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On induced representations for finite groups
Authors: Backhouse Nigel B. , Gard Patricia
Published: March 2003

Two new notions are introduced as tools for the study of representations of finite groups. First, in the spirit of duality, a basis set of class orientated characters is shown to possess nice properties with respect to induction and subduction, which lead to simple proofs of some well-known theorems. Secondly, as a useful device in constructive representation theory, a study is made of the subrepresentations which often naturally occur when a representation is induced from a subgroup to a supergroup.
Expansion formulas and addition theorems for Gegenbauer functions
Authors: Durand Loyal , Fishbane Paul M. , Simmons L. M.
Published: March 2003

We give a systematic summary of the properties of the Gegenbauer functions Cαλ(x) and Dαλ(x) for general complex degree and order, with emphasis on the functions of the second kind, Dαλ(x), and on results useful in scattering theory. The results presented include Sommerfeld–Watson type expansion formulas and two reciprocal addition formulas for the functions of the second kind.
Bifurcation of solutions with crystalline symmetry
Authors: Raveché Harold J. , Stuart C. A.
Published: March 2003

We consider the BBGKY equation for the single particle probability density in a hard sphere system. We investigate whether there is bifurcation from the fluid phase to functions which have crystalline symmetry. We find that as the density of the fluid increases from zero, there is bifurcation in one, two, and three dimensions. The bifurcation is shown to be characteristic of metastability and in general it does not occur at the equilibrium coexistence of two phases. The direction of branching and the stability of solutions near bifurcation is also discussed.
Higher indices of group representations
Authors: Patera J. , Sharp R. T. , Winternitz P.
Published: March 2003

The nth order index of an irreducible representation of a semisimple compact Lie group, n a nonnegative even integer, is defined as the sum of nth powers of the magnitudes of the weights of the representation. It is shown, in many situations, to have additivity properties similar to those of the dimension under reduction with respect to a subgroup and under reduction of a direct product. The second order index is shown to be Dynkin’s index, multiplied by the rank of the group. Explicit formulas are derived for the fourth order index. A few reduction problems are solved with the help of higher indices as an illustration of their utility.
Continuum calculus and Feynman’s path integrals
Authors: Lee L. L.
Published: March 2003

An operational calculus is set up with the specific aim of resolving the problem of the integral of functionals in a general complex Banach algebra. The functionals that occur frequently in physics are Feynman’s path integrals for quantum mechanics which usually appear in the form of an exponential of an integral. Through the establishment of two operations, the r differentiation and p integration, we succeed in constructing a formula, in closed form, for the integrals of Feynman type functionals. Applications to known problems (quantum harmonic oscillator, the electron–phonon system) corroborate conventional results. This formula is found to be at once consistent and more general than the method of projection into cylinder functionals of Frederichs type. It does not require an additive Gaussian measure, and it admits integration with finite limit functions. The methodology developed is general and applicable to other branches of mathematics. It is particularly suited to the study of infinitely divisible distributions in probability theory. We rederive, with facility, Lévy’s formulas for continuous sums, and Lévy–Khintchine and Kolmogorov formulas. We also find it applicable to continuum matrix algebra, where the formula for the determinant of matrices of continuous indices is given as a p integral. As to algebraic identities, we give a continuum version of the binary expansion, and retrieve Stirling’s formula of factorials by p integration. The idée-clef lies in the concept of infinitesimal ratio of a function in the same way that differential calculus deals with infinitesimal differences. Then the functional integral appears to be a natural product of the interaction between the conventional integration and the proposed p integration. It also heralds the possibility of a generalized measure theory for integrals where the basic operation between the measure and the integrand is not bilinear.
Nonlinear response of equilibrium strongly coupled Fermi fluids. I. Formal development
Authors: Reichl L. E.
Published: March 2003

This is Paper I of a series of three papers in which a self-consistent propagator resummation of self-energy effects in a strongly coupled Fermi fluid, in the presence of an external magnetic field, is performed. In the present paper, an exact expression is obtained for the grand potential in the presence of an external magnetic field which has a constant and a spatially varying part. The grand partition function and grand potential are written in terms of antisymmetrized cluster expansions. The cluster functions are then expanded in terms of a binary expansion. And, finally, an expression for the grand potential is obtained in terms of the reaction matrix for two-body scattering and in a form suitable for the subsequent propagator resummation.
Nonlinear response of equilibrium strongly coupled Fermi fluids. II. Fourier expansion and partial resummation of expectation values—polarization diagrams
Authors: Reichl L. E.
Published: March 2003

This is Paper II of a series of three papers in which a self-consistent propagator resummation of self-energy effects in a strongly coupled Fermi fluid is performed. In the present paper, a generating function for the expectation values of arbitrary one- and two-body operators is introduced and written in the form of a cluster expansion. An explicit expression is written for the magnetization of a strongly coupled Fermi fluid in the presence of a constant and a spatially varying external field, and rules are given for evaluating it in terms of a reaction matrix expansion. A Fourier expansion is performed on a subclass of the diagrams contributing to the grand potential and the magnetization, and a self-consistent resummation of self-energy effects due to both the medium and the external spatially varying field is performed. It is found that traditional perturbation theory techniques for summing self-energy effects cannot be applied to all terms in a reaction matrix expansion. The effect of the external fields on the polarization diagrams is discussed.
Tensor spherical harmonics and tensor multipoles. II. Minkowski space
Authors: Daumens M. , Minnaert P.
Published: March 2003

