bluuebunny/crossref_metadata_2025 huggingface dataset:
Special Functions and the Complex Euclidean Group in 3-Space. I
Authors: Miller Willard
Published: February 2004

It is shown that the general addition theorems of Gegenbauer, relating Bessel functions and Gegenbauer polynomials, are special cases of identities for special functions obtained from a study of certain local irreducible representations of the complex Euclidean group in 3-space. Among the physically interesting results generalized by this analysis are the expansion for a plane wave as a sum of spherical waves and the addition theorem for spherical waves. This paper is one of a series attempting to derive the special functions of mathematical physics and their basic properties from the representation theory of Lie symmetry groups.
Degenerate Representations of the Symplectic Groups. I. The Compact Group Sp(n)
Authors: Pajas P. , Rączka R.
Published: February 2004

The degenerate, irreducible, unitary representations of the compact group Sp(n), characterized by one and two invariant numbers, are considered. The explicit expressions for the basis functions spanning the corresponding representation spaces and the decomposition with respect to the maximal subgroup are given.
Irreducible Representations of the Five-Dimensional Rotation Group. I
Authors: Kemmer N. , Pursey D. L. , Williams S. A.
Published: February 2004

Explicit matrix elements are found for the generators of the group R(5) in an arbitrary irreducible representation using the ``natural basis'' in which the representation of R(5) is fully reduced with respect to the subgroup R(4)=SU(2)⊗SU(2). The technique used is based on the well-known Racah algebra. The dimension formula is derived and the invariants are found. A family of identities is established which relates various polynomials of degree four in the generators and which holds in any representation of the group.
Irreducible Representations of the Five-Dimensional Rotation Group. II
Authors: Williams S. A. , Pursey D. L.
Published: February 2004

A systematic study is made of the relationship between the generators of R(5) expressed in the ``natural basis,'' as discussed in I [J. Math. Phys. 9, 1224 (1968)], and the same generators in the ``physical basis'' in which representations of R(5) are fully reduced with respect to the physical three-dimensional rotation group. In this paper, attention is confined to the traceless symmetric tensors of R(5) which are the representations appropriate to the discussion of quadrupole vibrations of the nuclear surface. For these representations, one quantum number in addition to the angular momentum and its projection is required to specify a state within a representation. The required extra label is found through the definition of ``intrinsic states'' in the natural basis, and a complete set of states in the physical basis is projected out of these intrinsic states by integrations over the physical rotation group manifold. Members of this set of physical states are not orthonormal; however, the overlap integrals are presented in two simple algebraic forms convenient for computer programming. The construction of the explicit representation matrices for the generators of R(5) is completed by giving the reduced matrix elements of the octopole generator between physical states in terms of the overlap integrals.
Convergence of the Sudarshan Expansion for the Diagonal Coherent-State Weight Functional
Authors: Miller Marvin M.
Published: February 2004

The mathematical properties of the original expansion derived by Sudarshan for the diagonal coherent-state weight functional are discussed. It is shown that, for stationary fields, the expansion is a generalized function in the space Z′(R2). The validity of this method of defining the weight functional in the case of arbitrary density operators and its relationship to other approaches to the problem of the diagonal representation is briefly considered.
Two-Center Coulomb and Hybrid Integrals
Authors: Miller Kenneth J.
Published: February 2004

Coulomb and hybrid integrals are shown to be related to overlap integrals. They are expressed by an integral whose integrand is an overlap integral plus a finite sum of overlap integrals.
Quasibinomial Representations of Clebsch-Gordan Coefficients. I. Square Symbol
Authors: Ansari S. M. Razaullah
Published: February 2004

Quasibinomial representation of Clebsch-Gordan coefficients is symbolized by a Regge-like square. Rules for exchange of rows and columns of this square are derived. Thus our various quasibinomial forms can be read off from this square directly.
Construction of Invariants for Lie Algebras of Inhomogeneous Pseudo-Orthogonal and Pseudo-Unitary Groups
Authors: Rosen Joe
Published: February 2004

A method of constructing invariants for the Lie algebras of the inhomogeneous pseudo-orthogonal and pseudo-unitary groups from invariants of the (homogeneous) pseudo-orthogonal and pseudo-unitary groups, respectively, is presented. The method is based on the ``expansion'' (or ``deformation'') of the inhomogeneous algebras to homogeneous ones. Several examples are worked out.
Functional Integrals for Many-Boson Systems
Authors: Casher A. , Lurié D. , Revzen M.
Published: February 2004

We express the grand canonical partition function (GCPF) for a system of interacting bosons as a functional integral over one complex function. Our derivation is based on the so-called coherent-state representation [R. J. Glauber, Phys. Rev. 136, 2766 (1963)]. We show how to extract the perturbative expansion of the GCPF and the various Green's functions from our functional-integral representation and we indicate the relevance of our formalism to the theory of superfluidity.
Generalization of the ``Schwarzschild Surface'' to Arbitrary Static and Stationary Metrics
Authors: Vishveshwara C. V.
Published: February 2004

