Volume 19, Issue 12, December 1978
Index of content:

Punctual Padé approximants as a regularization procedure for divergent and oscillatory partial wave expansions of the scattering amplitude
View Description Hide DescriptionPrevious theorems on the convergence of the [n,n+m] punctual Padé approximants to the scattering amplitude are extended. The new proofs include the cases of nonforward and backward scattering corresponding to potentials having 1/r and 1/r ^{2} long‐range behaviors, for which the partial wave expansions are divergent and oscillatory, respectively. In this way, the ability of the approximation scheme as a summation method is established for all of the long‐range potentials of interest in potential scattering.

Spectral and scattering inverse problems
View Description Hide DescriptionThe reconstruction of a differential operator form discrete spectra is reduced to its reconstruction from an S‐matrix. This method makes it possible to solve the singular Sturm–Liouville problems which determine certain modes of a sphere. The results pave the way for handling studies in which information on modes and scattering results would all be taken into account. They are applied to the earth inverse problem and partial answers are given to a well‐known conjecture. Finally the relevance of the JWKB approximation in this kind of problem is briefly discussed.

Renormalized plane wave‐projected and Coulomb‐projected T‐matrices
View Description Hide DescriptionThe plane wave‐projected formulation of the half‐shell T‐matrix for ionization and the Coulomb‐projected formulation of the half‐shell T‐matrices for excitation and ionization are shown to converge to zero in the energy‐shell limit. ’’Renormalized’’ plane wave‐projected and Coulomb‐projected half‐shell T‐matrices are defined and are shown to have physical energy‐shell limits.

An illustration of the Lie group framework for soliton equations: Generalizations of the Lund–Regge model
View Description Hide DescriptionThe Lie group framework for soliton equations is illustrated. It is shown that the original Lund–Regge model is one of an infinite family of similar relativistically invariant models that possess associated eigenvalue problems and isospectral flows. The models are explicitly found and their associated structures displayed. The group theoretic significance of the soliton equations and associated structures are given in accordance with the general theory.

Time‐reversal noninvariance of the quantum‐mechanical kinetic equation of Kadanoff and Baym
View Description Hide DescriptionApplying anticausal Green’s function technique, it has been shown that the origin of time‐reversal noninvariance of the quantum‐mechanical kinetic equation of Kadanoff and Baym lies in the selection of the boundary condition that the system be in thermodynamical equilibrium at time t=−∞.

Tensor spherical harmonics on S ^{2} and S ^{3} as eigenvalue problems
View Description Hide DescriptionTensor spherical harmonics for the 2‐sphere and 3‐sphere are discussed as eigenfunction problems of the Laplace operators on these manifolds. The scalar, vector, and second‐rank tensor harmonics are given explicitly in terms of known functions and their properties summarized.

Conformal two‐structure as the gravitational degrees of freedom in general relativity
View Description Hide DescriptionIn this paper, we suggest that what we shall call the conformal 2‐structure may, in an appropriate coordinate system, serve to embody the two gravitational degrees of freedom of the Einstein (vacuum) field equations. The conformal 2‐structure essentially gives information concerning the manner in which a family of 2‐surfaces is embedded in a 3‐surface. We show that, formally at least, this prescription works for the exact plane and cylindrical gravitational wave solutions, for the double‐null and null‐timelike characteristic initial value problems, and for the usual Cauchy spacelike initial value problem. We conclude with a preliminary consideration of a two‐plus‐two breakup of the field equations aimed at unifying these and other initial value problems; and a discussion of some aspirations and remaining problems of this approach.

Effective Lagrangian in spontaneously broken gauge theories
View Description Hide DescriptionIn gauge theories of weak and electromagnetic interactions, it is generally assumed that the addition of extra groups [simple or U(1)] commuting with the standard Weinberg–Salam SU(2) ‐U(1) group, generates new degrees of freedom for the model, simply because there are new coupling constants in the game. This assertion is not true in general. When looking at the effective Lagrangian of the physical system (−q ^{2} much smaller than any mass of the massive vector bosons), we see that a coupling constant completely disappears if the generators of its corresponding group do not enter in any surviving unbroken subgroup [U(1) for Weinberg–Salam model]. In those cases, the novelties are provided only by the quantum numbers of the fields and especially by the arbitrariness on the choice of ’’unphysical’’ Higgs fields. This effective Lagrangian is defined and constructed in the case where the initial and final symmetry groups are direct products of simple groups and U(1) groups. Some of its remarkable properties are investigated.

The retarded Josephson equation
View Description Hide DescriptionThe retarded Josephson equation, which takes into account, in a simplified form, retardation effects of the Werthamer equation, is a nonlinear, nonsimultaneous, causal integrodifferential equation. It will be solved for kernels which are essentially the asymptotic kernels from BCS theory. Of physical interest are the rotational solutions, especially the characteristics and dynamics, which describe the steady state of a Josephson junction.

Evolution of a stable profile for a class of nonlinear diffusion equations. II
View Description Hide DescriptionFirst, explicit formulas are found for all the eigenfunctions and eigenvalues of a Sturm–Liouville problem associated with the class of nonlinear diffusionequations studied previously. The formulas for the eigenfunctions are proportional to Gegenbauer polynomials whose argument depends on the separable solution shape function. Next, rigorous bounds on the asymptotic amplitude are found in terms of integrals of the initial data. These bounds are the best possible bounds of the given type since they produce the exact result for the separable solution. Finally, results of numerical experiments are reported for D∼n ^{δ} where δ=1, −1/3, −1/2, and −2/3. The rigorous bounds are compared to the perturbation estimates from the earlier work and to the computed values of the asymptotic amplitude.