The bases of tensor spherical harmonics and of tensor multipoles discussed in the preceding paper are generalized in the Hilbert space of Minkowski tensor fields. The transformation properties of the tensor multipoles under Lorentz transformation lead to the notion of irreducible tensor multipoles. We show that the usual 4-vector multipoles are themselves irreducible, and we build the irreducible tensor multipoles of the second order. We also give their relations with the symmetric tensor multipoles defined by Zerilli for application to the gravitational radiation.
Quantum field theory on incomplete manifolds
Authors: Candelas P. , Raine D. J.
Published: March 2003

A theory of the scalar quantum field on static manifolds is constructed using the language of Feynman Green’s functions. By means of examples in which the manifolds are parts of Minkowski space, we show how the ’’method of images’’ can be used to solve for the Green’s functions. In particular, we consider the Rindler wedge and the space outside a uniformly accelerated conducting sheet. As an example in which the manifold is nonstatic, we consider the region exterior to a conducting sheet which is accelerated impulsively from rest to the speed of light. Finally, we study the steady-state part of de Sitter space where we do not obtain a unique result.
Systems of differential inequalities and stochastic differential equations. III
Authors: Ladde G. S.
Published: March 2003

Consider the system of stochastic differential equations x′ (t,ω) =f (t,x (t,ω),ω), x (t0,ω) =x0(ω), where f (t,x (t,ω),ω) is a product measurable n-dimensional random vector function whenever x (t,ω) is a product measurable random function, and it satisfies the desired regularity conditions to ensure the existence of solution process. By developing systems of random differential inequalities, a very general comparison theorem in the framework of a vector Lyapunov function is developed, and furthermore sufficient conditions are given for the stability of solutions in probability, in the mean and with probability one.
Some properties of canonical invariantly relaxing (CIR) systems
Authors: Chong Tak-Yuen
Published: March 2003

In this paper, two general properties of the conserved discrete states CIR systems have been found: (1) the equal spacing of and a general expression for the characteristic roots of their transition-rates-matrices; (2) a general formulation for their conditional probabilities. The discussion is also extended to the disappearing discrete-states CIR systems which include the infinite level harmonic oscillators as a special case.
Uniqueness of perturbation of a Reissner–Nordström black hole
Authors: Lee Chul Hoon , McGlinn W. D.
Published: March 2003

Coupled gravitational and electromagnetic perturbations of a Reissner–Nordström black hole are analyzed using the Newman–Penrose formalism. It is shown that χB1(≡3ψ2φB0−2φ1ψB1) or χB−1(≡3ψ2φB2−2φ1ψB3) determines the perturbations except for those corresponding to an infinitesimal change in the mass, charge, and angular momentum parameters of the balck hole.
Radial integrals with finite energy loss for Dirac–Coulomb functions
Authors: Sud Krishan , Wright L. E. , Onley D. S.
Published: March 2003

Analytic results for radial integrals over products of Dirac–Coulomb functions and the radial part of the electromagnetic Green’s function are expressed in terms of a matrix generalization of the gamma function. This matrix gamma function has many useful properties, including a recurrence relation similar to that of the gamma function, and provides a compact easily manipulated method of evaluating the Dirac–Coulomb radial integrals. These results can be used to calculate the virtual and real photon spectra associated with electron scattering from the nucleus.
The frequency dependent electrical conductivity for disordered alloys: Application of an abstract Hilbert space generalization of Feenberg’s perturbation theory
Authors: Berryman James G. , Bowen Samuel P.
Published: March 2003

An abstract Hilbert space with a particularly convenient scalar product is introduced to permit a generalization of Feenberg’s rearrangement method of perturbation theory to be applied to thermal Green’s function calculations. This method has the advantage of treating averages (either thermal or configuration) rigorously from the start. Explicit calculations are done for the frequency dependent electrical conductivity for alloys with diagonal disorder at zero temperature. Three practical approaches are discussed: (1) the Gram–Schmidt orthogonalization procedure, (2) a trick which depends on the Hermitian character of the polarization operator, and (3) a general procedure for using nonorthogonal basis vectors to expand the Feenberg formulas. To second order in the scattering strength, a new expression for the conductivity is found which is valid for all frequencies. This expression agrees with earlier perturbation theory results when the frequency is very small or very large.
On a solution of the Einstein–Maxwell equations admitting a nonsingular electromagnetic field
Authors: McLenaghan Raymond G. , Tariq Nessim
Published: March 2003

Solutions of the Einstein–Maxwell equations are investigated for which the electromagnetic field is nonsingular and weakly parallelly propagated along its principal null congruences. It is shown that a subclass of these solutions admits an invertible two-dimensional Abelian group of motions. A weaker characterization of a recently found solution is thereby obtained.
Lie series and invariant functions for analytic symplectic maps
Authors: Dragt Alex J. , Finn John M.
Published: March 2003