A generalization of the r = 2m ``Schwarzschild surface'' is defined for static metrics which are not necessarily spherically symmetric. This surface exhibits simultaneously the properties of being a ``one-way membrane'' for causal propagation and of being a surface of infinite red shift. The necessary and sufficient condition that these two phenomena take place on the same surface in an arbitrary stationary metric is also obtained. The distinctions between the static and stationary cases are shown to be essential by examples from the Kerr metric.
Exactly Solvable Many-Boson Model
Authors: Richardson R. W.
Published: February 2004

A many-boson model is formulated and expressions for its exact eigenstates and energies are obtained for both an arbitrary finite and an infinite number of bosons. The Hamiltonian of the model contains interactions between bosons whose momenta have equal magnitudes but opposite directions. The matrix elements of this interaction are taken to be a constant over a range of momenta surrounding k = 0. The ground state of the 2N-particle system is shown to be a product of N pair-creation operators acting on the vacuum state. Each of these pair-creation operators depends upon one of N parameters which are called pair energies. The N pair energies are shown to satisfy a coupled system of nonlinear algebraic equations. The energy of the state is the sum of the pair energies and the occupation probabilities of the single-particle levels are given as simple functions of the pair energies. Similar results are derived for the excited states of the system and for the states of an odd number of particles. These results are valid for both a repulsive and an attractive interaction, since they only depend upon the form of the interaction. The equations are solved algebraically for two model systems. The first of these is one whose single-particle kinetic energy takes on only one value. The equations for this system are solved for an arbitrary interaction strength and it is shown that the pair energies are proportional to the zeros of certain Laguerre polynomials. The second system is one in which the single-particle kinetic energy can take on two values. The equations for this system are solved in the strong repulsive-interaction limit and it is shown that the pair energies are proportional to the zeros of certain Jacobi polynomials. The excitation energies of this second system are shown to be proportional to 1/n and the occupations of the two single-particle levels in the ground state are shown to be proportional to n, where n is the total number of particles. For a repulsive interaction and an arbitrary single-particle spectrum, the algebraic equations for the pair energies are converted into an approximate integral equation for the density of roots which is accurate to order 1/n. This integral equation is solved for a strong interaction which, in the context of this model, means an interaction whose strength is greater than a constant times 1/V⅔ in the limit of a large volume. From this solution, the following results are obtained: (1) the lowest two single-particle levels have occupations of order n; (2) the excitation spectrum is that of a set of noninteracting quasiparticles; and (3) the quasiparticle spectrum has two zeros corresponding to the lowest two single-particle levels. Apart from the presence of two zeros, the quasiparticle spectrum does not differ significantly from that of the noninteracting particles.
Isotropic Solutions of the Einstein-Liouville Equations
Authors: Ehlers J. , Geren P. , Sachs R. K.
Published: February 2004

The gravitational field generated by a gas whose one-particle distribution function obeys the Liouville equation is examined under the following assumptions: First, the distribution is locally isotropic in momentum space with respect to some world-velocity field; second, if the particles have rest-mass zero, the gas is irrotational. It is shown that the model is then either stationary or a Robertson-Walker model. The time dependence of the radius in the Robertson-Walker models is given in terms of integrals containing the distribution function.
Some Variational Principles for Integral Equations
Authors: Robinson P. D. , Arthurs A. M.
Published: February 2004

Complementary variational principles are developed for the solution of Fredholm integral equations with symmetric positive-definite kernels. In particular, the theory is applied to linear equations of the type φ(r)=f(r)+λ∫K(r,s)φ(s)ds,and bounds are obtained for ∫fφdr. When λ is negative, the bounds are complementary upper and lower ones. When λ is positive, the bounds are one-sided, but an improvement is made on a result of Strieder and Prager [J. Math. Phys. 8, 514 (1967)]. A condition given by these authors for the existence of bounds does not seem to be strictly necessary, and alternative conditions are derived. Systematic improvement of bounds by iterative and scaling procedures is discussed.
Analytic Renormalization
Authors: Speer Eugene R.
Published: February 2004

Renormalized Feynman amplitudes are defined by a method of analytic continuation in subsidiary parameters. The results are shown to belong to the class of renormalized amplitudes defined by Boguliubov, Parasiuk, and Hepp.
Violation of the Quantum Ordering of Propositions in Hidden-Variable Theories
Authors: Turner James E.
Published: February 2004

A general definition of hidden-variable theories in terms of the dual structure of states and propositions is proposed. As a consequence of a theorem due to Zierler and Schlessinger, this definition implies a violation of the quantum ordering of propositions in the corresponding hidden-variable theory. This violation is shown explicitly for the theory of measurement due to Bohm and Bub.
Lorentz Covariant Distributions
Authors: Ynduráin F. J.
Published: February 2004

