which is outlined in this paper, a unitary representation of the Galilei group is constructed which is shown to contain all the physical Galilei representations. The quantum mechanical description which has been obtained in this way gives rise to an algebra of observables by means of a general procedure which connects a unitary representation of a Lie group with a Hilbert representation of the corresponding Lie algebra. The inner energy is shown to be a superobservable for the system under consideration. Moreover, the kinematical properties of such a system are independent of the values of the inner energy.
On the interaction of the electromagnetic field with heat conducting deformable semiconductors
Authors: de Lorenzi H. G. , Tiersten H. F.
Published: March 2003

The differential equations and boundary conditions describing the behavior of a finitely deformable, polarizable and magnetizable heat conducting and electrically semiconducting continuum in interaction with the electromagnetic field are derived by means of a systematic application of the laws of continuum physics to a well−defined macroscopic model. The model consists of five suitably defined interpenetrating continua. The relative displacement of the bound electronic continuum with respect to the lattice continuum produces electrical polarization, and electrical conduction results from the motion of the charged free electronic and hole fluids. Since partial pressures are taken to act in the conducting fluids, semiconduction boundary conditions arise, which have not appeared previously. The resulting rather cumbersome system of equations is reduced to that for the quasistatic electric field and static homogeneous magnetic field. In the absence of heat conduction, for the n−type semiconductor, nonlinear equations quadratic in the small field variables, for small fields superposed on a bias, are obtained from the latter, more tractable, system of equations. These small field equations reduce to four equations in four dependent variables. The linear portion of the small field equations is applied in the analysis of the propagation of both plane and surface waves in piezoelectric semiconductors subject to a static biasing electric field. On account of the aforementioned semiconduction boundary condition, the assumption of zero electric surface charge employed in previous treatments of the surface wave problem is not employed here.
Heat conduction model with finite signal speed
Authors: DeFacio B.
Published: March 2003

A simple physical model is derived which has a finite signal speed for a heat pulse in a linear or nonlinear medium. The assumptions which are required include continuum mechanics, thermodynamics of adiabatic processes, and a generalized Fourier heat law for the energy flux ℰ. Then we use singular perturbation theory with the finite signal speed u as a parameter to reduce the hyperbolic heat equation to the usual parabolic heat conduction equation.
Studies in the Kerr−Newman metric
Authors: Bose S. K.
Published: March 2003

The Kerr−Newman metric is analyzed according to the null tetrad formalism. The components of the Weyl and the Ricci tensors are calculated and these tensors are then projected on a suitable null−tetrad basis. The spin coefficients of Newman and Penrose are also calculated. These results are applied to obtain the equations of gravitational and neutrino perturbations in the Kerr−Newman metric.
Killing inequalities for relativistically rotating fluids
Authors: Hansen R. O. , Winicour Jeffrey
Published: March 2003

For rigidly rotating fluids in general relativity, it is shown that the angular momentum density is everywhere positive. This result depends on a global inequality satisfied by the Killing scalars. The inequality follows, via the Hopf theorem, from an elliptic equation (essentially one of the field equations) on the scalars. A derivation of the field equations in terms of the manifold of Killing orbits is presented. Possible generalizations of the result to systems with differential rotation or interior event horizons are discussed.
Lattice Green’s function for the body−centered cubic lattice
Authors: Inoue Michiko
Published: March 2003

We have shown that the lattice Green’s function at an arbitrary site with nearest neighbor interactions for the body−centered cubic lattice is expressed as a finite sum of products of the complete elliptic integrals of the first and the second kinds with real values of moduli for the entire range of energy.
Conformally flat solutions of the Einstein−Maxwell equations for null electromagnetic fields
Authors: McLenaghan R. G. , Tariq N. , Tupper B. O. J.
Published: March 2003

The spin coefficient formalism of Newman and Penrose is employed to obtain a direct derivation of the most general conformally flat solution of the source−free Einstein−Maxwell equations for null electromagnetic fields.
Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation
Authors: Habetler G. J. , Matkowsky B. J.
Published: March 2003