Symplectic maps (canonical transformations) are treated from the Lie algebraic point of view using Lie series and Lie algebraic techniques. It is shown that under very general conditions an analytic symplectic map can be written as a product of Lie transformations. Under certain conditions this product of Lie transformations can be combined to form a single Lie transformation by means of the Campbell–Baker–Hausdorff theorem. This result leads to invariant functions and generalizes to several variables a classic result of Birkhoff for the case of two variables. It also provides a new approach since the connection between symplectic maps, Lie algebras, invariant functions, and Birkhoff’s work has not been previously recognized and exploited. It is expected that the results obtained will be applicable to the normal form problem in Hamiltonian mechanics, the use of the Poincaré section map in stability analysis, and the behavior of magnetic field lines in a toroidal plasma device.
On the stationary axisymmetric Einstein–Maxwell field equations
Authors: Catenacci Roberto , Diaz Alonso Joaquin
Published: March 2003

We show the existence of a formal identity between Einstein’s and Ernst’s stationary axisymmetric gravitational field equations and the Einstein–Maxwell and the Ernst equations for the electrostatic and magnetostatic axisymmetric cases. Our equations are invariant under very simple internal symmetry groups, and one of them appears to be new. We also obtain a method for associating two stationary axisymmetric vacuum solutions with every electrostatic known.
A new topology for curved space–time which incorporates the causal, differential, and conformal structures
Authors: Hawking S. W. , King A. R. , McCarthy P. J.
Published: March 2003

A new topology is proposed for strongly causal space–times. Unlike the standard manifold topology (which merely characterizes continuity properties), the new topology determines the causal, differential, and conformal structures of space–time. The topology is more appealing, physical, and manageable than the topology previously proposed by Zeeman for Minkowski space. It thus seems that many calculations involving the above structures may be made purely topological.
Kerr black holes in a magnetic universe
Authors: Ernst Frederick J. , Wild Walter J.
Published: March 2003

We present exact expressions for the electromagnetic fields associated with arbitrarily charged Kerr–Newman black holes in a magnetic universe. In the particular case of charge Q=2 B0 J, where B0 is the magnetic field parameter and J=ma is the angular momentum, exact expressions for the gravitational field are also presented, while for arbitrarily charged black holes metrical corrections of order B0 are evaluated.
The semi-Euclidean approach in statistical mechanics. I. Basic expansion steps and estimates
Authors: Federbush Paul
Published: March 2003

The semi-Euclidean formulation, developed in constructive quantum field theory to handle boson–fermion models, is adapted to the statistical mechanics setting.
The semi-Euclidean approach in statistical mechanics. II. The cluster expansion, a special example
Authors: Federbush Paul
Published: March 2003

A form of the Glimm–Jaffe–Spencer cluster expansion, adapted to the statistical mechanics setting, is shown to converge for certain two-body potential interactions. The theory treated corresponds to negatively charged fermions and positively charged bosons interacting by a modified Coulomb interaction—the 1/r potential, cutoff at high and low momenta, becoming (1/r)(e−αr−e−γr).
A geometric formulation of the Taylor theorem for curves in affine manifolds
Authors: Bażański Stanisl/aw L.
Published: March 2003

The paper contains a theorem from differential geometry which exhibits the geometrical content of the classical Taylor expansion theorem applied to curves in a differentiable manifold. It also presents a review of a slightly modified version of the calculus of affine extensions, necessary for the proof of the theorem.
Multiple time scale analysis of an anharmonic crystal
Authors: Khoo I. C. , Wang Y. K.
Published: March 2003

We use the multiple time scale perturbation method to study the lattice dynamics of an anharmonic crystal. The Heisenberg equations of motion for the creation and annihilation operators are solved, and the frequency shift and the decay constants are found to the second order. We also discuss briefly how to apply our solution to calculate the correlation function.
Bilinear quantum field theories and their coherent states
Authors: DeFacio B. , Hammer C. L.
Published: March 2003

The quantization method developed by Hammer and Tucker, which is based upon a set of equations of motion and their conserved currents rather than a canonical formalism, is extended to interacting systems. The operators of the theory are bilinear and are essentially self-adjoint on a dense domain which is spanned by a suitably chosen subset of the coherent states. Both proper and improper gauge transformations of the second kind are discussed. For the proper case, the connection is given between these transformations and coherent states, which are discussed in detail. One interesting result is that a ’’smeared’’ Fock space can be constructed for a system where the particles have the same average quantum numbers. For the improper case, the gauge transformation of the second kind is related to the purely absolutely continuous measure. The formalism is applied to two examples. One is a Dirac field minimally coupled to a massive vector field, and the other is Klauder’s ultralocal models.
The Weyl tensor and stationary electrovac space–times
Authors: Esposito F. Paul , Glass E. N.
Published: March 2003

It is shown that static electrovac spaces admit only static electric type Maxwell fields, and it is proved that a necessary and sufficient condition for a stationary electrovac space–time to be static is that the Weyl tensor and the Maxwell field tensor both be pure electric type.
Transition probability spaces
Authors: Belinfante Johan G. F.
Published: March 2003