Tensor distributions of several four-vector variables which transform according to a finite representation of the Lorentz group are considered. We give a canonical classification of all possible forms of such objects; this is used to show that, in the relevant cases, it is sufficient to regularize them with respect to the invariants that may be formed out of the variables to obtain analytic functions. We apply this result to Wightman functions, showing a result similar to a theorem proved in position space by Borchers under different assumptions.
Background Dependence of Local States
Authors: Licht A. L.
Published: February 2004

We consider states prepared by a series of actions performed in a finite space-time region in the presence of a background state described by a density matrix. It is shown that there will always be some observable whose expectation value in such a state will depend to some extent on the background. There will, however, be a certain set of observables whose expectation values will be independent of the background. We obtain a characterization of such sets of observables.
Bethe-Salpeter Equation
Authors: Broido Michael M. , Taylor John G.
Published: February 2004

We treat the Bethe-Salpeter equation as a problem in singular integral equations. As such, it has three outstanding features: its algebraic structure, the fixed propagator singularities in the direct channel, and the possible singularities in the potential, which are usually moving singularities. We exploit the algebraic structure in order to give insight into the possible correctness classes for the equation. We give explicit prescriptions for the removal of fixed singularities in a wide class of equations. We show under what circumstances these prescriptions can be adapted to maintain such desirable features as symmetry of the kernel. Moving singularities arise in physically realistic kernels; they are the crossed-channel singularities. The basic mathematics of such singularities is well known and is related to the Riemann-Hilbert problem, but this is useless in off-shell methods because it cannot cope with the integration over the space parts of 4-momenta. Instead, we adopt a method (proposed by one of us elsewhere) based on analyticity in energy variables. The resulting formalism is too complicated to be applied in full generality. We therefore consider the example of the single-particle exchange potential in detail, and show how the moving singularities can be eliminated, exhibiting the resulting equations explicitly in a form to which our theory of fixed singularities can immediately be applied. All our arguments are exact.
Configuration-Space Approach to the Four-Particle Problem
Authors: Zickendraht W.
Published: February 2004

The configuration-space approach to the three-particle problem is generalized to the case of four particles. Special coordinates are defined which have simple symmetry properties with respect to the exchange of identical particles. The construction of a suitable orthogonal system is discussed. Some of these functions are given explicitly. It is pointed out that the use of this orthogonal system leads to a considerable simplification for a large number of four-particle problems, namely, the approximate reduction of the Schrödinger equation to a finite system of coupled differential equations for functions that depend on one variable only.
Note on Two Binomial Coefficient Identities of Rosenbaum
Authors: Gould H. W.
Published: February 2004

This paper gives rapid proofs of two binomial coefficient identities found by Rosenbaum [J. Math. Phys. 8, 1977 (1967)] who obtained the identities from rather involved considerations of commutation relations. The present proofs make use of the Vandermonde convolution, or addition, theorem and a well-known fact that the kth difference of a polynomial of degree k − 1 is zero. In a sense the two special cases are not essentially new.
Killing Horizons and Orthogonally Transitive Groups in Space-Time
Authors: Carter Brandon
Published: February 2004

Some concepts which have been proven to be useful in general relativity are characterized, definitions being given of a local isometry horizon, of which a special case is a Killing horizon (a null hypersurface whose null tangent vector can be normalized to coincide with a Killing vector field) and of the related concepts of invertibility and orthogonal transitivity of an isometry group in an n-dimensional pseudo-Riemannian manifold (a group is said to be orthogonally transitive if its surfaces of transitivity, being of dimension p, say, are orthogonal to a family of surfaces of conjugate dimension n - p). The relationships between these concepts are described and it is shown (in Theorem 1) that, if an isometry group is orthogonally transitive then a local isometry horizon occurs wherever its surfaces of transitivity are null, and that it is a Killing horizon if the group is Abelian. In the case of (n - 2)-parameter Abelian groups it is shown (in Theorem 2) that, under suitable conditions (e.g., when a symmetry axis is present), the invertibility of the Ricci tensor is sufficient to imply orthogonal transitivity; definitions are given of convection and of the flux vector of an isometry group, and it is shown that the group is orthogonally transitive in a neighborhood if and only if the circulation of convective flux about the neighborhood vanishes. The purpose of this work is to obtain results which have physical significance in ordinary space-time (n = 4), the main application being to stationary axisymmetric systems; illustrative examples are given at each stage; in particular it is shown that, when the source-free Maxwell-Einstein equations are satisfied, the Ricci tensor must be invertible, so that Theorem 2 always applies (giving a generalization of the theorem of Papapetrou which applies to the pure-vaccuum case).
High-Frequency Scattering by a Transparent Sphere. I. Direct Reflection and Transmission
Authors: Nussenzveig H. M.
Published: February 2004

