Volume 11, Issue 11, November 1970
Index of content:

Radiation Transport along Curved Ray Paths
View Description Hide DescriptionThe transport of radiation in a turbulent, refracting medium is studied. It is shown that the conventional transportequation must be generalized. Path integrals are taken along curved ray trajectories. When these ray paths have torsion, a rotation of the polarization vectors needs to be taken into account. Two derivations of the transportequation are given. One is phenomenological and one is based on Maxwell's equations. Some discussion is given of cross polarization of radarbackscatter.

Structure of the Point Spectrum of Schrödinger‐Type Tridiagonal Operators
View Description Hide DescriptionThe main result in the present work concerns a criterion on the existence and the structure of proper values in a class of bounded operators (Schrödinger‐type tridiagonal operators) on an abstract separable Hilbert space. The realization of these operators in the space of square summable sequences l _{2}(1, ∞) represents a boundary‐value problem of difference equations of the following form: f(n + 1) + f(n − 1) + a(n)f(n) = Ef(n). In our case f(n) ∈ l _{2}(1, ∞), and the condition f(2) + a(1)f(1) = Ef(1) must hold. The approach followed is based on the reduction by Deliyannis and Ifantis [J. Math. Phys. 10, 421 (1969)] of the above boundary‐value problem to an abstract operator form, which makes possible the application of the methods of functional analysis. It is shown that for every monotonically convergent and real‐valued sequence a(n) ≠ 0, n = 1, 2, ⋯, there exist proper values, the greatest of which can be determined by the Ritz approximation method.

The Lorentz Group and the Sphere
View Description Hide DescriptionA direct connection between the spin and conformally weighted functions on the sphere and geometric objects in Minkowski space is established through the isomorphism of the conformal group of the sphere to the restricted Lorentz group. It is shown that with the use of these functions one can duplicate all the standard work on the representations of the Lorentz group. It is shown further that these functions can be used to obtain a generalization of the classical equations of motion in which internal degrees of freedom arise naturally.

New Approach to the Motion of a Pole‐Dipole Particle
View Description Hide DescriptionA new approach to the equations of motion of the pole‐dipole particle in linearized general relativity is presented. It is based on earlier work for the monopole particle [E. T. Newman and R. Posadas, Phys. Rev. Letters 22, 1196 (1969); Phys. Rev. 187, 1784 (1969)]. Though the results are not new, the novelty of the method holds out the serious hopes that the work can be extended to the full theory.

Model of Phase Transitions for Interacting Fermi Systems
View Description Hide DescriptionThe behavior of a previously discussed model of phase transitions for Fermi systems is analyzed in the region very near the critical point. It is shown that for a very general class of 2‐body interactions the chemical potential is analytic in the temperature and density at the critical point, so that the model is in this sense equivalent to the classical theory of phase transitions. An extension of the model to include certain 3‐body and higher interactions leaves this conclusion unchanged.

Method of Constructing General Solutions of Certain Nonlinear Differential Equations Describing Plasma Oscillations
View Description Hide DescriptionA general solution is derived for a system of nonlinear partial differential equations describing longitudinal plasma oscillations. The method of construction of the solution involves the imposition of arbitrary functional relationships among the integrals of an associated system of ordinary differential equations.

Fluids of Particles with Short‐Range Repulsion and Weak Long‐Range Attractive Interaction. II. The Two‐Particle Distribution Function
View Description Hide DescriptionIn Paper I, we presented an expansion of the pressure and density in grand canonical form and corrections to the Maxwell rule for a system of particles with short‐range repulsion and weak long‐range attraction. These expansions can be ordered in powers of γ, the inverse range of the attractive potential. It was assumed that the thermodynamic functions and the molecular distribution functions of the reference system, i.e., the system with only the repulsive interaction, are given. In the present paper we have calculated the γ expansion of the pair distribution function, under the same assumption. The result is obtained by functional differentiation of the series for the pressure and presented in the form of a series of diagrams. The dominant order in γ of each diagram is the same as the order of that diagram in the series for the pressure, from which it is derived.

Complete Sets of Functions on Homogeneous Spaces with Compact Stabilizers
View Description Hide DescriptionWe formulate and solve the problem of determining a complete set of generalized functions for a wide class of homogeneous spaces with compact stabilizers. This allows us to say precisely what unitary irreducible representations can be realized on a given homogeneous space. The techniques are applied to the n‐dimensional orthogonal and unitary groups.

Two‐Dimensional Hydrogen Bonded Crystals without the Ice Rule
View Description Hide DescriptionModels of 2‐dimensional hydrogen bonded crystals obeying the ice rule, which previously have been solved exactly, are generalized by removing the ice rule. Many of the peculiar and unique properties of the solutions for the constrained models are now explained by showing that these models, above critical temperature, are equivalent to new unconstrained models at critical temperature. In addition to locating the critical temperature for the general but unsolved models, we locate the singularities of the ground state energy of a related ring of interacting spins.

