bluuebunny/crossref_metadata_2025 huggingface dataset:
Off-energy-shell results for scattering by a nonlocal potential. I
Authors: Talukdar B. , Das U. , Mukhopadhyaya S.
Published: March 2003

The importance of analyzing the off-shell effects due to scattering by nonlocal potentials is emphasized. Analytical expressions for l-wave off-shell wavefunctions associated with Jost (irregular) and physical (outgoing wave) boundary conditions are derived for an N-term separable potential by using a differential equation approach. The half-off-shell and fully off-shell T matrices are expressed in terms of appropriate Jost functions, Fredholm determinants, and transforms of the form factors of the potential. The general results presented are then used to construct exact expressions for T matrices for Tabakin, Beregi, and Mongan potentials. In limiting cases, each of our results can be seen to yield the off-shell T matrix for the one-term Yamaguchi potential calculated by other techniques.
Off-energy-shell results for scattering by a nonlocal potential. II
Authors: Talukdar B. , Das U. , Mallick N. , Sen Gupta M.
Published: March 2003

The wavefunction approach to off-shell scattering on nonlocal potentials is used to obtain half-off-shell and off-shell K matrices in terms of Jost functions and the Fredholm determinant associated with the off-shell principal value wavefunction. The results are employed to construct exact expressions for K matrices for Tabakin, Beregi, and Mongan case IV potentials.
Extension theorems for operator-valued measures
Authors: Christensen Mark J.
Published: March 2003

Benioff [J. Math. Phys. 13 (1972)] has shown that every consistent, countable family of positive, normalized operator-valued (PNOV) measures μn over Rn can be extended to a PNOV measure over R∞. In this paper we show that the same result holds for arbitrary, consistent families of PNOV measures over complete, separable metric spaces. Further we show that, while there may be no extension if the topological conditions are relaxed, it is always possible to construct a related family of PNOV measure spaces which: (1) Are measures theoretically indistinguishable from the original spaces; (2) Have an extending PNOV measure. These results use developments in the theory of algebraic models of measures as initiated by Dinculeanu and Foias [Ill. J. Math. 12 (1968)] and applied to stochastic processes by Schreiber, Sun, and Bharucha-Reid [Trans. AMS 158 (1973)].
The quantum harmonic oscillator revisited: A new look from stochastic electrodynamics
Authors: de la Peña L. , Cetto A. M.
Published: March 2003

We apply the theory of stochastic electrodynamics to the study of the (nonrelativistic) harmonic oscillator by using the Fokker–Planck method. It is demonstrated that the equilibrium distribution in phase space is exactly equal to that given by quantum statistical mechanics, i.e., the corresponding Wigner distribution, and that analysis of this distribution by means of a decomposition in terms of canonical densities leads automatically to the usual description of quantum mechanics in terms of excited states. All fundamental equations of quantum mechanics are recovered as aprproximations to zero order in the radiation terms; the first-order terms lead to the radiative corrections predicted by quantum electrodynamics, namely, the decay of states and the Lamb shift of the energy levels. The necessary differences between both treatments of the oscillator and their implications are briefly discussed.
Higher Euler operators and some of their applications
Authors: Aldersley S. J.
Published: March 2003

A set of operators which are associated with the Euler–Lagrange operator is introduced. An analysis of the commutation properties of these new operators, which will be referred to as the higher Euler operators, leads to a generalization of the necessary conditions for an expression to be an Euler–Lagrange expression. A product rule is derived for the higher Euler operators. In the special case of the Euler–Lagrange operator this product rule is basic to simple proofs of sufficiency theorems for the existence of a Lagrangian given the potential Euler–Lagrange expressions. By considering a certain homogeneity property, a characterization of Lagrangians in terms of their Euler–Lagrange expressions is established. Examples of applications of this characterization are given. A general procedure is given for constructing equivalent (not necessarily scalar density) Lagrangians when the field functions are tensorial and the Euler–Lagrange expressions are tensor densities. These results give particular significance to one of the higher Euler operators.
Quasisteady primitive equations with associated upper boundary conditions
Authors: Gordon Paul
Published: March 2003

This paper presents another approach to the problem of modeling large scale atmospheric flow. The major thrust of the method is to search for quasi-steady-state phenomena. This leads to sets of diagnostic and predictive equations that differ from those presently in use. Another important feature of the analysis is the introduction of a slowly floating upper boundary. In addition to simplifying the question of boundary conditions at the upper boundary, the floating top requires a highly significant change in the set of diagnostic variables. Two possible upper boundary conditions are derived in conjunction with the floating top. The first assumes continuous flow at the upper boundary, while the second assumes a compression-wave type discontinuity. Two specific criteria are formulated for checking the validity of the quasi-steady-state model. One is a scale assumption, between the physical scale and the time scale. The other is the requirement that the solution of the diagnostic equations be the steady-state limit of the original time-dependent equations. Various examples are given in order to attempt to clarify the techniques and philosophy of this approach. In addition, a specific test case is solved numerically with three models: The fixed top quasi-steady-state model, the floating top quasi-steady-state model, and a hydrostatic model. At the same time various upper boundary conditions are tested and compared. The results of the investigation indicate several significant advantages in favor of the floating top quasi-steady-state model.
Clebsch–Gordan coefficients: General theory
Authors: Dirl R.
Published: March 2003