We consider initial−boundary value and boundary value problems for transport equations in inhomogeneous media. We consider the case when the mean free path is small compared to typical lengths in the domain (e.g., the size of a reactor). Employing the boundary layer technique of matched asymptotic expansions, we derive a uniform asymptotic expansion of the solution of the problem. In so doing we find that in the interior of the domain, i.e., away from boundaries and away from the initial line, the leading term of the expansion satisfies a diffusion equation which is the basis of most computational work in reactor design. We also derive boundary conditions appropriate to the diffusion equation. Comparisons with existing results such as the asymptotic and P1 diffusion theories, the PN approximation, and the extrapolated end point condition for these approximations, are made. Finally the uniform validity of our expansions is proved, thus yielding the desired error estimates.
On the strength of a system of partial differential equations
Authors: Schutz Bernard F.
Published: March 2003

It is shown that Einstein’s concept of the ’’strength’’ of a system of differential equations is directly related to the number of dynamical degrees of freedom the equations permit. It may become a useful tool for investigating the structure of the system.
Some restrictions on algebraically general vacuum metrics
Authors: Brans Carl H.
Published: March 2003

Vacuum Einstein metrics of Petrov type I, general, are considered. It is shown that the only solution of this sort in which one of the Petrov scalars is zero is the trivial flat−space one. Further, it is shown that the point at which the four Petrov scalars vanish simultaneously (zero curvature tensor) cannot be included as a regular point of a neighborhood over which the scalars are functionally independent. In fact, for type I all derivatives of the Petrov scalars must vanish at a point at which the curvature tensor does so that this point cannot be a regular point of any nontrivial analytic solution.
Theory of Regge poles for 1/r2 potentials. III. An exact solution of Schrödinger’s equation for arbitrary l and E
Authors: Mastalir Robert O.
Published: March 2003

We give the exact solution of the radial Schrödinger equation and derive the S matrix for arbitrary energy and angular momentum for a particular 1/r2 potential with the behavior r2V (r) = −V0 at r = 0 and r2V (r) = −V2 at r = ∞. We obtain an expression for the Regge poles and study their properties and distribution at low energy. The present results are in agreement with those obtained in a previous paper in this series.
Second order error in variational calculation of matrix elements
Authors: Gerjuoy Edward
Published: March 2003

The second order error δF(2), made by employing the Schwartz−Dalgarno−Delves variational principle for the diagonal matrix element φ†Wφ of an arbitrary Hermitian operator W, is examined in the case that φ is the bound ground state eigenfunction of some given Hamiltonian H. This variational principle characteristically involves not only a trial estimate φt of φ, but also a trial estimate Lt(φt) Lt(φt) of a well−defined but generally not exactly known auxiliary function L (φ). It has previously been shown that, for certain special choices of a trial Hamiltonian Ht, the trial Lt(φt) can be found from a minimum principle. The present work finds that for these same special Ht it is possible to express δF(2) in comparatively simple closed form, depending only on known quantities, so that δF(2) should be calculable when the system described by H is not too complicated. However, these results for δF(2) are obtained on the assumption that Lt(φt) is known essentially exactly for any given φt; the practical utility of the formulas derived still must be tested, therefore. If Lt(φt) can be determined to this necessary accuracy, one expects that combining the computed δF(2) with the usual variational estimate of φ†Wφ will be a significant improvement over the usual variational estimate alone. Under the same circumstances, when W is a positive definite operator, the expression for δF(2) can provide nonrigorous but nonetheless potentially useful second order (variational) upper and lower bounds on the exact φ†Wφ.
A note on the unified Dirac−von Neumann formulation of quantum mechanics
Authors: Tjo/stheim D.
Published: March 2003

It is demonstrated that the mathematical model of a rigged Hilbert space is ideally suited for obtaining a unified Dirac−von Neumann formulation of quantum mechanics. It is shown that the eigenbras of an observable A can be interpreted as weak derivatives of certain functionals associated with the resolution of identity Eu, u - (−∞, ∞), associated with A.
Mathematical aspects of kinetic model equations for binary gas mixtures
Authors: Greene David E.
Published: March 2003