Calculation of the Inner Multiplicity of Weights by Means of the Branching‐Law Method and by Racah's Recurrence Relation
View Description Hide DescriptionWe discuss here in detail the validity of the branching‐law method suggested in a previous paper [M. K. F. Wong, J. Math. Phys. 11, 1489 (1970)] for the calculation of the inner multiplicity of weights in an irreducible representation of a classical group. It is found that the method works for all the irreducible representations of SU(n) and SO(2n + 1), but that in the case of SO(2n) and Sp(2n) the method does not always give complete solutions except in some simple cases. It is then suggested that Racah's recurrence relation be used in these cases so that complete solutions may be obtained. It is also noted that Racah's recurrence relation alone is sufficient to obtain the inner multiplicity of all weights. This fact is utilized in the calculation of inner multiplicities in another paper [B. Gruber, J. Math. Phys. 11, 3077 (1970)]. The method suggested in this paper is illustrated through the calculation of some typical examples of the inner multiplicity of weights in the two classical groupsSO(2n) and Sp(2n).

Coefficients Connecting the Stark and Field‐Free Wavefunctions for Hydrogen
View Description Hide DescriptionA general expression for the coefficients connecting the Stark (parabolic coordinates) and field‐free (spherical coordinates) wavefunctions for hydrogen is obtained. The result, which involves a generalized hypergeometric function, has been numerically evaluated through principal quantum number n = 10.

The Proper Vibration of the Space
View Description Hide DescriptionThe proper vibrations of homogeneous and isotropic space are examined on the basis of the equations of the de Broglie wave field (field equations). The time‐dependent part of the wavefunction, which is a solution of the Klein‐Gordon equation, satisfies the differential equation which coincides with the differential equation derived from field equations for the time‐dependent part of the Robertson‐Walker metric.

Equations of the de Broglie Wave Field and Their Relationship to Riemann's Curvature Tensor
View Description Hide DescriptionThe equations of the de Broglie wave field (field equations) [J. Kulhanek, Nuovo Cimento Supp. 4, 172 (1966)] under special conditions require a very particular geometry together with a specific interpretation of the curvature scalar. The purpose of the present paper is to show that the same condition turns the conservation law (which is a consequence of the field equations) into an identity and that the Rainich [Nature 115, 498 (1925)] decomposition of Riemann's curvature tensor gives only one component.

Calculation of Correlation Functions of Solutions of a Stochastic Ordinary Differential Equation
View Description Hide DescriptionIn this paper we use the ``smoothing method'' to calculate the correlation functions of the solutions of the equation,satisfying nonstochastic initial conditions, where N(z) is a real, wide‐sense stationary stochastic process with zero mean and β_{0} and η ≪ 1 are positive constants. It is shown that an appropriate application of the smoothing method leads to the exact results in the case when N(z) is the random telegraph process. Moreover, under appropriate conditions on the general process N(z), approximate expressions are obtained for the correlation functions in terms of the first‐ and second‐order moments of the solutions, and approximate expressions are given for these moments.

Mixed‐Basis D Functions and Clebsch‐Gordan Coefficients of Noncompact Groups
View Description Hide DescriptionMixed‐basis D functions are introduced as a tool for deriving Clebsch‐Gordan coefficients of induced representations of semisimple groups. The Clebsch‐Gordan coefficients of SU(1, 1) and SL(2, C) are computed as examples.

N‐Representability Problem: Particle‐Hole Equivalence
View Description Hide DescriptionIt is shown that every p matrix has a dual matrix which describes the p‐hole properties of the system. A general procedure is given for using particle‐hole equivalence to obtain new N‐representability conditions. In particular, necessary and sufficient conditions are given for both pure and ensemble N‐representability of p matrices whose 1‐rank is N + p.

Projection Techniques
View Description Hide DescriptionThe twofold multiplicity problem associated with the Wigner supermultiplet reduction is resolved by spin‐isospin projection techniques analogous to the angular momentum projection technique introduced by Elliott to resolve the multiplicity problem. The projection quantum numbers, which furnish either an integer or half‐integer characterization of the multiplicity, are assigned according to an (ST)‐multiplicity formula derived from a consideration of the symmetry properties of spin‐isospin degeneracy diagrams. An expression is obtained for the coefficients which relate the projected basis states to states labeled according to the natural chain. General expressions for coupling coefficients and tensorial matrix elements are given in terms of the corresponding quantities.

Theorem of Uniqueness and Local Stability for Liouville‐Einstein Equations
View Description Hide DescriptionWe prove, by use of energy inequalities, a theorem of uniqueness and local (i.e., for finite time) stability for the solution of Cauchy problem relative to the integro‐differential system of Einstein and Liouville. A global theorem of geometrical uniqueness follows from a general method, previously given. We will prove elsewhere an existence theorem.

Realization of the Lie Group G(0, 1) by the Function of Landau Levels
View Description Hide DescriptionThe irreducible basis of the Lie groupG(0, 1) are obtained in connection with the quantum physical problem: a motion of a free electron in a magnetic field. The differential operators are shown to be the infinitesimal operators defined by Brown.

Regge Trajectories for the Inverse Square Potential
View Description Hide DescriptionThe analytic properties of the Sfunction in the complex angular momentum plane for regular potentials with inverse square tails are discussed. Special attention is given to the determination of the poles of S in the limits of low and high energies. Two soluble examples are considered in detail.