A general method is given for obtaining Clebsch–Gordan coefficients for finite groups, by considering the columns of the Clebsch–Gordan matrices as G-adapted vectors and by identifying the multiplicity index as special column indices of the Kronecker product. The matrix representations are assumed to be projective ones, however not necessarily belonging to equivalent factor systems.
The classical limit of quantum dissipative generators
Authors: Pulè J. V. , Verbeure A.
Published: March 2003

For finite quantum systems the classical limit of the general dissipative generator of a semigroup of completely positive maps is obtained, yielding a generalized Fokker–Planck generator.
First order estimates of the error in approximate calculations of scattering phase shifts
Authors: Darewych J. W. , Sokoloff J.
Published: March 2003

We discuss the question of error estimation in approximate calculations of scattering phase shifts. The Kato integral identity between the exact and approximate solutions is used as the starting point to determine an upper bound to the absolute value of the difference between the exact and approximate result. This bound involves the maximum value of the modulus of the exact wavefunction as a factor. For the potential scattering case it is shown that this maximum occurs in the asymptotic region, if the potential is monotonically decreasing (with decreasing r). For more general potentials simple calculable bounds to the maximum of the wavefunction are derived, for energies which are everywhere higher than the potential. The results are illustrated for scattering by an (attractive) exponential potential and are compared to bounds obtained previously by Bardsley, Gerjuoy, and Sukumar, who have used the Lippman–Schwinger equation to determine bounds to the maximum modulus of the wavefunction.
The Diophantine moment problem and the analytic structure in the activity of the ferromagnetic Ising model
Authors: Barnsley M. , Bessis D. , Moussa P.
Published: March 2003

We show that the intensity of magnetization I (z,x) where z=e−2βH and x=e−2βJ, for the ferromagnetic Ising model in arbitrary dimension, reduces, for rational values of x, to a Diophantine moment problem I (z) =∑∞0nkzk, where nk=∫Λ0σ (λ) λkdλ, σ (λ) is a positive measure, n0=1/2, and nk is integer for k≠0. The fact that the nk are positive integers puts very stringent constraints on the measure σ (λ). One of the simplest results we obtain is that for Λ<4, σ (λ) is necessarily a finite sum of Dirac δ functions whose support is of the form 4 cos2(pπ/m), p=0,1,2,...,m−1, with m a finite integer. For Λ=4, which correspond to the one-dimensional Ising model, we have the result that either I (z) is a rational fraction belonging to the previous class Λ<4, or I (z) = (1/2)(1−4z)−1/2 which corresponds precisely to the exact answer for dimension 1. For Λ≳4, which is associated with Ising models in dimension d⩾2 we show that all cases are reducible to Λ=6, by a quadratic transformation which transforms integers into integers and positive measures into positive measures. The fixed point of this type of transformation is analyzed in great detail and is shown to provide a devil’s staircase measure. Various other results are also discussed as well as conjectures.
Geometrical theory of contractions of groups and representations
Authors: Arnal Didier , Cortet Jean-Claude
Published: March 2003

The contractions of Lie groups and Lie algebras and their representations are studied geometrically. We prove they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act. These deformations are given by Dirac constraints which induce on C∞ functions on the deformed manifold an associative twisted product, characterizing the contracted group or its representations. We treat the contractions of SO(n) to E(n) and apply this theory to thermodynamical limits in spin systems.
An attempt to separate the long and short range forces by Gaussian method
Authors: Malo Otieno
Published: March 2003

In the study of phase transition problems, short range forces (SRF) play a dominant role. A constructive and rigorous study of the effects of short range forces has yet to be given. It is suggested in the present paper that by separating long range forces (LRF) from the short range forces, it would be possible to estimate contributions to the virial expansion of the collective oscillations due to short range forces. The method of Stratonovich or the functional integration technique is employed in the treatment of the interaction term of the partition function.
Geometry of spacetime founded on spacelike metric
Authors: Montesinos Angel
Published: March 2003

The first part of this paper contains new mathematical techniques for describing a spacetime anisotropy as suggested by the violation of parity conservation. Geometric measures of spacetime involve both the laboratory doing them and the events upon which these measures are done. The time form c and the spacelike length γ are the basic issues of those measures. Both depend on events and also on the timelike direction of the laboratory. Relativity tells that the field γ−c ⊗c depends, on the contrary, on events only; in this sense, relativistic spacetime is isotropic. If γ and c do not have that property, the manifold where the observable geometry takes place must be the set of timelike directions. The geometric structure of this manifold given by c and γ is studied in detail. The second part of the paper contains the study of a line of thought opposite to chronogeometry: Building the geometry from lengths instead of times. The datum is γ; through the conditions of stationary spacelike volume and of stationary proper time, a class of time forms and a gauge are obtained under some weak restrictions. Newtonian and relativistic spacelike metrics fulfill these restrictions. Standard connections are induced; they define the absolute derivative of physical fields and the geometric structure of the manifold of timelike directions. The paper ends with some comments about the remaining problem: to suggest and justify field equations.
The Coulomb and Coulomb-like off-shell Jost functions
Authors: van Haeringen H.
Published: March 2003