This is Paper I of a series on high-frequency scattering of a scalar plane wave by a transparent sphere (square potential well or barrier). It is assumed that (ka)⅓≫1,|N−1|½(ka)⅓≫1, where k is the wave-number, a is the radius of the sphere, and N is the refractive index. By applying the modified Watson transformation, previously employed for an impenetrable sphere, the asymptotic behavior of the exact scattering amplitude in any direction is obtained, including several angular regions not treated before. The distribution of Regge poles is determined and their physical interpretation is given. The results are helpful in explaining the reason for the difference in the analytic properties of scattering amplitudes for cutoff potentials and potentials with tails. Following Debye, the scattering amplitude is expanded in a series, corresponding to a description in terms of multiple internal reflections. In Paper I, the first term of the Debye expansion, associated with direct reflection from the surface, and the second term, associated with direct transmission (without any internal reflection), are treated, both for N > 1 and for N < 1. The asymptotic expansions are carried out up to (not including) correction terms of order (ka)−2. For N > 1, the behavior of the first term is similar to that found for an impenetrable sphere, with a forward diffraction peak, a lit (geometrical reflection) region, and a transition region where the amplitude is reduced to generalized Fock functions. For N < 1, there is an additional shadow boundary, associated with total reflection, and a new type of surface waves is found. They are related to Schmidt head waves, but their sense of propagation disagrees with the geometrical theory of diffraction. The physical interpretation of this result is given. The second term of the Debye expansion again gives rise to a lit region, a shadow region, and a Fock-type transition region, both for N > 1 and for N < 1. In the former case, surface waves make shortcuts across the sphere, by critical refraction. In the latter one, they excite new surface waves by internal diffraction.
Cluster Expansion in a Bose System
Authors: Lee M. Howard
Published: February 2004

When a trial function for Bose systems contains both one-particle and two-particle functions, the cluster expansion is no longer simply related to the classical expansion of Mayer and Mayer. But by observing that the permutations in one-particle functions can be represented exactly as in the high-temperature expansion of the spin-½ XY model, the cluster expansion is developed analogously to the classical case.
Formal Solutions of Inverse Scattering Problems
Authors: Prosser Reese T.
Published: February 2004

Formal solutions of inverse scattering problems for scattering from a potential, a variable index of refraction, and a soft boundary are developed using a method devised by Jost and Kohn.
Solutions of the Einstein and Einstein-Maxwell Equations
Authors: Debney G. C. , Kerr R. P. , Schild A.
Published: February 2004

Algebraically degenerate solutions of the Einstein and Einstein-Maxwell equations are studied. Explicit solutions are obtained which contain two arbitrary functions of a complex variable, one function being associated with the gravitational field and the other mainly with the electromagnetic field.
Remarks on Lie-Algebra-Invariant Equations. I
Authors: Humi Mayer
Published: February 2004

We construct first-order-invariant equations with respect to the inhomogeneous Euclidean algebra IE(n) (isomorphic to the n-dimensional Galilei algebra) by embedding its Euclidean subalgebra E(n) in O(n + 1, 1). This embedding allows us to utilize Gel'fand's method [I. Gel'fand et al., Representations of the Rotation and Lorentz Groups and Their Applications (Pergamon Press, Inc., New York, 1963)] adapted to the construction of invariant equations on semisimple algebras to nonsemisimple ones also.
Finite Transformations and Basis States of SU(n)
Authors: Holland Douglas Francis
Published: February 2004

A convenient parameterization for finite transformations of SU(n) is developed which explicitly exhibits the special unitary subgroups. This is also used to parameterize the defining space. Higher-dimensional representations are discussed. The question of which representations carry the trivial representation of SU(n − 1) is considered, as well as the parameterization of these states. Application is made to the transformations and basis states of SU(3).
Soluble Model of Condensation
Authors: Girardeau M. D.
Published: February 2004

A soluble model of gas-liquid condensation is constructed by a quantum-mechanical extension of van Kampen's classical analysis. The Hamiltonian of a system of identical particles is separated into a ``core'' and a ``tail'' Hamiltonian. Following van Kampen, we simulate the actual core Hamiltonian by an effective Hamiltonian with the same free energy. The tail Hamiltonian is replaced by its diagonal part in a single-particle basis chosen self-consistently so that the thermodynamics is exactly soluble. The resultant free energy agrees with that of Lieb's quantum-mechanical extension of the van der Waals-Maxwell theory, without the necessity of taking the van der Waals limit of a suitably infinite-ranged and infinitely-weak attractive tail. However, the exchange contributions omitted from the model become negligible only in that limit.
Alternative Dynamics for Classical Relativistic Particles
Authors: Katz Amnon
Published: February 2004

Equations of motion for interacting classical relativistic particles (different from Van Dam-Wigner equations) are derived from an action principle. In the classical limit they reduce to Newton's equations with an interparticle potential depending on distance only.
New Methods for Reduction of Group Representations. IV
Authors: Gabriel J. R.
Published: February 2004

