Volume 15, Issue 7, July 1974
Index of content:

Scalar waves in the exterior of a Schwarzschild black hole
View Description Hide DescriptionUsing the method of separation of variables, we study rigorously scalar waves due to a point source in the exterior of a Schwarzschild black hole. First, a Fourier analysis gives general formulas for the interior and exterior radial wave functions and their relations to solutions of special cases, the Green's function, and the frequency spectrum. Three special cases are examined. Second, Laplace transforms of the field are obtained and their properties are studied. Using the results of the Laplace transformation and some general properties of timelike curves, we prove the following theorem: The time‐dependent scalar field of a point source goes to zero outside the horizon as the source falls into the black hole.

Singular perturbation method in neutron transport theory
View Description Hide DescriptionWe consider equations of evolution with a small parameter in a Banach space. The method of singular perturbations is applied to derive inner and outer asymptotic solutions. It is shown that the neutron transport equation coupled with the equation for the concentration of delayed neutron precursors, if considered in the Hilbert space of square integrable functions, satisfies all the requirements set up in the paper.

On mass zero indecomposable representations of the Poincaré group
View Description Hide DescriptionConsidering the space of massless‐particle wavefunctions transforming locally according to the irreducible representation D ^{(m,n)} of the homogeneous Lorentz group, we determine explicitly, by a direct and elementary method, the matrices constituting the (reducible but indecomposable) representation of Lorentz transformations with respect to a helicity basis.

Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. I. General theory
View Description Hide DescriptionRoberts' proposal of a rigged Hilbert space for a certain class of quantum systems is reinvestigated and developed in order to exhibit various properties of this kind of rigged Hilbert spaces which might be of interest for the application of this formalism to special quantum systems. It is shown that on the basis of this proposal one also obtains a satisfactory solution for a rigged Hilbert space for composite systems. Another part is concerned with topological properties of the so‐called eigenoperators γ(t) belonging to an A‐eigen‐integral decomposition of Φ with respect to a self‐adjoint operator A on . We derive a representation of γ(t) in terms of the generalized eigenvectors of A and in the same context give a rough topological characterization of these eigenvectors.

Rigged Hilbert space formalism as an extended mathematical formalism for quantum systems. II. Transformation theory in nonrelativistic quantum mechanics
View Description Hide DescriptionResults of a previous paper are used to obtain a rigorous mathematical formulation of the transformation theory of nonrelativistic quantum mechanics within the framework of rigged Hilbert spaces.

On the inverse problem in radiative transfer
View Description Hide DescriptionThe inverse problem in monochromatic radiative transfer is considered for an infinite medium with anisotropicscattering. It is shown that each Legendre coefficient of the scattering function can be related independently of the others to an appropriate integral over space and angles of the intensity due to a monodirectional plane source. This result offers some advantage over the analogous one for an isotropic plane source if the medium is weakly absorbing.

On the geometrization of neutrino fields. I
View Description Hide DescriptionNecessary and sufficient algebraic and differential conditions are obtained for a geometry to have as its source a neutrino field whose energy density relative to any observer is positive or negative definite.

On the geometrization of neutrino fields. II
View Description Hide DescriptionIn a previous paper neutrino fields with positive (or negative) energy density were geometrized in the sense of Rainich, Misner, and Wheeler. The present paper deals with the geometrization of neutrino fields which do not satisfy such an energy condition.

Applications of infinite order perturbation theory in linear systems. I
View Description Hide DescriptionThe applicability of infinite order perturbation theory to linear systems is exhibited. The technique involves a generalization of the method developed by Wu and Taylor and can be used to study systems described by the equations of the following form V_{nn}u_{n} + V _{ n,n+1} u _{ n+1} + V _{ n,n−1} × u _{ n−1} = Eu_{n} , where the coupling coefficients depend on n. The wide range of application of the generalized method is demonstrated by using it to study systems as different as the plane rotator in an external field on the one hand and the dynamics of a disordered chain on the other.

Application of infinite order perturbation theory in linear systems. II. The frequency spectrum of disordered chains
View Description Hide DescriptionA formulation is presented such that the ensemble average of the diagonal elements of the Green's function for a one‐dimensional disordered system can be calculated to arbitrary accuracy. The method is illustrated with an application to an isotopically disordered chain. An approximate solution to the frequency spectrum of a disordered binary chain is also discussed.