The off-shell Jost functions are studied for a potential which is the sum of the Coulomb potential and an arbitrary local short-range central potential. We derive their singular on-shell behavior and their connection with the pure Coulomb off-shell Jost functions. For the latter we derive a large variety of interesting explicit analytic expressions which are useful for various purposes.
Matrix products with application to classical statistical mechanics
Authors: Novotny Mark A.
Published: March 2003

A matrix product that generalizes the Kronecker matrix product is introduced and its properties are documented. This product is applied to classical statistical mechanics where its trace properties lead to both the transfer matrix and graphical expansion methods of evaluating the partition function.
Jost solutions and Green’s functions for the three-dimensional Schrödinger equation
Authors: Moses Harry E.
Published: March 2003

In an earlier paper, in which the minimal scattering data needed to reconstruct the scattering potential was found for the three-dimensional inverse problem, a Green’s function appeared very naturally in this context. We use this Green’s function to construct Jost solutions for the three-dimensional problem which are closely analogous to those for the one-dimensional problem. The Green’s function and the Jost solutions differ from those given by Faddeev. The completeness relations of the new Jost solutions are given simply in terms of the scattering amplitude. They are the same as those given in an earlier attempt to obtain an algorithm of the Gelfand–Levitan type for the three-dimensional problem. However, it is not yet clear that the Jost solutions of the present paper are the same as those of the earlier paper, since different methods are used to define the two sets of Jost solutions. The one-dimensional problem is discussed in some detail to motivate our definition of the Jost solutions for the three-dimensional problem. The present paper is the first of three papers which report on research arising from the three-dimensional inverse problem.
Adiabatic expansions of solutions of coupled second-order linear differential equations. II
Authors: Fulling S. A.
Published: March 2003

The ’’phase-integral’’ approach to higher-order WKB approximations, associated with the names of Fröman and Chakraborty, is generalized to systems of equations, written in vector notation as h″ (t)+u2 M(t)h(t) =0, where M is positive definite and u→∞. Expansions are constructed in the form h∼p−1/4e exp (−iuℱt p−1/2 dt′), where p and e are power series in u−1 and e is (asymptotically) a unit vector. Expanding out the higher-order terms in the exponential yields the approximations studied in the first paper of this series, which are less uniform in t. (A search for still greater uniformity leads to nonlinear differential equations for the leading term in e, which can be explicitly solved only in special cases, notably that in which all eigenvalues of M are distinct.) The expansions are proved to be valid asymptotic approximations on finite intervals where the eigenvalues of M are strictly positive and do not cross (i.e., the multiplicity does not depend on t).
Some exceptional electrovac type D metrics with cosmological constant
Authors: Plebański Jerzy F. , Hacyan Shahen
Published: March 2003

We investigate all type D solutions of the Einstein–Maxwell equations (with cosmological constant) such that the Debever–Penrose vectors are aligned along the two eigenvectors of the electromagnetic field, in the special case when a direct generalization of the Goldberg–Sachs theorem is not possible. A solution is found which admits no Killing vectors. We also present an extension of the Golberg–Sachs theorem valid for type D metrics.
SU(4) Clebsch–Gordan coefficients for the formation of baryonium and exotic baryons
Authors: Anderson Ronald , Joshi G. C.
Published: March 2003

We give the SU(4) Clebsch–Gordan coefficients required for the formation of exotic hadronic states of the form (QQQ̄Q̄) and (QQQ̄QQ).
SU(2,1) generation of electrovacs from Minkowski space
Authors: Hauser I. , Ernst F. J.
Published: March 2003

For every nonnull Killing vector K of any given electrovac, there exists a group of tranformations ℋK of the gravitational and electromagnetic potentials of Ernst. This is the group which is a nonlinear representation of SU(2,1) and was developed by Kinnersley on the basis of work by Ehlers, Harrison, and Geroch. For every K of Minkowski space (MS), we compute the set ℋK(MS) of all electrovacs derived from MS by noniterative application of HK; the results include appropriate null tetrads, the connection forms, the conform tensors, and (in the discussion) the group of all motions of every member of every ℋK(MS). Each conform tensor is type Npp (plane gravitational wave) or type D, and the principal null vector(s) are also eigenvectors of the Maxwell field. Except for those K which represent infinitesimal rotations about a timelike 2-surface of MS followed by null translations in that 2-surface, each K has a corresponding MS Killing vector L such that the G2 generated by K and L has nonnull surfaces of transitivity and is invertible. The discussion covers properties of the principal null rays and the Maxwell fields, Killing tensors of the results (one of the Npp families admits an irreducible Killing tensor of Segre characteristic [(11)(11)]), and the precise conditions under which a Killing vector of an electrovac is also an MS Killing vector. Also, some deductions are made concerning the Petrov class and principal null ray properties of the second generation electrovacs which would result from further applications of SU(2,1). Those points of MS which are possible singularities of electrovacs in ℋK(MS) are classified. The conditions under which an electrovac in ℋK(MS) has all of R4 (except for curvature singularities) as its domain are found; in particular, such an extension to R4 exists whenever the one-parameter group generated by K has no fixed points or whenever one restricts ℋK to the Ehlers or Harrison transformations.
An algebraic approach to Coulomb scattering in N dimensions
Authors: Rasmussen W. O. , Salamó S.
Published: March 2003

