Volume 15, Issue 10, October 1974
Index of content:

The recursive perturbation method and its application to the study of nuclear shell models
View Description Hide DescriptionThe recursive perturbation method given previously by the author is generalized and applied to the calculation of the energy levels of nuclear shell models. Specifically, we consider a numerically exactly soluble model of interacting fermions with S U(3) symmetry studied by Li, Klein, and Dreizler. The method can be obviously extended to the study of more general models with S U(n) symmetry.

Local bounded perturbations of KMS states
View Description Hide DescriptionLocal bounded perturbations of an infinite equilibrium state are studied in the C*‐algebraic framework. It is assumed that for the unperturbed state the pressure exists in the thermodynamic limit and that the Dubin‐Sewell hypotheses are fulfilled. The following is then shown: At constant temperature the perturbed state is analytic in the perturbation Q, the infinite volume pressure does not depend on Q, and the new state is KMS with respect to the time evolution corresponding to the adiabatic perturbation, as treated in a previous paper.

Mathematical theory of the R matrix. I. The eigenvalue problem
View Description Hide DescriptionThis is the first paper in a two part series aimed at placing the theory of Wigner's R matrix on a mathematically rigorous footing. In Paper I of the series, we will show that the eigenvalue problem associated with the R matrix can be solved for a large class of potentials, including Coulomb‐like potentials. We will do this for the case in which the boundary of the internal region is a smooth surface—although the results remain true for a much larger class of surfaces. In Paper II of the series, we will show that the R matrix exists for the class of potentials mentioned, is a compact operator, and can be approximated uniformly (i.e., normwise) by the usual expansions associated with the R matrix.

The mathematical theory of the R matrix. II. The R matrix and its properties
View Description Hide DescriptionIn this paper, it is shown that Wigner's R matrix, for a certain class of unbounded potentials which may be nonlocal or have Coulomb‐type singularities, exists, is a compact operator, and that the expansions associated with the R‐matrix converge. For the same class of potentials, a perturbation theory is constructed and conditions are given for the convergence of the resulting Born‐type expansions.

The Clebsch‐Gordan problem and coefficients for the three‐dimensional Lorentz group in a continuous basis. III
View Description Hide DescriptionAlong the lines of two previous papers, the Clebsch‐Gordan problem for products of representations of SU(1, 1) of the form is related to the properties of the Lorentz groupO(3, 1). The structure of the Clebsch‐Gordan series for this case is understood in a new way as being due to the properties of O(3, 1) spherical harmonics on the timelike and spacelike hyperboloids in Minkowski space. The Clebsch‐Gordan coefficients in a continuous basis are then evaluated.

The Clebsch‐Gordan problem and coefficients for the three‐dimensional Lorentz group in a continuous basis. IV
View Description Hide DescriptionThis is the last of four papers describing a new approach to the Clebsch‐Gordan problem for the group SU(1, 1). Here we have related the Clebsch‐Gordan series for products of the type to properties of the group O(2, 2) and the structure of the series is thus seen to arise out of the properties of O(2, 2) spherical harmonics in an basis. The Clebsch‐Gordan coefficients in a continuous basis are also evaluated.

Multiregion criticality in general geometries
View Description Hide DescriptionA general transform technique is developed for multiregion critical problems. The equivalence of a replication procedure and a derived boundary condition approach is demonstrated for the general multiregion geometries. An exact representation for the particle density may be obtained using this approach in the form of singular integral equations or equivalent Fredholm equations for expansion coefficients which arise from the superposition of the normal modes representing the particle density. The method is specifically demonstrated in the determination of solutions to the two‐region critical cylinder problem.

Lattice gas with nearest‐neighbor interaction in one dimension with arbitrary statistics
View Description Hide DescriptionWe define a quantum lattice gas with arbitrary statistics. For a one‐dimensional system with nearest‐neighbor interaction, we show that the problem is exactly soluble by use of Bethe's hypothesis when the interaction Δ=±1. The ground state energy is then obtained for the fermions of spin 1/2. Two phases are found in the case Δ=‐1.

Must quantum theory assume unrestricted superposition?
View Description Hide DescriptionA conjecture that the (n −1)^{2} independent moduli and (2n −1) unphysical phases completely specify all n‐dimensional unitary matrices is shown to be true in two and three dimensions, but false in four or more. The implications for quantum theory are discussed.

Spectrality, cluster decomposition and small distance properties in Wightman field theory
View Description Hide DescriptionWe apply the results of a previous paper by Screaton and Truman to the truncated vacuum expectation values in Wightman field theory and, using spectrality, translational invariance, and Lorentz invariance, we derive the best bounds for the truncated vacuum expectation values at the real Jost points. In a local field theory these bounds include as a special case Araki's result on the exponential decrease of the truncated vacuum expectation value for large spacelike separations and the cluster decomposition property. The bounds also establish a connection between the small distance and high energy behaviors of the theory. In addition we evaluate the bounds in a nonlocal field theory and discuss some of their ramifications.