Some of the methods given in an earlier paper are extended; in particular, it is shown how to construct the central from the matrix representatives of the group generators and how to apply the techniques given for the central to simplify the reduction of the representation.
Conformal-Invariant Scattering Amplitude for Spinless Particles
Authors: Bali Naren F. , Coon Darryl D. , Katz Amnon
Published: February 2004

The implications of exact conformal symmetry for the S matrix of spinless massless particles are studied. Explicit forms for the scattering amplitudes are obtained in configuration space. Differential conditions for conformal-invariant amplitudes are derived in momentum space. These are solved for the case of elastic two-body scattering and lead to an integral representation of the general two-body amplitude in terms of Lorentz scalars. The integral representation involves two arbitrary functions of two variables and four explicit quadratures. Further study of the four quadratures appears in appendices.
On the Uncoupling of Wiener-Hopf Equations for Half-Plane Diffraction Problems in a Gyrotropic Medium
Authors: Tan H. S. , Cheng D. K.
Published: February 2004

Integral equations of the Wiener-Hopf type appear in the study of electromagnetic diffraction problems by a half-plane. When the environment is gyroelectric and characterized by an arbitrary permittivity tensor and a scalar permeability, the equations become coupled and no general method of solution is available unless a certain specific criterion is met. Hurd formulated this criterion and examined a number of solvable cases. The present paper extends Hurd's work to include oblique incidence on a half-plane in a gyrotropic medium. Media which are characterized by an arbitrary permittivity tensor and a diagonal permeability tensor are considered. An analogous but more general class of situations is obtained which permits the uncoupling of the Wiener-Hopf equations.
Numerical Solution of Integral Equations Using Legendre Polynomials
Authors: Kschwendt H.
Published: February 2004

This paper deals with the solution of certain integral equations of displacement type. A previously developed theory, which uses an expansion of the solution in Legendre polynomials, is applied to two special cases. One is a singular equation encountered in polymer physics. The results of the Legendre expansion are compared with calculations based on two other recently proposed methods, and the merits of an expansion in Gegenbauer polynomials are discussed. For the second case, which arises in electrodynamics, a very accurate solution is given for a large range of parameter values.
Canonical Transformations and Spectra of Quantum Operators
Authors: Leaf Boris
Published: February 2004

The claim that unitary transformations in quantum mechanics correspond to the canonical transformations of classical mechanics is not correct. The spectra of operators produced by unitary transformation of Cartesian coordinate and momentum operators (q, k) are necessarily continuous over the entire real domain of their eigenvalues. Operators with spectra which are not everywhere continuous are generated from (q, k) by one-sided unitary transformations U for which U†U = 1 but for which UU† commutes with either q or k (but not both). If UU† commutes with k, the new coordinates and momenta (r, s) satisfy commutation relations [sm, rn] = 2πi1δm,n, [sm, sn] = 0, but [rm, rn] ≠ 0; (r, s) are canonical only for one-dimensional systems. The properties of one-sided unitary transformations are described; they are characterized by φ(K), the eigenvalue of UU†. The one-dimensional case for which the one-sided unitary transformation is canonical is discussed in detail. A prescription is given for obtaining the operator canonically conjugate to any one-dimensional observable. Generalization to higher dimensions is also discussed.
Canonical Operators for the Simple Harmonic Oscillator
Authors: Leaf Boris
Published: February 2004

Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.
Perturbation Method for a Nonlinear Wave Modulation. II
Authors: Asano Naruyoshi , Taniuti Tosiya , Yajima Nobuo
Published: February 2004

A perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein-Gordon equation. In these systems, and probably in most physical systems, an assumed condition for a mode of l = 0 is not valid. Consequently, the direct application of the method is impossible. In the present paper, we shall illustrate by these examples how this difficulty can be overcome to allow us to use the method. As a result we shall find that, in either case, the original equation can be reduced to the nonlinear Schrödinger equation.
Boson Formalism in Superconductivity
Authors: Leplae L. , Umezawa H.
Published: February 2004

After a brief summary of the results obtained previously concerning the boson formalism in superconductivity, the formalism is generalized to include finite temperature and Coulomb interactions. Finally, as an illustration, the formalism is used to derive the Landau-Ginsburg equations and to study the vortices in type II superconductors.
Possible Characterization of Short-Range Interactions
Authors: Finley J. D.
Published: February 2004

In the preceding paper [J. Math. Phys. 10, 2047 (1969)], we investigated possible characterizations of ``short-range'' interactions by means of conditions on the total transition probability ∥Tψ∥2 which were valid only for certain types of sequences of wavepackets. In this paper we generalize the assumption on the transition probability so that it will apply to an arbitrary wavepacket. We do this by utilizing a measure G[R, ψ] of how much of a wavepacket ψ is within a sphere of radius R about the scattering center at all times. We then characterize a ``short-range'' interaction by the existence of a positive constant A and positive functions B(p) and C(p) such that for all R ≥ 0, ‖Tψ‖2≤AG[R,ψ]+〈C(p)e−RB(p)〉ψ.We then show that for such interactions the square of the scattering amplitude is analytic, in the cosine of the scattering angle, in an ellipse the size of which depends upon B(p). We compare our results with the similar results obtained by the method in the preceding paper and show that these new results much more closely approximate the behavior of the Yukawa interaction.
Irreducibility of the Ladder Representations of U(2, 2) when Restricted to the Poincaré Subgroup
Authors: Mack G. , Todorov I.
Published: February 2004