Using purely algebraic techniques, based on the larger symmetry group of the Kepler problem, the phase shifts and the scattering amplitude for Coulomb scattering in N dimensions are derived.
Coupled inertial and gravitational effects in the proper reference frame of an accelerated, rotating observer
Authors: Li Wann-Quan , Ni Wei-Tou
Published: March 2003

To the second order in metric and the first order in equations of motion in the local coordinates of an accelerated rotating observer, the inertial effects and gravitational effects are simply additive. To look into the coupled inertial and gravitational effects, we derive the third-order expansion of the metric and the second-order expansion of the equations of motion in local coordinates. Besides purely gravitational (purely curvature) effects, the equations of motion contain, in this order, the following coupled inertial and gravitational effects: redshift corrections to electric, magnetic, and double-magnetic type curvature forces; velocity-induced special relativistic corrections; and electric, magnetic, and double-magnetic type coupled inertial and gravitational forces. An example is provided with a static observer in the Schwarzchild spacetime.
Generation of the Lewis solutions from the Weyl solutions
Authors: Tanabe Yukio
Published: March 2003

A method is presented which can be used to generate an infinite series of new solutions of stationary axially symmetric Einstein–Maxwell equations. It is shown that if we start with the Weyl solutions the Lewis solutions are generated as a member of this series.
Transport scattering coefficients from reflection and transmission measurements
Authors: McCormick N. J.
Published: March 2003

Moments of the emerging distributions for the one-speed homogeneous slab albedo problem are manipulated to provide a scheme for calculating the expansion coefficients of the scattering kernel of the medium. Simple equations for the highest-order and next-highest-order coefficients are given, and suggestions for their possible use in experiments are provided.
Nth-order multifrequency coherence functions: A functional path integral approach
Authors: Rose Conrad M. , Besieris Ioannis M.
Published: March 2003

A functional (or path) integral applicable to a broad class of randomly perturbed media is constructed for the nth-order multifrequency coherence function (a quantity intimately linked to nth-order pulse statistics). This path integral is subsequently carried out explicitly in the case of a nondispersive, deterministically homogeneous medium, with a simplified (quadratic) Kolmogorov spectrum, and a series of new results are derived. Special cases dealing with the two-frequency mutual coherence function for plane and beam pulsed waves are considered, and comparisons are made with previously reported findings.
Current responses of all orders in a collisionless plasma. II. Homogeneous plasma
Authors: Larsson J.
Published: March 2003

Expressions for the admittance tensors of all orders are obtained for a relativistic magnetized Vlasov–Maxwell plasma. Symmetries, from which the Manley–Rowe relations follow, are explicit in the expressions obtained. The treatment is covariant.
Representations of the Poincaré group for relativistic extended hadrons
Authors: Kim Y. S. , Noz Marilyn E. , Oh S. H.
Published: March 2003

Representations of the Poincaré group are constructed from the relativistic harmonic oscillator wave functions which have been effective in describing the physics of internal quark motions in the relativistic quark model. These wave functions are solutions of the Lorentz-invariant harmonic oscillator differential equation in the ’’cylindrical’’ coordinate system moving with the hadronic velocity in which the time-separation variable is treated separately. This result enables us to assert that the hadronic mass spectrum is generated by the internal quark level excitation, and that the hadronic spin is due to the internal orbital angular momentum. An addendum relegated to PAPS contains discussions of detailed calculational aspects of the Lorentz transformation, and of solutions of the oscillator equation which are diagonal in the Casimir operators of the homogeneous Lorentz group. It is shown there that the representation of the homogeneous Lorentz group consists of solutions of the oscillator partial differential equation in a ’’spherical’’ coordinate system in which the Lorentz-invariant Minkowskian distance between the constituent quarks is the radial variable.
Theory of strings and membranes in an external field. II. The string
Authors: Aurilia A. , Christodoulou D.
Published: March 2003

We apply the general formulation of the theory of extended systems developed previously to the case of a relativistic string in an external field. This problem is relevant for the construction of models of strongly interacting particles. When the external field is static and uniform, explicit solutions are presented and classified according to their symmetry properties.
A Lie group framework for composite particles and mass spectrum
Authors: Beau Daniel , Horchani Salah
Published: March 2003

We propose a concept of internal structure and a relativistically covariant method of unifying the external and internal structures, leading to a dynamical Lie algebra without superfluous generators. In this framework we study in more detail a Lie algebra unifying external space with an internal 3-space, and several representations which describe models of composite particles and give rise to various mass formulas capable of describing the hadron spectrum; we make use of both unitary irreducible global representations and partially integrable, Schur-irreducible, symmetric local representations.
A uniqueness result in the Segal–Weinless approach to linear Bose fields
Authors: Kay Bernard S.
Published: March 2003