The set of pure quantum states is described as an abstract space with a geometry determined by transition probabilities. We describe all possible structures for three-dimensional transition probability spaces with less than ten states, as well as some even larger spaces of a certain symmetric type. It is shown that the orthoclosed subspaces of a transition probability space form an atomistic orthomodular poset.
Einstein–Maxwell metrics admitting a dual interpretation
Authors: Tariq N. , Tupper B. O. J.
Published: March 2003

Conditions are given under which the metric part of a solution of the source-free Einstein–Maxwell equations may be interpreted as the metric part of a solution with sources. Examples are given of space–times which admit this dual interpretation and also of space–times admitting one interpretation only.
An effective-potential approach to stationary scattering theory for long-range potentials
Authors: Masson D. , Prugove?ki E.
Published: March 2003

It is shown that previously derived integral equations for two-body scattering with long-range potentials (equations which replace the Lippmann-Schwinger equations) can be reduced to a form which is solvable by iterative methods. The method is applicable to potentials V (r) which behave asymptotically as r−α, 1/2<α?1, and in particular to Coulomb-like potentials.
Gauge-covariant differentiation and Green’s functions for the Yang–Mills field
Authors: Frey Heinz
Published: March 2003

A gauge-independent definition of differentiation is given for non-Abelian gauge fields in terms of parallel translation. This is achieved by a suitable definition of time-ordered operator products. Equal time commutation relations are used to derive the differential equations for the related Green’s functions. The Green’s functions are discussed for general linear gauges. In comparison with electrodynamics the Green’s functions have the well-known ghost-loop terms.
On the conductivity of fiber reinforced materials
Authors: Schulgasser Kalman
Published: March 2003

A two-phase material in which the phase boundaries are cylindrical surfaces is considered. A technique exists for finding upper and lower bounds on the effective thermal conductivity (or electrical conductivity, permittivity, or magnetic permeability) of the composite in the direction perpendicular to the generators of the phase boundaries in terms of two different three point correlation functions. It is shown how a phase interchange theorem can be introduced into these bounds enabling us to express them in terms of a single geometrical constant of phase geometry. We determine what range of values of this factor is realizable for real phase geometries, and we show that the bounds thus obtained span exactly all realizable effective conductivities for such composites. Finally, we show that the bounds as expressed here enable us to use a knowledge of the effective conductivity of a composite for one ratio of constituent conductivities to narrow the bounds for some other ratio.
Long-wavelength normal mode vibrations of infinite, ionic crystal lattices. II
Authors: Davies John A.
Published: March 2003

In an earlier paper [J. Math. Phys. 16, 1156 (1975)] we presented a mathematical theory of the long-wavelength normal mode vibrations of infinite crystal lattices whose particles interact with Coulomb forces. (Retardation was neglected.) The paper showed how the eigenvalues and eigenvectors of the complete long-wavelength dynamical matrix are related to the eigenvalues and eigenvectors of the dynamical matrix obtained by neglecting the contribution of the macroscopic electric field. Rules were obtained for determining whether or not the various branches of the dispersion relations for a lattice approach definite frequencies in the long-wavelength limit. The paper was restricted to the rigid ion approximation. In this paper we show that the above treatment can be easily extended to include lattices with polarizable and deformable atoms.
On the equality of S operators corresponding to unitarily equivalent Hamiltonians in single channel scattering
Authors: Sáenz A. W. , Zachary W. W.
Published: March 2003

Some years ago, conditions in order that unitarily equivalent Hamiltonians H and W*HW yield the same S operator were investigated by Ekstein and by Coester and his collaborators in the framework of nonrelativistic time-dependent single-channel quantum scattering theory. This subject has turned out to be of considerable importance for nuclear physics, since it constitutes the foundation of a widely used method for constructing phase-equivalent nucleon–nucleon potentials. In the present paper we derive a rigorous and concise necessary and sufficient condition for a pair of unitarily equivalent Hamiltonians, governing the dynamics of a pair of nonrelativistic particles in the center-of-mass system, to yield the same S operator. Our theory, based on a time-dependent approach, applies to very general types of short-range potentials, with or without hard cores, and to an extensive class of long-range potentials. Our necessary and sufficient condition simplifies when certain strong limits W±, related to W, are unitary and when W+=W−. Requirements sufficient for each of these properties to hold are determined. Various examples of operators W such that H and W*HW have the same S operator are discussed.
Bounds on the admittance for KMS states
Authors: Naudts J. , Verbeure A.
Published: March 2003

Upper and lower bounds are proved for the static admittance of observables in a KMS state on a von Neumann algebra. As an application some exact results for the transverse Ising model are derived.
A theorem on Ruch’s principle of nonequilibrium statistics
Authors: Lesche Bernhard
Published: March 2003

It is proved that whenever two probability distributions, on a finite set, p and p′ are given such that p′ has a bigger mixing character than p, it is possible to find a unitary matrix U, such that p′i =Σj‖Uij‖2pj. This theorem ensures that Ruch’s principle of increasing mixing character provides the strongest assertion to be made upon the diagonal of the density matrix of some quantum mechanical system at time t≳0, without one’s knowing the Hamiltonian operator but given that the density matrix was initially diagonal with a known but arbitrary diagonal.
Correspondence rules and path integrals
Authors: Cohen Leon
Published: March 2003