A system of integrodifferential equations, which has a structure similar to the Boltzmann equations for a binary gas mixture and which qualitatively describes wave propagation, is investigated. The Oppenheim model is used and a linear initial−value problem is considered. The initial−value problem is shown to be well set mathematically with certain specifications on the initial distribution functions. Justification is made for the use of Fourier−Laplace transforms. A discussion is made of the dispersion relation and its analytic continuation. The roots σ (k) of the dispersion relation are shown to lie in three distinct regions of the σ plane: the hydrodynamic region, the semihydrodynamic region, and the rarefied region. It is established that the roots σ (k) are bounded by −1 + δ < Reσ ⩽ 0 under the assumption of plane−wave solutions which implies that the system is stable and that plane waves cease to exist if Reσ ⩽ −1 + δ.
Vestigial effects of singular potentials in diffusion theory and quantum mechanics
Authors: Ezawa H. , Klauder J. R. , Shepp L. A.
Published: March 2003

Repulsive singular potentials of the form λV (x) =λ‖x−c‖−α, λ≳0, in the Feynman−Kac integral are studied as a function of α. For α≳2 such potentials completely suppress the contribution to the integral from paths that reach the singularity, and thus, unavoidably, certain vestiges of the potential remain even after the coefficient λ↓0. For 2⩾α⩾1 careful definition by means of suitable counterterms at the point of singularity (similar in spirit to renormalization counter terms in field theory) can lead to complete elimination of effects of the potential as λ↓0. For α<1 no residual effects of the potential exist as λ↓0. In order to prove these results we rely on the theory of stochastic processes using, in particular, local time and stochastic differential equations. These results established for the Feynman−Kac integral conform with those known in the theory of differential equations. In fact, a variety of vestigial effects can arise from suitable choices of counter terms, and these correspond in a natural way to various self−adjoint extensions of the formal differential operator.
Pyramidal composition rules for Wronskians upon Wronskians
Authors: Brunet Robert C.
Published: March 2003

We give composition rules for Wronskians which have Wronskians as arguments. These pyramids of Wronskians are shown to reduce to products of Wronskians of different order. Many symmetric and antisymmetric regroupings of functions are then possible. The arithmetic of Wronskians can efficiently be reduced to the application of the ℏ⇊ of the LSZ formalism.
Extension of the Case formulas to L
p. Application to half and full space problems
Authors: Larsen E. W. , Sancaktar S. , Zweifel P. F.
Published: March 2003

The singular eigenfunction expansions originally applied by Case to solutions of the transport equation are extended from the space of Hölder-continuous functions to the function spaces Xp = {f‖μf (μ) - Lp}, where the expansions are now to be interpreted in the Xp norm. The spectral family for the transport operator is then obtained explicitly, and is used to solve transport problems with Xp sources and incident distributions.
An iterative method for solving a two-point boundary-value problem
Authors: Pao C. V.
Published: March 2003

The purpose of this paper is to study the existence, uniqueness, and method of construction of a nonnegative solution as well as the question of criticality for a two-point boundary-value problem arising in the transport process of n different types of particles in a rod of finite length subjecting incident fluxes and internal source. A recursion formula is derived for the calculation of the maximal and the minimal solution which are the respective limits of a monotonically nonincreasing sequence and a monotonically nondecreasing sequence. The behavior of these sequences leads to a characterization for the criticality question of the transport problem. It is shown under some physically reasonable conditions that the minimal and maximal solutions coincide so that it leads to a uniqueness theorem.
An explicit model for the renormalization of field equations
Authors: de Mottoni P. , Tesei A.
Published: March 2003

We provide a simple model where a precedure for renormalizing field equations yields nontrivial distribution solutions.
On the curvature dynamics for metric gravitational theories
Authors: Camenzind M.
Published: March 2003