We prove a theorem, which, while it fits naturally into the Segal–Weinless approach to quantization seems to have been overlooked in the literature: Let (D,σ) be a symplectic space, and 𝒯 (t) a one parameter group of symplectics on (D,σ). Let (ℋ, 2Im〈⋅ ‖ ⋅〉) be a complex Hilbert space considered as a real symplectic space, and U(t) a one-parameter unitary group on ℋ with strictly positive energy. Suppose there is a linear symplectic map K from D to ℋ with dense range, intertwining 𝒯 (t) and U(t). Then K is unique up to unitary equivalence.
Settling the question of the high-energy behavior of phase shifts produced by repulsive, strongly singular, inverse-power potentials
Authors: Fröman Nanny , Thylwe Karl-Erik
Published: March 2003

For repulsive, strongly singular, inverse-power potentials it is rigorously shown that the JWKB expression for the nonrelativistic phase shift tends to exactness in the high-energy limit. The hitherto open question as to the correct expression for the leading term in the high-energy expansion of the phase shift for these potentials is thus definitely settled, and it is further confirmed that even the next term in the expansion yielded by the JWKB expression is significant.
Classical, cross-section generating solutions of field equations
Authors: Suranyi Peter
Published: March 2003

The symmetry properties of classical, cross-section generating solutions of field equations, sectons, are investigated. It is shown that under general conditions on the interaction Lagrangian of the field theory the symmetry group of the solutions can only be 0(1,1) ×0(2) for finite momenta. Such solutions generate inclusive cross sections with Feynman scaling.
Characteristic surfaces and characteristic initial data for the generalized Einstein–Maxwell field equations
Authors: Horndeski Gregory Walter
Published: March 2003

The characteristic hypersurfaces of the source-free generalized Einstein–Maxwell field equations are investigated. It is shown that such hypersurfaces can be null, and it appears as though they may also be spacelike or timelike. Examples of characteristic initial data on spacelike and timelike hypersurfaces are presented, and it turns out that these examples involve very intense gravitational and electromagnetic fields when the coupling constant k is small in magnitude.
Evaluation of SU(6) ⊇ SU(3) ⊗ SU(2) Wigner coefficients
Authors: Strottman D.
Published: March 2003

A technique is described which allows the calculation of any SU(6) Wigner coefficient (isoscalar factor) from known one-body SU(6) coefficients. The technique is applied to calculate the SU(6) coefficients for q3×q3 leading to SU(3) singlet, octet, and decuplet states of six quarks. A sum rule of SU(3) 9−λμ symbols is given.
Solutions of the nonlinear 3-wave equations in three spatial dimensions
Authors: Cornille H.
Published: March 2003

Recently Ablowitz and Haberman have shown that, in three spatial dimensions, the nonlinear 3-wave evolution equation results from the compatibility condition between two well-defined first-order linear differential 3×3 systems having common solutions. We construct inversionlike integral equations (I.E.) associated with both these two linear differential systems so that the solutions of the I.E. embody their compatibility conditions. The scalar kernels of these I.E. depend upon three independent variables in such a way that there exist degenerate kernels confined in the three-dimensional coordinate space. Consequently we exhibit, for the nonlinear 3-wave evolution equations, an infinite number of solutions which, at fixed time, are confined in the three-dimensional coordinate space.
Comments on certain divergence-free tensor densities in a 3-space
Authors: Aldersley S. J.
Published: March 2003

It is well known that a necessary and sufficient condition for the conformal flatness of a three-dimensional pseudo-Riemannian manifold can be expressed in terms of the vanishing of a third-order tensor density concomitant of the metric which has contravariant valence 2. This was first discovered by Cotton in 1899. It is shown that Cotton’s tensor density is not the Euler–Lagrange expression corresponding to a scalar density built from one metric tensor. This tensor density is shown to be uniquely characterized by its conformal properties coupled with the demand that it be differentiable for arbitrary metrics.
The nondiverging and nontwisting type D electrovac solutions with λ
Authors: Plebański Jerzy F.
Published: March 2003

Assuming the type D of the metric and the alignment of the EM field along the double D–P directions all solutions of Einstein–Maxwell equations free of complex expansion in the presence of cosmological constant are studied. All solutions of this type are found equivalent to Carter’s branch [B̄ (−)] in a previous paper and are derivable from the general 7-parametric D’s by contractions. Various special cases are examined, and at least a 4-parameter group of symmetries of these solutions is exhibited and studied.
Birman–Schwinger bounds for the Laplacian with point interactions
Authors: Thomas Lawrence E.
Published: March 2003

Birman–Schwinger bounds on the eigenvalues of −Δ are described, where −Δ is the self adjoint operator obtained from the three-dimensional Laplacian by imposing boundary conditions at N distinct points in R3.
Explicit solution of the wave equation for arbitrary power potentials with application to charmonium spectroscopy
Authors: Müller-Kirsten H. J. W. , Hite G. E. , Bose S. K.
Published: March 2003