It is shown that the most degenerate discrete series of unitary irreducible representations of U(2, 2), the so-called ladder representations, remain irreducible when restricted to representations of the Poincaré subgroup ISL(2, C). They correspond to representations of this subgroup with mass zero and arbitrary integer or half-integer helicity λ. The basis vectors of the canonical basis are calculated as functions of a lightlike 4-vector, which is formed by the simultaneous eigenvalues of the generators of the subgroup of translations.
On a Generalized Distribution of the Poles of the Unitary Collision Matrix
Authors: Ullah Nazakat
Published: February 2004

An expression for the joint distribution of the complex poles of the unitary collision matrix is derived for the single-channel case, which is valid for all values of the ratio of the width to the spacing. The derivation uses the statistical distribution of the parameters of the real R-matrix theory. We find that unitarity gives rise to the statistical correlations between the width and the spacing of the collision matrix. It is shown that the distribution of the poles of the unitary collision matrix using Feshbach's unified theory of nuclear reactions is the same as the one obtained using R-matrix theory, provided we make a particular choice of the arbitrary boundary condition in the latter theory. A remark is made about the use of the random complex orthogonal matrix in the study of the parameters of the statistical collision matrix.
Treatment of Degeneracies in the Schrödinger Perturbation Theory by Partitioning Technique
Authors: Choi Jong H.
Published: February 2004

The degenerate case in the Schrödinger perturbation theory has been treated by use of the partitioning technique developed by Löwdin. In order to simplify the concept and treatment, the repeated partitioning technique is utilized. This repeated partitioning allows us to use a one-dimensional reference space and to determine the correct zero-order wavefunction φA.
Physical Regions of Six-Particle Processes
Authors: McNeil Richard P. , Morrow Richard A.
Published: February 2004

The physical regions of six-particle processes are constructed in all planes of pairs of Lorentz-invariant variables. As a matter of course, the permissible ranges of the eight independent variables are established. Thus, one application is the determination of the integration limits in phase-space integrals that occur in calculations involving two-to-four and one-to-five particle processes.
Solution of a Three-Body Problem in One Dimension
Authors: Calogero F.
Published: February 2004

The problem of three equal particles interacting pairwise by inversecube forces (``centrifugal potential'') in addition to linear forces (``harmonical potential'') is solved in one dimension.
Vacancy Annihilation for One-Dimensional Dumbbell Kinetics
Authors: McQuistan R. B.
Published: February 2004

Expressions are developed which describe the ensemble average of the annihilation of groups of contiguous vacant compartments when spatially random attempts are made to place dumbbells on a linear array of N compartments. It is shown that in the limit, as the number of compartments tends to infinity, <θp(t)>, the ensemble average of the fraction of the compartments which is composed of p contiguous vacant compartments, is given by 〈θp(t)〉=pe−2(1− ∑ n=2∞Cne−nvt)e−(p−1)vt,where v is the striking frequency of the dumbbells, t is time, and the Cn's are appropriately defined coefficients.
Infinite Renormalization of the Hamiltonian Is Necessary
Authors: Glimm James , Jaffe Arthur
Published: February 2004

We show that the unrenormalized Hamiltonian in quantum field theory is unbounded from below whenever lowest-order perturbation theory indicates that this is true. We conclude that perturbation theory is an accurate guide to the divergence of the vacuum energy in quantum field theory.
Relativistic Effects of Strong Binding on Slow Particles
Authors: Katz Amnon
Published: February 2004

The relativistic effect of strong binding on the equations of motion of slow particles is derived by taking the appropriate limit of classical relativistic equations of motion of interacting particles. The expected effect on the total mass of the system is verified. The relative motion is also affected-in a modeldependent way.
Half-Space Multigroup Transport Theory
Authors: Pahor S. , Shultis J. K.
Published: February 2004

A method for solving various half-space multigroup transport problems for the case of a symmetric transfer matrix is explained. This method is based on the full-range completeness and orthogonality properties of the infinite-medium eigenfunctions. First, the albedo problem is considered. A system of Fredholm integral equations is derived for the emergent distribution of the albedo problem, and it is shown that this system has a unique solution. Then, by using the full-range eigenfunction completeness, the inside angular distribution is obtained from the emergent distribution. Finally, the Milne problem and the half-space Green's function problem are solved in terms of the emergent distribution of the albedo problem and the infinite-medium eigenfunctions.
Spectral Analysis of Classical Central Force Motion
Authors: Prosser Reese T.
Published: February 2004