Orbits of the rotation group on spin states
View Description Hide DescriptionA simple theorem on projective spaces generalizes the concept of the Riemann sphere. This leads us to a generalized interpretation of the ray space associated with a finite‐dimensional Hilbert space. An application is given about the way the rotation group acts on states of given spin j.

Free‐particle‐like formulation of Newtonian instantaneous action‐at‐a‐distance electrodynamics
View Description Hide DescriptionFrom the infinite order equations of motion of conventional electrodynamics, one can extract by order depression a subclass of second order equations of motion parametrized only by initial positions and velocities. This article presents, with a view toward possible later quantization, a canonical formulation of this electrodynamics. It happens to have the same aspect as for free particles: . The p's are constant, and the canonical variables q's describe straight lines (particle positions cannot be canonical). The extension to the many‐body problem is given.

Radial charged particle trajectories in the extended Reissner‐Nordstrom manifold
View Description Hide DescriptionIt is shown that the trajectory of a charged particle on the extended Reissner‐Nordström manifold can be such as to carry it into regions of the manifold where the definition of energy at infinity is different from the one at its point of origin. The various types of radial trajectories are classified. In the event one considers the manifold as having been produced by a collapsed star, there exist trajectories which go through both horizons, reach a minimum value of r, and go through two more horizons to a copy of the space in which it originated (flat at r = + ∞) without colliding with the matter of the collapsed star.

A general method for obtaining Clebsch‐Gordan coefficients of finite groups. I. Its application to point and space groups
View Description Hide DescriptionA general method is developed for obtaining Clebsch‐Gordan coefficients of finite groups. With this method Clebsch‐Gordan coefficients are obtained in a matrix form, whereas the so‐called basis‐function generating machine generates these coefficients one by one. The method is applied to double point group D̄ _{3}, the point group T, and the nonsymmorphic space group . It will be shown that the method can be simplified by the conservation law of the reduced wave vectors when applied to space groups.

A general method for obtaining Clebsch‐Gordan coefficients of finite groups. II. Extension to antiunitary groups
View Description Hide DescriptionA general method is presented for obtaining Clebsch‐Gordan coefficients, in a matrix form, of finite antiunitary groups, as a direct extension of a general method for unitary groups. It is shown that there is an essential difference as well as apparent similarities between two methods for unitary and antiunitary groups.

Phase transitions of a multicomponent Widom‐Rowlinson model
View Description Hide DescriptionWe study a multicomponent version of the ``A−B'' model of Widom and Rowlinson, generalized in a symmetric way: There is an infinite repulsive interaction between any two unlike particles. We consider both lattice and continuum versions of the model and show that the ``demixing'' transition occurs for any finite number M of components, all having the same activity. No conclusion can be drawn about this transition in the limit M→∞. It is shown, however, that another transition, in which the density is greater on one of the sublattices, appears at a finite value of M which persists for all larger M at any fixed value of the activity. In the limit M→∞, z→0, M z=ζ, const, this system apparently becomes ``equivalent'' to a one‐component system with activity ζ in which there is an exclusion for occupancy of nearest neighbor sites. The latter transition then becomes the ``hard square'' transition.

Matrix mechanics approach to a nonlinear oscillator
View Description Hide DescriptionThe system with Hamiltonian p ^{2} + x ^{4} is discussed. An approximation scheme is given for matrices x and p that satisfy the canonical commutation relation and diagonalize the Hamiltonian.

A new method for the evaluation of slowly convergent series
View Description Hide DescriptionA new method is presented which sums certain slowly convergent series. It is based on the use of the Hankel integral transform and Schlömilch series. This method is applied with great success to the computation of lattice sums in ionic crystals. In particular, the Madelung constant is calculated with great accuracy through rather simple calculations: The final results only involve elementary functions so that the numerical evaluation is quite easy.

Lie theory and separation of variables. 5. The equations iU_{t} + U_{xx} = 0 and iU_{t} + U_{xx} −c/x ^{2} U = 0
View Description Hide DescriptionA detailed study of the group of symmetries of the time‐dependent free particle Schrödinger equation in one space dimension is presented. An orbit analysis of all first order symmetries is seen to correspond in a well‐defined manner to the separation of variables of this equation. The study gives a unified treatment of the harmonic oscillator (both attractive and repulsive), Stark effect, and free particle Hamiltonians in the time dependent formalism. The case of a potential c/x ^{2} is also discussed in the time dependent formalism. Use of representation theory for the symmetry groups permits simple derivation of expansions relating various solutions of the Schrödinger equation, several of which are new.

The distribution of the zeroes of the Jost function: The s ‐wave attractive exponential potential
View Description Hide DescriptionWe show how the zeroes of the Jost function for an s ‐wave attractive exponential potential are distributed. In particular, we use known results, especially some of Coulomb's, on the zeroes of Bessel functions to demonstrate that there are no zeroes for complex momentum k = k _{1} + ik _{2} (k _{1}≠0, k _{2}≠0).