We present an explicit and almost complete series solution of the Schrödinger equation for an arbitrary quark-confining power potential with or without a weak Coulomb component or other corrections. In particular, we derive two pairs of high-energy asymptotic expansions of the bound-state eigenfunctions together with a corresponding expansion of the eigenvalue determined by the secular equation. We also obtain a pair of uniformly convergent expansions and discuss other types of solutions. Various properties of the solutions and eigenvalues are examined including the scattering problem of the cutoff potential and the behavior of Regge trajectories. Finally, the relevance of these investigations to the spectroscopy of heavy quark composites is discussed. In particular, we derive approximate expressions for leptonic decay rates. Examples are given to demonstrate the usefulness of these results for theoretical discussion and as alternatives for numerical integration techniques. A subsequent paper will deal with the normalization of the bound-state wavefunctions and the corresponding derivation of explicit series expressions for certain decay rates.
A kernel of Gel’fand–Levitan type for the three-dimensional Schrödinger equation
Authors: Moses H. E.
Published: March 2003

In a previous paper we introduced a Green’s function for the three-dimensional Schrödinger equation analogous to the Green’s function used to obtain the integral equation for the Jost wave functions in one dimension. The three-dimensional Green’s function was used to define Jost wave functions for the three-dimensional problem and the completeness relations for these wave functions were obtained. In the present paper we use the three-dimensional Green’s function to construct influence functions for the 3+3 ultrahyperbolic partial differential equation which have analogs to the causal properties of the corresponding influence functions for the 1 + 1 hyperbolic partial differential equation. Just as the 1 + 1 influence function can be used to obtain an integral equation for the one-dimensional Gel’fand–Levitan kernel in terms of the scattering potential, we use the 3 + 3 influence function to obtain an analogous integral equation for our proposed Gel’fand–Levitan kernel for the three-dimensional problem. Though much of the formalism for finding the properties of the kernel for the three-dimensional problem can be carried out in a straightforward manner, the interpretation of the triangularity properties is more difficult than in the one-dimensional case because of the complicated geometrical picture associated with the notion of causality. In addition to its use in obtaining a Gel’fand–Livitan kernel, the 3+3 influence function can be used to simplify the second term in an expansion of the potential in terms of the minimal scattering data. This simplification is also given. In the Appendix the asymptotic form of the three-dimensional Jost wave function is given in a form which is analogous to the asymptotic form for the one-dimensional Jost wave function and which is compatible with our notion of triangularity for the Gel’fand–Levitan kernel.
The rosette of rosettes of Hilbert spaces in the indefinite metric state space of the quantized Maxwell field
Authors: Gessner W. , Ernst V.
Published: March 2003

The indefinite metric space 𝒪M of the covariant form of the quantized Maxwell field M is analyzed in some detail. 𝒮M contains not only the pre-Hilbert space 𝒳0 of states of transverse photons which occurs in the Gupta–Bleuler formalism of the free M, but a whole rosette of continuously many, isomorphic, complete, pre-Hilbert spaces ℒq disjunct up to the zero element o of 𝒮M. The ℒq are the maximal subspaces of 𝒮M which allow the usual statistical interpretation. Each ℒq corresponds uniquely to one square integrable, spatial distribution jo(x) of the total charge Q=0. If M is in any state from ℒq, the bare charge j0(x) appears to be inseparably dressed by the quantum equivalent of its proper, classical Coulomb field E(x). The vacuum occurs only in the state space ℒ0 of the free Maxwell field. Each ℒq contains a secondary rosette of continuously many, up to o disjunct, isomorphic Hilbert spaces ℋgq related to different electromagnetic gauges. The space ℋoq, which corresponds to the Coulomb gauge within the Lorentz gauge, plays a physically distinguished role in that only it leads to the usual concept of energy. If M is in any state from ℋgq, the bare 4-current j0(x), j(x), where j(x) is any square integrable, transverse current density in space, is endowed with its proper 4-potential which depends on the chosen gauge, and with its proper, gauge independent, Coulomb–Oersted field E(x), B(x). However, these fields exist only in the sense of quantum mechanical expectation values equipped with the corresponding field fluctuations. So they are basically different from classical electromagnetic fields.
Stability of streets of vortices on surfaces of revolution with a reflection symmetry
Authors: Hally David
Published: March 2003

Helmholtz’s theory of ideal vortex motion in two dimensions is generalized to flows on curved surfaces. The existence of a generalized vortex stream function is proved and used to generate conservation laws. In particular, the angular moment of circulation is related to invariance under scale transformations. The theory is used to derive criteria for stability of vortex streets on surfaces of revolution having symmetry under reflection in a plane whose normal is the axis of revolution. For the special case of the sphere it is found that only those vortex streets having six or fewer vortices per ring can be stable and that, in contradistinction to the results of von Karman, both symmetric and staggered vortex streets can be stable.
Einstein spaces and homothetic motions. I
Authors: Halford W. D. , Kerr R. P.
Published: March 2003