It is shown here that the Liouville operator, which governs the development in time of a classical one-particle system, has an absolutely continuous spectrum for a large class of attractive central force potentials. It follows that every absolutely continuous initial distribution of a monatomic ideal gas enclosed in a spherical container must approach a steady-state distribution in time.
Continuous Representation Theory Using the Affine Group
Authors: Aslaksen Erik W. , Klauder John R.
Published: February 2004

We present a continuous representation theory based on the affine group. This theory is applicable to a mechanical system which has one or more of its classical canonical coordinates restricted to a smaller range than − ∞ to ∞. Such systems are especially troublesome in the usual quantization approach since, as is well known from von Neumann's work, the relation [P, Q] = −iI implies that P and Q must have a spectrum from − ∞ to ∞ if they are to be self-adjoint. Consequently, if the spectrum of either P or Q is restricted, at least one of the operators, say Q, is not self-adjoint and does not have a spectral resolution. Thus Q cannot generate a coordinate representation. This leads us to consider a different pair of operators, P and B, both of which are self-adjoint and which obey [P, B] = −iP. The Lie group corresponding to this latter algebra is the affine group, which has two unitarily inequivalent, irreducible representations, one in which the spectrum of P is positive. Using the affine group as our kinematical group, we have developed continuous representations analogous to those Klauder and McKenna developed for the canonical group, and have shown that the former representations have almost all the desirable properties of the latter.
Dirac Formalism and Symmetry Problems in Quantum Mechanics. II. Symmetry Problems
Authors: Antoine J.-P.
Published: February 2004

The quantum-mechanical formalism developed in a previous article and based on the use of a rigged Hilbert space Φ⊂H⊂Φ′ is here enlarged by taking into account the symmetry properties of the system. First, the compatibility of a particular symmetry with this structure is obtained by requiring Φ to be invariant under the corresponding representation U of the symmetry group in H. The symmetry is then realized by the restriction of U to Φ and its contragradient representation Ǔ in Φ′. This double manifestation of the symmetry is related to the so-called active and passive points of view commonly used for interpreting symmetry operations. Next, a general procedure is given for constructing a suitable space Φ out of the labeled observables of the system and the representation U describing its symmetry properties. This general method is then applied to the case where U is a semidirect product G=T[squared times]Δ, with T Abelian. Finally, the examples of the Euclidean, the Galilei, and the Poincaré groups are briefly studied.
Uniform Asymptotic Theory of Edge Diffraction
Authors: Lewis Robert M. , Boersma Johannes
Published: February 2004

Geometrical optics fails to account for the phenomenon of diffraction, i.e., the existence of nonzero fields in the geometrical shadow. Keller's geometrical theory of diffraction accounts for this phenomenon by providing correction terms to the geometrical optics field, in the form of a high-frequency asymptotic expansion. In problems involving screens with apertures, this asymptotic expansion fails at the edge of the screen and on shadow boundaries where the expansion has singularities. The uniform asymptotic theory presented here provides a new asymptotic solution of the diffraction problem which is uniformly valid near edges and shadow boundaries. Away from these regions the solution reduces to that of Keller's theory. However, singularities at any caustics other than the edge are not corrected.
Closed-Form Solution of the Differential Equation (∂2∂x∂y+ax∂∂x+by ∂∂y+cxy+∂∂t)P=0 Subject to the Initial Condition P(x, y, t = 0) = Φ(x, y)
Authors: Neuringer J. L.
Published: February 2004

A closed-form solution of the differential equation (∂2∂x∂y+ax∂∂x+by ∂∂y+cxy+∂∂t)P=0,subject to the initial condition P(x, y, t = 0) = Φ(x, y), is presented.
Symmetry of the Two-Dimensional Hydrogen Atom
Authors: Cisneros Arturo , McIntosh Harold V.
Published: February 2004

A paradox has arisen from some recent treatments of accidental degeneracy which claim that, for three degrees of freedom, SU(3) should be a universal symmetry group. Such conclusions are in disagreement both with experimentally observed spectra and with the generally accepted solutions of Schrödinger's equation. The discrepancy occurs in the transition between classical and quantum-mechanical formulations of the problems, and illustrates the care necessary in forming quantum-mechanical operators from classical expressions. The hydrogen atom in parabolic coordinates in two dimensions, for which the traditional treatment of Fock, extended by Alliluev, requires the symmetry group O(3), is a case for which the newer methods of Fradkin, Mukunda, Dulock, and others require SU(2). Although these groups are only slightly different, SU(2) fails to be the ``universal'' symmetry group on account of the multiple-valuedness of the parabolic representation. This conclusion extends a result of Han and Stehle: that, for rather similar reasons, SU(2) cannot be the classical symmetry group for the two-dimensional hydrogen atom.
Elementary Solutions of the Reduced Three-Dimensional Transport Equation
Authors: Kaper Hans G.
Published: February 2004

