Volume 21, Issue 11, November 1980
Index of content:

Homothetic motions and the Hauser metric
View Description Hide DescriptionThe Hauser metric is found in the canonical form obtained by Halford for a vacuum metric of Petrov type N admitting a two parameter group of homothetic motions. A result concerning the symmetries of vacuum metrics of Petrov type N possessing twisting geodesic rays is corrected.

Multiple scattering theory for discrete, elastic, random media
View Description Hide DescriptionA theory is presented for determining the ensembleaveraged Green tensor of a statistically homogeneous distribution of identical, randomly oriented elasticscatterers embedded in an infinite, homogeneous, and isotropic matrix. The theory is based on the selfconsistent formulation of Lax’s [Rev. Mod. Phys. 23, 287 (1951); Phys. Rev. 85, 621 (1952)] multiple scatteringtheory due to Gyorffy [Phys. Rev. B 1, 3290 (1970)] and Korringa and Mills [Phys. Rev. B 5, 1654 (1972)]. The average Green tensor is found to be characterized by three parameters which may depend on the momentum operator but which are otherwise analogous to the Lamé constants and density of an ideal, homogeneous, and isotropic medium. These ’’effective’’ parameters are shown to be related in the usual way to the wave numbers of coherent compressional and shear planewave modes of the random composite. The ensemble averaged Green tensor and the d i s p e r s i o n r e l a t i o n s satisfied by the wave numbers of the coherent modes are found to depend on the single and joint probability density functions for the scattering centers and on the transition operator of a discrete scatterer embedded in the effective (average) medium. The dispersion relations are evaluated explicitly for the limiting case of a completely random ensemble of homogeneous and isotropic scatterers whose elastic parameters and density differ very little from those of the matrix medium.

Lateral boundary conditions for quasisteady atmospheric flows
View Description Hide DescriptionThe quasisteady model, derived by the author in an earlier paper, is extended to include lateral boundary conditions. The approach is to first specify a complete time‐dependent problem, including boundary conditions; it is stressed that to each boundary condition there should be associated a precise physical assumption. Assuming the time‐dependent problem is well‐posed, it is then shown that the quasisteady assumption can be applied consistently to both the partial differential equations and the boundary conditions, thereby obtaining a well‐posed mathematical model with time scales suitable for large scale atmospheric flows. Three types of conditions are considered at the lateral boundaries: (1) outflow, (2) inflow (driven)—velocity is specified and is essentially independent of the internal flow, (3) inflow (passive)—inflow is created primarily by the internal flow configuration. The upper boundary conditions include the two derived by the author in the earlier paper, the continuous and discontinuous boundary conditions, and a third condition, which is designed to allow the flow to propagate independently of the height of the region. Numerical solutions are obtained for various test cases, and convergence of the calculations is demonstrated. One sees that the calculations are reasonable, both from a mathematical and physical standpoint.

Time evolution operator for N interacting quantum harmonic oscillators
View Description Hide DescriptionWe discuss time evolution operator for N interacting quantum harmonic oscillators and determine certain conditions under which an explicit closed‐form expression can be found for this time‐ordered operator. The special case of N=2 is discussed, as an example.

Shift operator techniques for the classification of multipole‐phonon states. VI. Properties of nonscalar R(3) product operators in the R(2λ+1) groups
View Description Hide DescriptionExpressions connecting nonscalar R(3) products of operators which shift the eigenvalues of the R(3) Casimir operator L ^{2} are constructed within the R(2λ+1) groups (λ=2 or 3).

Shift operator techniques for the classification of multipole‐phonon states. VII. Self‐consistent single step algorithm for R(5) O ^{0} _{ l } eigenstate and eigenvalue determination
View Description Hide DescriptionA previously derived set of quadratic relations in the R(5) shift operators O ^{ k } _{ l } (‖k‖?3), is shown to be in such a way complete, that any O ^{0} _{ l }eigenvector and corresponding eigenvalue can be unambiguously obtained in a step by step calculation which starts at the highest angular momentum state. Such a calculation strictly follows the pattern of an algorithm, which unlike the tree generating mechanism, has unlimited applicability. Previous knowledge of the existence and of the degeneracy of a state not being required the algorithm itself accounts for the l multiplicity of states and is therefore called self‐consistent.