Using the Weyl correspondence, Mizrahi has derived an expression for the transition amplitude as a path integral in phase space. It is shown that the same result follows if any correspondence rule is used.
The infinitely renormalized field in the scalar field model
Authors: Hamilton Eugene
Published: March 2003

The system consisting of a relativistic scalar boson field interacting with a single spinless nucleon with kinetic energy taken to be independent of momentum is studied in d space dimensions. The interaction Hamiltonian is taken to be HI(f) =ℱRdφf(x) ψ* (x) ψ (x) dx, where f is a momentum cutoff. The physical Hilbert space 𝒦 corresponding to the case f≡1 in d space dimensions is discussed. The time smoothed nucleon annihilation operator is constructed as a closable operator on 𝒦. First order estimates are established for ψ (h) in terms of the local (in momentum space) number operators on 𝒦 for the case d=3. It is shown that the union of the ranges of the adjoints ψ* (h) is dense in 𝒦. The one particle Hamiltonian is related to the nucleon creation operator on 𝒦.
SU(6) isoscalar factors for the product 405×56→56, 70
Authors: Machacek M. , Tomozawa Y.
Published: March 2003

SU(6) isoscalar factors for the product 405×56→56, 70 are calculated. SU(3) isoscalar factors for the products 27×10→10, 8 and 1̄0̄×8→8 are also tabulated.
Clebsch–Gordan coefficients of magnetic space groups
Authors: Rudra P. , Sikdar M. K.
Published: March 2003

We have obtained sets of homogeneous linear equations in the Clebsch–Gordan coefficients for magnetic space groups in terms of the matrix elements of the irreducible representations of the little cogroup of the linear subgroup of index 2. Depending on the types of the co-representations in the triple product, 18 cases arise. These 18 cases can be divided into six categories. We have given explicit forms for one case in each category and have indicated how the other cases are to be treated. The formalism has been developed for projective co-representations so that both the vector and the spinor case can be treated.
Comment on the reduction of an important 9-j symbol
Authors: Stassis C. , Williams S. A.
Published: March 2003

An error in the formula for the reduction of an important 9-j symbol is pointed out, and the correct relationship is given.
On the Majorana transformation
Authors: Weaver D. L.
Published: March 2003

Some properties of the Dirac equation in its four- and two-component forms suggested by the Majorana representation of Dirac matrices are derived. Extension of the ideas to higher spin is also given.
Numerical experiments on the Calogero lattice
Authors: Casati Giulio , Ford Joseph
Published: March 2003

This paper presents the results of computer experiments performed on one-dimensional, classical mechanical, N-body systems whose point particles interact pairwise via the potential V (r) =ar2+br−2, where r is interparticle distance and where a and b are positive constants. When each particle interacts with all other particles, the numerical experiments indicate that the system is mathematically integrable for either free-end or fixed-end boundary conditions. On the other hand, when each particle interacts with only its nearest neighbors, the computer detects a transition from near-integrable to stochastic behavior again for either free-end or fixed-end boundary conditions. Our results thus support the conjecture that integrability is highly sensitive to changes in the total interaction potential but insensitive to modification of boundary conditions.
Solution of the Dirac equation with Coulomb and magnetic moment interactions
Authors: Barut A. O. , Kraus J.
Published: March 2003

The Dirac equation for a charged spin 1/2 particle with an anomalous magnetic moment in the Coulomb field is solved. A new phenomenon of formation of very narrow resonances of very high mass at small distances is demonstrated.
Spinor representations and projective factor systems of crystallographic point groups
Authors: Rudra P.
Published: March 2003

We have given the correspondence between the factor system of the spinor representations of the different crystallographic point groups and the factor system given by Doering and Hurley. It turns out that for the groups C1, Ci, C2, Cs, C2h, C4, S4, C4h, C3, C3i, C3v, D3, D3d, C6, C6h, C3h the factor system of the spinor representations are associated with that of the vector representations.
Probability measures on fuzzy events in phase space
Authors: Prugove?ki Eduard
Published: March 2003

The notion of fuzzy sample point is introduced, and generalized probability measures on fuzzy events are defined. This leads to the concept of spectral measure on fuzzy events. It is shown that such measures can be associated with quantum-mechanical states when the fuzzy elementary events are represented by Gaussian distributions on phase space.
Hilbert spaces of analytic functions and generalized coherent states
Authors: Arik M. , Coon D. D.
Published: March 2003

Generalized coherent states which are associated with a generalization of the harmonic oscillator commutation relation are investigated. It is shown that these states form an overcomplete basis in a Hilbert space of analytic functions. The generalized creation and annihilation operators are bounded except in a limit in which they reduce to the usual boson creation and annihilation operators. In this limit the Hilbert space of analytic functions reduces to the Bargmann–Segal Hilbert space of entire functions and in another limit it reduces to the Hardy–Lebesgue space.
Neutron transport operators in C and L2 spaces
Authors: Busoni G. , Mangiarotti L. , Frosali G.
Published: March 2003