Any ’’metric gravitation theory’’ (including general relativity) is shown to determine transport equations for the connection and curvature of the Lorentz frame bundle P4 defined by the metric g. Observers are generally defined as curves in P4 which project down to timelike trajectories in space–time. The transport of curvature along an observer trajectory is then given by a Lorentz Lie algebra-valued current composed of an internal and external part. Einstein’s equations are shown to define one part of the self-dual limit of the usual Yang–Mills gauge equations, here called a particular form of curvature dynamics. As a consequence, the Yang–Mills-like energy–momentum tensor, introduced for the Lorentz connection, vanishes identically under Einstein’s vacuum conditions.
The inverse scattering transform: Semi-infinite interval
Authors: Ablowitz M. J. , Segur H.
Published: March 2003

Certain nonlinear evolution equations can be solved on the semi-infinite interval by the method of inverse scattering. These equations are a subset of those which can be solved on the full interval. The equations have even dispersion relations when linearized, and are subject to appropriate homogeneous boundary conditions at the origin.
A note on coherent state representations of Lie groups
Authors: Onofri Enrico
Published: March 2003

The analyticity properties of coherent states for a semisimple Lie group are discussed. It is shown that they lead naturally to a classical ’’phase space realization’’ of the group.
The de Donder coordinate condition and minimal class 1 space–time
Authors: Krause Jorge
Published: March 2003

By means of immersion techniques a set of ’’adapted coordinates’’ are introduced as preferred coordinates for class 1 space–time. It is proved that the necessary and sufficient condition for the adapted coordinates to be harmonic coordinates is that class 1 space–time be a minimal variety. Some interesting features of the embedding approach to
Physical applications of multiplicative stochastic processes. III. Nonequilibrium entropy
Authors: Fox Ronald Forrest
Published: March 2003

It is argued that the expression − KB Trace [〈ρ (t) 〉 ln〈ρ (t) 〉], which appears in a stochastic treatment of the dynamics of the density matrix is indeed the nonequilibrium entropy. The reasoning involves consideration of the time evolution of the free energy for a relaxing magnetic moment in a fluctuating magnetic environment. It is shown that the H theorem, and the monotonic decrease of the free energy, as described by Pauli’s master equation can be generalized to the full density matrix, at least for the case of magnetic relaxation, which requires the presence of off-diagonal density matrix elements.
Cross sections in quantum mechanics
Authors: Campesino-Romeo Enrique , Taylor John R.
Published: March 2003

The definition of scattering cross sections requires an averaging over wavepackets with random impact parameters ρ; this leads to an integral of the scattering probability over all ρ in a plane perpendicular to the incident beam. We show that, for scattering off a potential which is O (1/rβ) as r→∞, the scattering probability is O (1/ρ2β−4) as ρ→∞. Thus for any β ≳ 3, the integral over impact parameters is well-defined and convergent.
Harmonic oscillator Green’s function from a BCH formula
Authors: Berendt G. , Weimar E. , Gilmore R.
Published: March 2003

An integral operator expression is formulated for the n-dimensional harmonic oscillator by exploiting the U (n) symmetry group of the oscillator Hamiltonian. The operator expression is disentangled using a Baker–Campbell–Hausdorff formula appropriate to the dynamical group Sp (4;R) of the Green’s function. The BCH formula is computed in a faithful matrix representation of Sp (4;R). It is sufficient to compute the disentangling theorem for the more restricted dynamical group SO (2,2).
A Goursat problem for the fourth moment equation
Authors: Oğuztöreli M. N.
Published: March 2003

The solution of a Goursat problem for the pseudoparabolic equation satisfied by the fourth statistical moment of an initially plane wave propagating in a random medium is presented, using an integro–differential equation technique. Two-dimensional propagations are considered.
On the stability of equilibrium states of finite classical systems
Authors: Lebowitz Joel L. , Aizemann Michael , Goldstein Sheldon
Published: March 2003