Algebraically special, nonflat vacuum Einstein spaces with an expanding and/or twisting geodesic principal null congruence are considered. These spaces are assumed to possess locally a homothetic symmetry as well as two or more Killing vectors. All metrics of such spaces are determined along with the form of the homothetic Killing vector admitted. All but one of the metrics are twist free. It is proved that two of the NUT metrics do not admit a homothetic motion.
The Planck integral cannot be evaluated in terms of a finite series of elementary functions
Authors: Pavelle Richard
Published: March 2003

It is shown that the Planck integral cannot be evaluated in terms of a finite series of elementary functions. A relation is first established between the Planck and dilogarithm integrals. To prove nonintegrability the Risch decision procedure for elementary functions is then applied to the dilogarithm integral.
Dust cylinders in static space-times
Authors: Lathrop J. D. , Orsene M. S.
Published: March 2003

We argue that the work of Vishveshwara and Winicour suggests that there is an upper limit to the linear mass densities of infinite cylinders in static, cylindrically symmetric spacetimes. We investigate the class of static cylinders havong two oppositely directed circular flows of dust, and show that the above conjecture is true for these cylinders.
Linearization stability and a globally singular gange of variables
Authors: Arms Judith M.
Published: March 2003

An example is given showing that global properties of a change of field variables can affect the validity of linear perturbation theory.
Variational principles for particles and fields in Heisenberg matrix mechanics
Authors: Klein Abraham , Li Ching Teh , Vassanji Moyez
Published: March 2003

For many years we have advocated a form of quantum mechanics based on the application of sum rule methods (completeness) to the equations of motion and to the commutation relations, i.e., to Heisenberg matrix mechanics. Sporadically we have discussed or alluded to a variational foundation for this method. In this paper we present a series of variational principles applicable to a range of systems from one-dimensional quantum mechanics to quantum fields. The common thread is that the stationary quantity is the trace of the Hamiltonian over Hilbert space (or over a subspace of interest in an approximation) expressed as a functional of matrix elements of the elementary operators of the theory. These parameters are constrained by the kinematical relations of the theory introduced by the method of Lagrange multipliers. For the field theories, variational principles in which matrix elements of the density operators are chosen as fundamental are also developed. A qualitative discussion of applications is presented.
The Dirac inverse spectral transform: Kinks and boomerons
Authors: Leon Jérôme JP.
Published: March 2003

The inverse spectral transform (IST) is derived when using the eigenvalue problem for the one-dimensional Dirac operator: (D)=iσ3(d/dx)+i(r ̃0 0-q̃), σ3=(01-10), where the potentials q̃ and r̃ have nonzero asymptotic values. The method used is of AKNS type. It is shown that the nonlinear evolution equations (NEE) obtained are of differential type at any order (and not of integro-differential type). Some particular solutions are studied, and it is shown that their special behavior is a direct consequence of the nonzero boundary condition on (D).
Multiple scattering theory for discrete, elastic, random media
Authors: Devaney A. J.
Published: March 2003

A theory is presented for determining the ensemble-averaged Green tensor of a statistically homogeneous distribution of identical, randomly oriented elastic scatterers embedded in an infinite, homogeneous, and isotropic matrix. The theory is based on the self-consistent formulation of Lax’s [Rev. Mod. Phys. 23, 287 (1951); Phys. Rev. 85, 621 (1952)] multiple scattering theory due to Gyorffy [Phys. Rev. B 1, 3290 (1970)] and Korringa and Mills [Phys. Rev. B 5, 1654 (1972)]. The average Green tensor is found to be characterized by three parameters which may depend on the momentum operator but which are otherwise analogous to the Lamé constants and density of an ideal, homogeneous, and isotropic medium. These ’’effective’’ parameters are shown to be related in the usual way to the wave numbers of coherent compressional and shear plane-wave modes of the random composite. The ensemble averaged Green tensor and the dispersion relations satisfied by the wave numbers of the coherent modes are found to depend on the single and joint probability density functions for the scattering centers and on the transition operator of a discrete scatterer embedded in the effective (average) medium. The dispersion relations are evaluated explicitly for the limiting case of a completely random ensemble of homogeneous and isotropic scatterers whose elastic parameters and density differ very little from those of the matrix medium.
Shift operator techniques for the classification of multipole-phonon states. VI. Properties of nonscalar R(3) product operators in the R(2λ+1) groups
Authors: Vanden Berghe G. , De Meyer H. E.
Published: March 2003

Expressions connecting nonscalar R(3) products of operators which shift the eigenvalues of the R(3) Casimir operator L2 are constructed within the R(2λ+1) groups (λ=2 or 3).
A concept of spin 1/2 approximation in the quantum theory of lattice Bose systems
Authors: Garbaczewski Piotr
Published: March 2003

We give a model independent description of criteria under which expectation values of observables associated with a finite part of the lattice Bose system can be made to converve to those of the associated Fermi or finite spin system.
Bloch electrons in a magnetic field: Reduction to one dimension
Authors: Wannier G. H.
Published: March 2003