On the exact scattering solution of the Schrödinger and Dirac equations with a short range potential
View Description Hide DescriptionThe total and partial wave scattering amplitudes in the Schrödinger equation with a short range potential have been derived. The Dirac equation for a short range potential has been exactly solved analytically for all partial waves for positive energies, and an expression for the S _{1/2} wave phase shift has been explicitly deduced.

Analytic continuations of the Lauricella function
View Description Hide DescriptionThe Laplace transform of the product of three confluent hypergeometric functions is expressed in terms of Lauricella’s function F _{ A }(α,a _{1},a _{2},a _{3}, b _{1},b _{2},b _{3}; x,y,z). Two analytic continuation relations of the F _{ A } function are obtained by making use of its Barnes integral representation. One analytic continuation leads to a set of one term transformation relations and in the second, F _{ A } is expressed in terms of eight Lauricella F _{ B } series. Analytic continuations are given for the F _{ B } series, thereby allowing one to obtain a new analytic continuation for the F _{ A } series. Our result is useful for calculating the F _{ A } function when ‖x‖+‖y‖+‖z‖=2, which occurs in the analysis of the electron scattering from the nucleus.

Yang–Mills equations in Maxwell form
View Description Hide DescriptionThe Yang–Mills field equations are written in a form analogous to Maxwell’sequations. The inherent nonlinearities are to be thought of as arising from a medium: The gauge fields then look like waves propagating in a medium. Some well‐known solutions are considered in this approach.

Note on the entropy production in a discrete Markov system
View Description Hide DescriptionThe Prigogine inequalities on the rate of entropy production are derived by information theoretic methods for a discrete open Markov system. A new inequality is proposed. A comparison is made with similar results due to Levine and co‐workers.

Criticality of neutron transport in a slab with finite reflectors
View Description Hide DescriptionThe purpose of this paper is to investigate the subcriticality and the supercriticality for the neutron transport in a slab which is surrounded by two finite reflectors. The mathematical problem is to determine when the coupled boundary‐value problem has or has no positive solution. It is shown under some explicit conditions on the material properties of the transport mediums and the size of the slab length that the coupled problem has a unique solution which insures the subcriticality of the system. It is also shown under some different conditions on the same physical quantities that the system cannot have a nonnegative solution when there is an external source, and it only has the trivial solution when there is no source in the system. This conclusion leads to the supercriticality of the system. Both upper and lower bounds for the critical length of the slab are explicitly given.

Exact solitary ion acoustic waves in a magnetoplasma
View Description Hide DescriptionIt is shown that finite amplitude ion acoustic solitary waves propagating obliquely to an external magnetic field can occur in a plasma.

A generalized eigenvalue distribution
View Description Hide DescriptionThe statistical properties of the eigenvalues of random unitary matrices may be determined from the joint probability density function of the matrix eigenvalues. Earlier theorems have derived the density function for the unitary and symplectic circular ensembles from that for the circular orthogonal ensemble. A method is presented here for successively eliminating variables from the probability density function for the orthogonal circular ensemble; the method generalizes an earlier result, and the resulting function appears to represent the behavior of eigenvalues from a new series of matrix ensembles.

Explicit decomposition of tensor products of certain analytic representations of symplectic groups
View Description Hide DescriptionFor any integer k≳1 let E=C^{2k×1}, E′ =C^{1×2k }, and G=Sp(k,C). If P _{ m }(E) denotes the linear space of all homogeneous polynomial functions of degree m on E, then the representation L _{ m } of G, obtained by left translation on P _{ m }(E), is irreducible with signature (m,0,⋅⋅⋅,0). Similarly, P _{ n }(E′) and R _{ n } are defined by replacing E by E′, m by n, and left translation by right translation. In this article, an explicit decomposition of the tensor product representation R _{ n }⊗L _{ m } on P _{ n }(E′) ⊗P _{ m }(E) is given in terms of the solid symplectic Stiefel harmonics.

The point form of quantum dynamics and a 4‐vector coordinate operator for a spinless particle
View Description Hide DescriptionWe construct the analog in the quantum mechanics of a free spinless particle, of Dirac’s formula for the generators of space–time translations in his point form of classical dynamics, where one takes as fundamental variables the generators of homogeneous Lorentz transformations and the coordinate 4‐vector of the point where the world line of the particle meets one sheet of a two‐sheeted hyperboloid in space–time. A 4‐vector coordinate operator is determined for such a particle, with commuting Hermitian components. The corresponding observable is the analog of the coordinate 4‐vector of the point on the hyperboloid. This operator bears the same relation to such a surface as the Newton–Wigner operator does to an instant.

Properties of the self dual equations for an SU(n) gauge theory
View Description Hide DescriptionThe Yang equations for all self dual solutions of SU(r,s) gauge theory are exhibited in simple form. Algebraic and Bäcklund transformations of the solutions of these equations are derived. The Bäcklund transformations change an SU(r,s) solution into an SU(r−1, s+1) solution.