We study the stationary neutron transport equation (in its integral form) in plane and spherical symmetry. The investigation is carried out in both L2 and C spaces, by means of standard methods of functional analysis. The equivalence between the eigenvalue problems in the two spaces is proved, and the space C seems often to be more appropriate than L2 for investigating the main properties of the eigenfunctions. Results of functional analysis are applied in establishing some properties of the eigenvalues and the eigenfunctions. Moreover, the continuous dependence of neutron flux on optical and spatial parameters is shown.
Null geodesic deviation. II. The Schwarzschild metric
Authors: Peters P. C.
Published: March 2003

The equation of geodesic deviation is solved in the Schwarzschild geometry in a covariant manner. The solution is exact for null geodesics, and is given as an integral equation otherwise. The solution is then used to evaluate second derivatives of the world function and derivatives of the parallel propagator, which need to be known in order to find the Green’s function for wave equations. The method of null geodesic limits is used to calculate higher order derivatives, and the results are applied to the scalar Green’s function in the Schwarzschild geometry.
Padé approximant bounds for the difference of two series of Stieltjes
Authors: Barnsley M. F.
Published: March 2003

Over the complex plane cut along the negative real axis let f (z) be an analytic function of the form f (z) =∫∞0dφ (u)/(1+uz), where φ (u) is of bounded variation on 0⩽u<∞. Suppose that the first (N+1) moments defined by fn=∫∞0undφ (u), n=0,1,...,N, exist and are known. Suppose further that dφ (u) undergoes at most p sign changes, where 0⩽p⩽N, as u goes from zero to infinity, to locations of an associated set of points u=ul, l=1,2,...,p, being known. Then, by means of a simple modification to the customary Padé approximant method, best possible upper and lower bounds can be imposed on f (x) for all 0<x<∞. Similarly, for given z in the cut complex plane, a best possible inclusion region can be imposed upon f (z). Finally it is shown that a best possible inclusion region can be imposed upon f (z) when the constraint that the set of points u=ul, l=1,2,...,p, is known is either relaxed or disposed of altogether.
Further heavenly metrics and their symmetries
Authors: Finley J. D. , Plebański J. F.
Published: March 2003

New developments in a continuing investigation of complex V4’s with purely self-dual conformal curvature are presented: (1) conformal and projective extensions of spaces with C̄ȦḂĊḊ=0 are discussed; (2) Killing vectors for general heavenly metrics are determined; (3) the solutions, in heavenly spaces, for (massless) D (0,s) spinor fields (in particular, the Maxwell field) are found; then (4) new examples of heavenly metrics of types G×[–] and D×[–] are provided; lastly, contraction of the D×D solutions of Plebański and Demiański to type D×[–] is performed, giving a complex prototype of the Kerr–Newman solution, and all solutions of the type N×[–] are given, which contain two arbitrary functions of two variables.
Scattering of scalar waves from a Schwarzschild black hole
Authors: Sanchez Norma G.
Published: March 2003

The scattering of scalar waves from a Schwarzschild black hole is investigated for wavelengths much less than the graviational radius (rs). Explicit expressions for scattering parameters are obtained for two cases: high angular momenta and low angular momenta. In the first case we obtain the phase shifts and absorption coefficient with the JWKB method. The elastic differential cross section and the total absorption cross section are also calculated. For low angular momenta we present a method based in the DWBA (distorted wave Born approximation). With this method, the phase shifts and the absorption coefficients are obtained.
A geometric proof of no-interaction theorems
Authors: Kracklauer Aloysius F.
Published: March 2003

No-interaction theorems are proven, using the methods of modern differential geometry, and an example of a Hamiltonian yielding relativistic canonical equations of motion with interaction is presented.
Canonical transforms, separation of variables, and similarity solutions for a class of parabolic differential equations
Authors: Wolf Kurt Bernardo
Published: March 2003

Using the method of canonical transforms, we explicitly find the similarity or kinematical symmetry group, all ’’separating’’ coordinates and invariant boundaries for a class of differential equations of the form [α∂2/∂q2+βq ∂/∂q+γq2+δq+ε∂/∂q+ζ] u (q,t) =−i (∂/∂t) u (q,t),or of the form [α′ (∂2/∂q2+μ/q2)+β′q∂/∂q+γ′q2] u (q,t) =−i (∂/∂t) u (q,t), for complex α,β,..., γ′. The first case allows a six-parameter WSL(2,R) invariance group and the second allows a four-parameter O(2) ⊗ SL(2,R) group. Any such differential equation has an invariant scalar product form which, in the case of the heat equation, appears to be new. The proposed method allows us to work with the group, rather than the algebra, and reduces all computation to the use of 2×2 matrices.
On the dynamics of particles in a bounded region: A measure theoretical approach
Authors: Marchioro C. , Pellegrinotti A. , Presutti E. , Pulvirenti M.
Published: March 2003

An existence theorem is proven for the solution of the differential equations of motion of a finite number of particles moving in a bounded piecewise regular region and mutually interacting via C1 forces. it is shown that the elastic reflection laws uniquely determine a Lebesque measure solution of the differential equations of motion (with elastic boundary conditions). The Lebesgue measure is invariant so that an extension of the Liouville theorem to non-Hamiltonian flows is obtained. A natural representation of the time evolution is given as a flow upward from a base under a ’’ceiling’’ function.
An approximant for democratic representation of all Born terms
Authors: Brunet Robert C.
Published: March 2003