By treating one of the space dimensions exactly and approximating the other two by the exp (−iB·r) assumption, which is suggested by asymptotic transport theory, it is possible to reduce the three-dimensional transport equation to an equation that is of one-dimensional form and that still contains details of the complete three-dimensional angular distribution. In this paper we develop the method of elementary solutions for the reduced transport equation in the case of time-independent, monoenergetic neutron transport in homogeneous media with isotropic scattering. The spectrum of the transport operator consists of a pair of discrete points if B2 is sufficiently small and a continuum which occupies a two-dimensional region in the complex spectral plane. The eigenfunctions possess full-range and half-range orthogonality and completeness properties, which are proved via the solution of two-dimensional integral equations using the theory of boundary-value problems for generalized analytic functions. As applications we solve the Green's function for an infinite homogeneous prism and the albedo operator for a semi-infinite homogeneous prism. Also discussed are possible generalizations of the method to more complicated forms of the reduced transport equation.
Classification and Complete Reducibility of Representations of Compact Inhomogeneous Groups and Algebras
Authors: Mirman R.
Published: February 2004

The representations of compact inhomogeneous groups are classified according to which type of operators are taken as diagonal. The question of complete reducibility of some of the different classes is discussed.
Crossing and Unitarity in a Multichannel Static Model. II
Authors: Ehrhardt P. O. G. , Fairlie D. B.
Published: February 2004

Constraints on the solution of the unitarity equation for a two-channel scattering problem arising from the requirements of crossing for the inelastic amplitude are shown to imply the relations imposed by crossing under SU(2) for the elastic amplitudes in the limit of vanishing coupling to the inelastic channel. This result is extended to the three-channel case, where at least two distinct classes of internal symmetry crossing relations can exist: the unitarity equations alone simulate the SU(2) case, but not that associated with higher symmetry groups.
Normal and Abnormal Diffusion in Ehrenfests' Wind-Tree Model
Authors: Hauge E. H. , Cohen E. G. D.
Published: February 2004

The Ehrenfest wind-tree model—a special case of the Lorentz model—where noninteracting point particles move in the plane through a random array of square scatterers, is used to study the divergences previously discovered in the density expansions of the transport coefficients. Two cases for which the results are qualitatively different are discussed. When the scatterers are not allowed to overlap, the diffusion of particles through the array of scatterers is normal, characterized by a diffusion constant D. The calculation of D−1 is carried to the second order in the density of the scatterers, and involves a discussion of the above-mentioned divergences and a resummation of all most-divergent terms in the straightforward expansion of D−1. If, however, the trees are allowed to overlap, the growth of the mean-square displacement with time is slower than linear, so that no diffusion coefficient can be defined. The origin and possible relevance of this new phenomenon to other problems in kinetic theory is discussed. The above results have stimulated molecular dynamics calculations by Wood and Lado on the same model. Their preliminary results seem to confirm the theoretical predictions.
Spectral Theory of the Difference Equation f(n + 1) + f(n − 1) = [E − φ(n)]f(n)
Authors: Deliyannis Platon C. , Ifantis Evangelos K.
Published: February 2004

In this work, the spectrum of the second-order difference equation f(n+1)+f(n−1)=[E−φ(n)]f(n)in the l2 Hilbert space is studied for the case in which the limit of the sequence {φ(n)}n=1∞ exists. By means of a simple representation the problem is transferred to one about the spectrum of an abstract operator in a separable Hilbert space. This operator T has a form analogous to the Schrödinger operator, namely T = T0 + A, where T0 is self-adjoint with a purely continuous spectrum but bounded, while A depends on the sequence {φ(n)}. In fact, A is of Hilbert-Schmidt type for any {φ(n)} in l2, and of trace class if the series ∑n=1∞|φ(n)| converges. Sufficient conditions for the existence of a discrete spectrum and more generally, of proper values, are found. Using the theory of the wave operators Ω±=s−lim lim t→±∞exp (iTt) range exp (−iT0t), results on the existence of a mixed spectrum are obtained.
Ray Velocity and Exceptional Waves: A Covariant Formulation
Authors: Boillat Guy
Published: February 2004

In a nonlinear field an accelerated wave sooner or later turns into a shock. When this is not the case, the wave is exceptional (e.g., Alfvén waves of magnetohydrodynamics). Then the normal speed of the wave remains undisturbed. This criterion is given a convenient covariant form in terms of the ray velocity. As an example a thermodynamical relativistic fluid is studied.
S-Matrix Singularity Structure in the Physical Region. I. Properties of Multiple Integrals
Authors: Bloxham M. J. W. , Olive D. I. , Polkinghorne J. C.
Published: February 2004

We obtain the real singularities and corresponding discontinuities of a class of multiple integrals over real contours. Our aim is to give a unified treatment, obtaining, by elementary mathematical methods, both previously known results and some new generalizations. In a subsequent paper the results are applied to unitarity integrals.