The state of a system is characterized, in statistical mechanics, by a measure ω on Γ, the phase space of the system (i.e., by an ensemble). To represent an equilibrium state, the measure must be stationary under the time evolution induced by the systems Hamiltonian H (x), x-Γ. An example of such a measure is ω (dx) = f (H) dx;dx is the Liouville (Lebesgue) measure and f (H (x)) is the ensemble density. For ’’nonergodic’’ systems there are also other stationary measures with ensemble densities, e.g., for integrable dynamical systems the density can be a function of any of the constants of the motion. We show, however, that the requirement that the equilibrium measure have a certain type of ’’stability’’ singles out, in the typical case, densities which depend only on H.
Self-interacting, boson, quantum, field theory and the thermodynamic limit in d dimensions
Authors: Baker George A.
Published: March 2003

By use of a finite volume, lattice approximation, we set up an approximation to the analytic continuation of a polynomial, self-interacting boson quantum field theory from Minkowski space to Euclidean space. The infinite volume limit for various boundary conditions is shown to exist and to be asymptotic to the perturbation expansion in the coupling constant g at g=0. For g:φ4:d theory we prove mass renormalizability and show how, by use of Nelson’s reconstruction theorem, the corresponding Minkowski space quantum field theory can be obtained. We discuss, at least for d?4, how statistical mechanical techniques, used to analyze the Ising model in the critical region just above the critical temperature, can be used to compute the properties of quantum field theory.
Higher symmetry of certain classical quadratic Hamiltonians
Authors: Wollenberg L. S.
Published: March 2003

It is proved explicitly that the classical n-dimensional isotropic harmonic oscillator is invariant under U (n). Two new examples of higher symmetry are found. One is the n-dimensional free particle which is shown to be invariant under a semidirect product which contains the n-dimensional Euclidean group as a proper subgroup.
Bäcklund transformation solutions of the Toda lattice equation
Authors: Chen Hsing-Hen , Liu Chuan-Sheng
Published: March 2003

Bäcklund transformation, superposition formula, and multisoliton solutions are constructed for the Toda lattice difference–differential equation with some discussion of its generalizations.
Is something missing in the Boltzmann entropy?
Authors: Forte B. , Sastri C. C. A.
Published: March 2003

A representation theorem for entropy functionals on the set of probability densities on the space Rn is proved. The important feature of the theorem is that the representation contains, in addition to the Boltzmann term, the continuous analog of the Hartley entropy as well as a term that, in statistical mechanics, corresponds to the chemical potential and is usually introduced ad hoc into the expression for entropy.
Evaluation of a class of lattice sums in arbitrary dimensions
Authors: Chaba A. N. , Pathria R. K.
Published: March 2003

Starting from the Poisson summation formula in m dimensions, a class of lattice sums is evaluated analytically. The resulting formulas are applicable to the electronic-structure studies of crystalline solids, the analysis of stability of quantized vortex arrays in extreme type-II superconductors and in rotating superfluid helium, and the investigation of Bose–Einstein condensation in finite systems.
Integrals of motion and resonances in a dipole magnetic field
Authors: Contopoulos G. , Vlahos L.
Published: March 2003

A method is developed for deriving a third integral of motion, besides the Hamiltonian and the angular momentum, of a charged particle, in a dipole magnetic field. This method is particularly useful in resonance cases, where the usual adiabatic invariants are not applicable. First the Hamiltonian is reduced to a ’’regular’’ form, i.e., its lowest order terms are written as H2=Φ12 + Φ22, where Φ12= (1/2)(a2+ p2a), Φ22= (1/2)(ω2b2+ p2b). Then the third integral can be constructed step by step as a series; in every resonance case a different form of the integral is derived. In the nonresonant cases the Hamiltonian is written in a normal form H=H* (Φ1, Φ2), where Φ1, Φ2 are canonical variables introduced by using von Zeipel’s method. The nonresonant orbits are quasiperiodic with frequencies ω1=∂H*/∂Φ1, ω2=ω∂H*/∂Φ2 and rotation number Rot=ω1/ω2. As an example the location of a particular resonance is found. The comparison with the numerical integrations is satisfactory.
Self-gravitating fluids with cylindrical symmetry
Authors: Letelier P. S.
Published: March 2003