We propose an approximant which attempts to reconstruct a solution starting from the Born terms of a formal power series. The approximant follows closely the Fredholm solutions to Born–Neumann series of completely continuous operators and their finite rank approximations. We show that if the Fredholm solution is written as a combination of Born terms, it tends to become independent of the expansion parameter λ asymptotically. The proposed approximants then appear as solutions to differential equations approximately satisfied by the formal power series, a feature they share with kernel of finite rank approximations. All Born terms are put on equal footing and their respective weight is determined independently of λ by initial conditions; thus knowledge of the solution and (N−1) derivatives at one point is necessary. Analytic and crossing-symmetry properties are preserved by the proposed approximant, but unitarity is not insured and has to be examined specifically. Its error structure and its properties are studied and compared to Padé approximants.
The Kirkwood–Salsburg equation and phase transitions
Authors: Klein W.
Published: March 2003

A perturbation expansion solution of the Kirkwood–Salsburg equation is used to investigate the hard sphere phase transition. A necessary condition for the phase transition identical in form to one obtained by Kirkwood is obtained. However, it is shown that this condition is not sufficient and that other conditions must also be satisfied.
Rigorous derivation of the Kirkwood–Monroe equation for small activity
Authors: Grewe N. , Klein W.
Published: March 2003

We show, for small values of z, that the solution of the Kirkwood-Salsburg equation approaches, in the norm topolgy, the solution of the Kirkwood-Monroe and van Kampen equations if the potential of interaction is the Kac potential φ (x12) =γsg(γx12) and the limit γ→0 is taken. We have to assume that the function g is bounded and absolutely integrable and that Σi≠j (γxij) ⩾−mB (B<∞), the the sum being performed over all pairs of the m particles.
Spontaneously broken symmetry and cosmological constant
Authors: Wang M. Y.
Published: March 2003

A solution of the Einstein equation with cosmological term produced by spontaneously broken symmetry is presented. The solution implies that the universe will recontract.
n-body scattering into cones with long-range time-dependent potentials
Authors: Hendrickson Joseph H.
Published: March 2003

Modified wave operators are shown to exist for n particles with long-range, time-dependent potentials. Bound states must be assumed to be ’’slightly better’’ than L2. As an application of these temporally inhomogeneous modified wave operators, results of Dollard on scattering into cones are extended to this context.
Stochastic wave-kinetic theory in the Liouville approximation
Authors: Besieris Ioannis M. , Tappert Frederick D.
Published: March 2003

The behavior of scalar wave propagation in a wide class of asymptotically conservative, dispersive, weakly inhomogeneous and weakly nonstationary, anisotropic, random media is investigated on the basis of a stochastic, collisionless, Liouville-type equation governing the temporal evolution of a phase-space Wigner distribution density function. Within the framework of the first-order smoothing approximation, a general diffusion–convolution-type kinetic or transport equation is derived for the mean phase-space distribution function containing generalized (nonloral, with memory) diffusion, friction, and absorption operators in phase space. Various levels of simplification are achieved by introducing additional constraints. In the long-time, Markovian, diffusion approximation, a general set of Fokker–Planck equations is derived. Finally, special cases of these equations are examined for spatially homogeneous systems and isotropic media.
Rigorous bounds on inclusive and exclusive cross sections. I. Lower bound on the integral of an analytic function
Authors: Scanio J. J. G. , Suranyi P.
Published: March 2003

A lower bound on the integral of an analytic function with certain properties is derived. The properties of the function are those expected of an inclusive or exclusive differential cross section. The integral bound can be used to derive rigorous bounds on these cross sections.
Solitons and simple pseudopotentials
Authors: Corones James
Published: March 2003

A simple type of pseudopotential is defined. It is shown how to compute this pseudopotential by classical means. When the construction is attempted based on the Hirota equation it is found that the condition necessary for the existence of the pseudopotential is the same as that for the existence of n-soliton solutions to the Hirota equation. It is also shown how to obtain Backlund transformations using this pseudopotential. It is also pointed out that without making any closure assumption this pseudopotential defines a Lie algebra.
Asymptotic approximations to angular-spectrum representations
Authors: Sherman George C. , Stamnes Jakob J. , Lalor Éamon
Published: March 2003