Scattering of plane longitudinal elastic wave by a large convex rigid object with a statistically corrugated surface. II. Far field solution
View Description Hide DescriptionThe far field solution for the scattering of plane longitudinal elastic wave by a large convex rigid object with a statistically corrugated surface is obtained and the effective reflection and diffraction coefficients are deduced.

Thermodynamics of a mixture of fermions and bosons in one dimension with a repulsive δ‐function potential
View Description Hide DescriptionThe thermodynamics of a mixture of fermions and bosons is derived on the basis of two ansätze about the roots of a set of algebraic equations.

Elastic scattering in the Kerr metric
View Description Hide DescriptionThe differential cross section for scattering from a source whose gravitational field is described by the Kerr metric is evaluated.

Theory of self‐reproducing kernel and dispersion inequalities
View Description Hide DescriptionThe theory of the self‐reproducing kernel by Aronszajn has been investigated for a Hilbert space with norm,where f(z) is a H ^{2} function and λ(x) is a nonnegative summable function. The self‐reproducing kernel of this space satisfies an integral equation. The dispersion inequalities for various problems in the high energy physics can be treated in unified and generalized manner by this theory.

Scattering theory, orthogonal polynomials, and the transport equation
View Description Hide DescriptionIt is shown that under rather weak restrictions the discrete eigenvalues occurring in one‐velocity transport theory are real or purely imaginary, simple, ≥ 1 in magnitude, finite in number, and occur in ± pairs. Proofs are obtained using methods of scattering theory applied to orthogonal polynomials.

Ising models derived from binary lattice gases
View Description Hide DescriptionTransformations of two‐component lattice gases are studied. The original binary lattice gas has the Widom‐Rowlinson type of interaction: infinite repulsion between nearby unlike particles and no interaction between like particles (except that multiple site occupancy is forbidden). The first transformation is to an equivalent one‐component lattice gas, with many‐body interactions‐some attractive, some repulsive. The second transformation is to the isomorphic Ising spin system, having many‐spin interactions‐some ferromagnetic and some antiferromagnetic. All interaction functions are obtained as explicit geometrical derivates of the mutual exclusion ``sphere'' of the original binary system. The results are discussed in light of recent theories of phase transitions of Ising models with symmetry‐breaking many‐spin interactions.

A scattering problem for a straight line segment
View Description Hide DescriptionAn explicit analytical solution is found for the scattering potential produced by a straight line segment. Integral transformations are introduced which map the mixed boundary value problem into one which can be solved by standard Green's function methods.

Expansion‐free electromagnetic solutions of the Kerr‐Schild class
View Description Hide DescriptionStarting with the general Kerr‐Schild form of the metric tensor, (where l is null and η is flat space‐time), a study is made for those solutions of the Einstein‐Maxwell equations in which l is geodesic, shear‐free, and expansion‐free. It is shown that all resulting solutions must be of Petrov type [4] or type [−] and the Maxwell field must be null. Because of the expansion‐free assumption there exist flat and conformally flat gauge conditions on all metrics in this class; i.e., there exist metrics of this Kerr‐Schild form which are flat (or conformally flat) but are not Lorentz‐related. A method is given for obtaining meaningful solutions to the field equations with the latter gauge equivalence class removed. A simple example of a radiative field of type [4] along a line singularity exhibits how solutions in this class may be generated.

A simple proof of certain FKG inequalities
View Description Hide DescriptionWe present a proof of a set of FKG inequalities, namely,.The proof is obtained using Gaussian random variables and holds for systems whose Hamiltonian contains a positive quadratic form and one‐body interactions on which no restrictions are placed.

Unification of the external conformal symmetry group and the internal conformal dynamical group
View Description Hide DescriptionThe two common applications of O(4,2) as a conformal group on external space‐time coordinates and as a dynamical group on internal relative coordinates are combined into a unified algebraic structure for composite systems. A method is given for obtaining from this structure infinite‐component wave equations and a discrete, linearly increasing mass spectrum.