A reduction to one dimension of the above problem, found by Schellnhuber and Obermair for a special model lattice, is shown to be valid for all lattices without restriction. As was the case in their problem, the field must be rational. If the rational number is the reciprocal of an integer a single equation results. This condition is well adapted to the study of fields of practically attainable magnitude. If the rational number is of the form q/p a system of q coupled equations is obtained.
Geometrical reinterpretation of Faddeev–Popov ghost particles and BRS transformations
Authors: Thierry-Mieg Jean
Published: March 2003

A classical geometrical interpretation of the ghosts fields is presented. BRS rules follow from the Cartan-Maurer fibration theorem. The statistics of ghosts are explained and the effective quantum Lagrangian is derived without factorizing the volume of the gauge group. Topologically nontrivial ghost configurations are defined.
Generalized groups as global or local symmetries
Authors: Taylor J. G.
Published: March 2003

We extend global particle symmetries from the traditional group framework to that of generalized groups. The nature of these latter are presented, and various invariants constructed for them. The problem of gauging generalized groups is discussed and a no–go theorem proved under reasonable conditions on the generalized group structure.
An infinity of polynomial conserved densities for a class of nonlinear evolution equations
Authors: McGuinness Mark J.
Published: March 2003

Noether’s theorem is applied to the infinity of polynomial conserved densities possessed by a general class of nonlinear evolution equations. The densities are identified on the solution sets of higher-order enveloping equations as canonical energy or momentum densities, and a new recursive formula is derived for these densities.
Operator methods for time-dependent waves in random media with applications to the case of random particles
Authors: Furutsu K.
Published: March 2003

The random medium is represented by the operator, constructed from the characteristic functional of the medium, and this representation is shown to considerably facilitate the formulation of various equations of waves in random media, as well as obtaining the physical insight into the equations. A specific application is made to waves in the medium of random particles, and the equations obeyed by the characteristic functional of wave are derived with the aid of the effective medium method. Here, the optical condition is exhibited by the condition of an operator in space and time. Independent of this operator method, the general theory is extended, in an unperturbative way, for the equations of the second-order coherence functions, being given in form of the Bethe–Salpeter equation, and the coherent potential equations are formulated for the basic matrices of two kinds appeared in the equations. The explicit expressions of these matrices are obtained, on utilizing the coherent potential approximation, and are shown to be exactly the same as those obtained by the effective medium method, in both cases of weak-scattering limit and of random particles. Finally, on employing the appropriate Fourier representations in space and time, the theory is presented in a few different forms, one being particularly suited to derive the equations of multifrequency coherence functions.
A note on the Schrödinger equation for the x2+λx2/(1+g
x2) potential
Authors: Bessis N. , Bessis G.
Published: March 2003

The energy levels and wave functions of the Schrödinger equation involving the potential x2+λx2/(1+gx2) are calculated by the variational method, for any range of λ and g, without having to resort to numerical quadrature. Using properly scaled (in λ and g) harmonic oscillator functions as a basis set, an easy to compute analytical expression of the current Hamiltonian matrix element is derived. Perturbative results are also given.
The use of anticommuting variable integrals in statistical mechanics. I. The computation of partition functions
Authors: Samuel Stuart
Published: March 2003

Integrals over anticommuting variables are use to rewrite partition functions as fermionic field theories. The method is used to solve the two-dimensional Ising model, the planar close-packed dimer problems, and the free-fermion eight vertex model.
The Jost solutions for general Gaussian potentials
Authors: Trlifaj L.
Published: March 2003

The explicit Born series which represent the regular solutions (of the Schrödinger equation), the Jost solutions, and the Jost functions analytic in some domains of complex wave numbers and angular momenta are found for general potentials of the Gaussian type expressed by the Stiltjes integral. When finding their form we make use of the analogy with the corresponding solutions for general potentials of the Yukawa type, which is based on the special representations of the Bessel and Hankel functions (purely kinematic solutions) in either case. Some other relations are also derived.
Partial inner product spaces. III. Compatibility relations revisited
Authors: Antoine J.-P.
Published: March 2003

This paper continues our systematic study of partial inner product spaces. We show here that a linear compatibility relation on a vector space V is characterized by special families of vector subspaces of V, called involutive coverings, and vice versa. This result provides the link between partial inner product spaces, defined in an intrinsic way, and various concrete structures, such as rigged or nested Hilbert spaces. Given a linear compatibility, generating sets (’’rich subsets’’) are discussed, and several examples are worked out. Finally, we introduce an order relation among all linear compatibilities on the same vector space.
A modified asymptotic Padé method. Application to multipole approximation for the plasma dispersion function Z
Authors: Martín Pablo , Donoso Guillermo , Zamudio-Cristi Jorge
Published: March 2003

An improved asymptotic Padé method is presented. The approximations expressed as rational functions are developed both in a power series and asymptotic expansions. Identifying these developments with those for the exact function, the approximated rational functions are obtained. Three- and four-pole approximations for the plasma dispersion function have been determined with this method. Our approximations (Zlm) give closer results to the exact function than all the published ones.
Two sided estimates of the scattering amplitude for low energy
Authors: Ramm A. G.
Published: March 2003

Estimates and iterative processes for calculating the scattering amplitude for k=0 are given.