Under rather general conditiosn a time-harmonic wave field u (x,y,z) can be represented in a half-space z≳0 by a double integral known as the angular spectrum of plane waves. The representation divides naturally into the sum of two double integrals, one of which (uH) is a superposition of homogeneous plane waves and the other (ui) is a superposition of inhomogeneous plane waves. We obtain asymptotic approximations to u (x,y,z), uH, and UI valid when the point of observation of the field recedes towards infinity in a fixed direction through a fixed point. The results apply when the spectral amplitude of the plane waves belongs to a specific class which arises frequently in applications. Our approach is based on the method of stationary phase, which we extend in order to permit the presence of inhomogeneous waves in the integrand. Although the analysis of u requires that we distinguish the directions that are perpendicular to the z axis from the directions pointing into the half-space z≳0, the results for the former case are the same as would be obtained by taking appropriate limit in the results of the latter case. We obtain the general form of the asymptotic sequence appropriate for expanding u and present explicit expressions for the first two terms. Our derivation justifies the results of previous heuristic treatments. The analysis of uH and UI requires separate treatments for directions that are (i) perpendicular to the z axis, (ii) parallel to the z axis, and (iii) neither perpendicular nor parallel to the z axis. In contrast to the behavior of u, the asymptotic behavior of uH (and of uI) differs in the different cases. In each case, we obtain the general form of the appropriate sequence and present the first term explicitly.
Completely positive dynamical semigroups of N-level systems
Authors: Gorini Vittorio , Kossakowski Andrzej , Sudarshan E. C. G.
Published: March 2003

We establish the general form of the generator of a completely positive dynamical semigroup of an N-level quantum system, and we apply the result to derive explicit inequalities among the physical parameters characterizing the Markovian evolution of a 2-level system.
On the nonuniqueness of the Lagrangian
Authors: Browne Seán
Published: March 2003

We describe a general method for comparing two theories which are based on Lagrangians which differ only by the divergence of a 4-vector terms, and we examine how the generators of symmetry transformations depend on this 4-divergence term. For a Poincaré invariant theory we find that the Poincaré generators are independent of this term. Corresponding results are obtained for the case of internal symmetries and for scale and conformal symmetry.
Series of Stieltjes, Padé approximants and continued fractions
Authors: Field David A.
Published: March 2003

Nested sequences of lens-shaped regions which contain Padé and continued fraction approximants to series of Stieltjes are investigated. It is shown by continued fraction methods that the recent results on Padé approximants by Baker and results on continued fractions by Gragg yield identical sequences.
Wave operators for long-range hard-core potentials
Authors: Sáenz A. W. , Zachary W. W.
Published: March 2003

In a previous paper, we have proposed a definition of wave operators for hard-core potentials of long range from the standpoint of time-dependent single-channel quantum scattering theory. In the present note, we prove the existence of these operators for a large class of such potentials. The proof involves a combination of methods previously used to establich the existence of wave operators for short-range hard-core potentials and long-range potentials without hard cores.
Subgroups of the Poincaré group and their invariants
Authors: Patera J. , Sharp R. T. , Winternitz P. , Zassenhaus H.
Published: March 2003

The continuous subgroups of the Poincaré group, classified into conjugacy classes in a previous article, are here classified into isomorphism classes. For each isomorphism class of Lie subalgebras all invariants are found, with a distinction made between Casimir operators (polynomials in the generators), rational invariants (rational functions of the generators), and general invariants (irrational and transcendental functions of the generators). All results are summarized in tables. The meaning of nonpolynomial invariants is briefly discussed and illustrated in examples.
Invariants of real low dimension Lie algebras
Authors: Patera J. , Sharp R. T. , Winternitz P. , Zassenhaus H.
Published: March 2003

All invariant functions of the group generators (generalized Casimir operators) are found for all real algebras of dimension up to five and for all real nilpotent algebras of dimension six.
Generalized molecular distribution functions for fluids and occupation probabilities for lattice gases
Authors: Allnatt A. R.
Published: March 2003

The generalized distribution functions σn(ω) specify the probability of finding a subset of n molecules in a specified configuration and simultaneously a subvolume empty of all other molecules, whereas the conventional distribution functions ρn specify only the configuration. The Mayer integral equation theory for the ρn is generalized assuming short-range intermolecular forces to express both the ρn and σn as sums of integrals over the σn and a kernel dependent on the forces. For the nearest-neighbor lattice gas, certain of these relations are equivalent to those obtained by Widom and Van Leeuwen for the probabilities fn that an empty site is surrounded by n filled sites. The intimate relation of these relations, the Kirkwood–Salsburg integral equations, and the generalized distribution functions is thus displayed.
The Weyl transform of distributions
Authors: Liu Kuang Chi
Published: March 2003

The Weyl transform is defined rigorously on the twisted product algebra of c∞ functions slowly increasing at ∞. The image set under the Weyl rule of correspondence is shown to contain linear unbounded operators. In particular, the momentum and the coordinate operators P and Q and the polynomial functions of P and Q are included.
On the structure of simple pseudo Lie algebras and their invariant bilinear forms
Authors: Nahm W. , Scheunert M.
Published: March 2003

By definition simple pseudo Lie algebras do not contain any nontrivial ideal. We show that ’’graded simplicity’’ implies ’’simplicity’’ and discuss the uniqueness of invariant bilinear forms on a simple pseudo Lie algebra. A lot of examples of simple pseudo Lie algebras is given together with their invariant bilinear forms. Under certain general assumptions we derive that the Lie algebra ℊ contained in a simple pseudo Lie algebra 𝒶 is reductive. Assuming that ℊ is reductive, we prove that the ’’adjoint representation of ℊ in the odd subspace of 𝒶’’ is completely reducible with at most two irreducible components. Finally we show that the pseudo Lie algebras with nondegenerate ’’generalized Killing form’’ are direct products of simple pseudo Lie algebras.