The general solution of the Einstein field equation is obtained under the assumptions that (1) the source of the gravitational field is a perfect fluid with pressure p, equal to energy density ρ, (2) the space–time is cylindrically symmetric, and (3) the metric is given by three functions of two variables. The coordinate transformation to comoving coordinates is discussed. The energy and the Hawking–Penrose inequalities are studied. The singularities of a class of solutions is studied using the concept of velocity-dominated singularity. A relation between Einstein–Rosen waves and a class of solutions is shown.
Canonical transforms. III. Configuration and phase descriptions of quantum systems possessing an s
l (2,R) dynamical algebra
Authors: Boyer Charles P. , Wolf Kurt Bernardo
Published: March 2003

The purpose of this article is to present a detailed analysis on the quantum mechnical level of the canonical transformation between coordinate-momentum and number-phase descriptions for systems possessing an sl (2,R) dynamical algebra, specifically, the radial harmonic oscillator and pseudo-Coulomb systems. The former one includes the attractive and repulsive oscillators and the free particle, each with an additional ’’centrifugal’’ force, while the latter includes the bound, free and threshold states with an added ’’centrifugal’’ force. This is implemented as a unitary mapping—canonical transform—between the usual Hilbert space ℒ2 of quantum mechanics and a new set of Hilbert spaces on the circle whose coordinate has the meaning of a phase variable. Moreover, the UIR’s D+k of the universal covering group of SL (2,R) realized on the former space are mapped unitarily onto the latter.
Isolating constants of motion for the homogeneous turbulence of two and three dimensions
Authors: Lee Jon
Published: March 2003

For the inviscid eddy motion in a finite-dimensioned Fourier space, it is stated that energy and enstrophy are the isolating constants of motion for the 2D homogeneous turbulence. In contrast, the 3D isotropic turbulence has energy as the only constant of motion. If we relax the reflexional invariance, however, helicity emerges as another invariant; hence energy and helicity are said to be the isolating constants of motion for the helical turbulence. Although these are the key assumptions in the construction of equilibrium distributions, they have heretofore been accepted, without proof, as a natural property of the Navier–Stokes dynamics. This paper provides the proof. We have shown here that quadratic constants of motion for the individual triad-interactions collapse to energy–enstrophy in 2D, but to energy and helicity in 3D.
On angular momentum and channel coupling for a meromorphic many-channel S matrix
Authors: Cox Joseph R.
Published: March 2003

We consider a class of possible (i.e., not known a priori to be unitary) nonrelativistic many-channel S matrices meromorphic on their energy Riemann surfaces whose general form is suggested by the inverse scattering problem. These possible S matrices are associated with n (spinless) coupled channels of the same angular momentem l, and they reduce, in the one-channel limit, to the quotient of two polynomials of degree 2m in the wavenumber k. It is shown that the assumptions which imply unitarity for real energies of the open-channel submetrix also imply that no channels can be coupled when m?l.
Representations of the three-dimensional rotation group by the direct method
Authors: Torruella A. J.
Published: March 2003

The irreducible representations of the three-dimensional rotation group are obtained directly from the irreducible representations of its infinitesimal generators (the spin matrices), parametrized in terms of the rotation angle and the direction of the rotation axis. Expressions are given for the rotation operator exp(iψn ⋅ S) in terms of two different bases of 2j+1 elements for spin j. The results are related to the spectral decomposition of the rotation operator and expressions obtained for spin projection operators along any spatial direction for arbitrary spin.
The three-body time-delay operator
Authors: Osborn T. A. , Bollé D.
Published: March 2003

Using Faddeev’s form of time-dependent scattering theory, we give an abstract definition of time delay valid for multichannel scattering. For the three-body scattering problem we find an explicit relation, that is valid on the energy shell, between the time-delay operator and the S operators and their energy derivatives.
On the density of the Breit–Wigner functions
Authors: Perry W. L. , Luning C. D.
Published: March 2003

It is shown, for certain sequences {λi} in the complex plane, that linear combinations of the Breit–Wigner functions {Bi} approximate, in the mean square, any function in L2 (0,∞). Implications and numerical use of this result are discussed.