On a connection between radial Schrödinger equations for different power‐law potentials
View Description Hide DescriptionA general correspondence is given between the radial Schrödinger equations (in arbitrary dimensions) for confining‐ and inverse‐power law potentials. The wave functions and Green’s functions of the two types of potentials are shown to differ by little more than a change of variable (a special case being the well‐known equivalence of the harmonic oscillator and Coulomb problems). This gives rise to relationships between the discrete state eigenvalues and matrix elements. Following the lines of a recent paper by Gazeau, the relevance of this correspondence to Sturmian representations for power law potentials is examined. Generalizations are considered for potentials containing a linear combination of powers of r.

Formal solutions of inverse scattering problems. III.
View Description Hide DescriptionThe formal solutions of certain three‐dimensional inverse scattering problems presented in papers I and II of this series [J. Math. Phys. 10, 1819 (1969); 17 1175 (1976)] are obtained here as fixed points of a certain nonlinear mapping acting on a suitable Banach space of integral kernels. When the scattering data are sufficiently restricted, this mapping is shown to be a contraction, thereby establishing the existence, uniqueness, and continuous dependence on the data of these formal solutions.

On the uniqueness of the equilibrium configurations of slowy rotating relativistic fluids
View Description Hide DescriptionWe consider the equations of a general relativistic space‐time that is stationary, asymptotically Euclidean, diffeomorphic to R^{4} and consists of an exterior vacuum solution and an interior perfect fluid in rigid motion. If one requires further that the solution be close to the static spherically symmetric ones (in the sense of a suitable topology on the set of stationary space–time metrics) it is shown that for a given equation of state ρ( p) and given total mass m and (small) angular momentumJ there are no smooth curves of physically distinct global axially symmetric solutions. In view of a recent result of Lindblom that all such space‐times are axisymmetric this result is quite general. The method is a generalization of the one used to prove (in a ’’local’’ sense) the uniqueness of the spherical solution in the static case.

Properties of the noise‐induced ‘‘spurious’’ drift. I.
View Description Hide DescriptionThe coefficients of Fokker–Planck equations associated to Langevin equations (LE) may be interrelated, since both the diffusion matrix D and the noise‐induced drift a are derived from the same coefficients of the LE. If D is regular and if furthermore its dimension M equals the number of independent noise sources (conditions to be dropped in the subsequent paper II), a is uniquely determined by D if M=1 and independent of D if M?3. For M=2, a splits into a nontensor part which is uniquely determined by D and a vector field with given divergence. The result for M?3 means that to any LE with noise terms specified by their covariance matrix only, there exists another stochastically equivalent LE with a fully arbitrary deterministic part. As a byproduct it is shown that any given a can be removed by a nonlinear change of the state variables.

Properties of the noise‐induced ‘‘spurious’’ drift. II. Simplifications of Langevin equations
View Description Hide DescriptionIn a previous paper (I) we studied the interrelations between the coefficients of Fokker–Planck equations associated to a restricted class of Langevin equations. We now discuss the general case of an arbitrary state‐dependent diffusion matrix D, which, in particular, may be singular. Moreover the number of noise sources, M, is now allowed to exceed the rank L of D and even the number N of state variables. The noise‐induced drift a is always found to split into a part which is determined by D alone and a contribution which must lie within the tangential subspaceT_{∥} spanned by the noise coefficients (T_{∥} is specified by D). This second contribution strongly depends on L: While for L=1 it vanishes, it is unrestricted (within T_{∥}) for L?3. If L=2, a simple characterization is only found under the assumptions of I; the condition on the divergence derived therein is now proved to be sufficient. Langevin equations can often be simplified in two respects: The number of noise sources is always reducible to L (unless L=2) without change of any terms in the Fokker–Planck equation, and if either L=1 or a set of conditions holds, new variables can be introduced in such a way that only L of them are directly affected by noise. (Both reductions together lead to the form considered in I.) For L?3 any given Langevin equation can be replaced by a stochastically equivalent one, the deterministic part of which has an arbitrarily chosen component in T_{∥}.

A concept of spin 1/2 approximation in the quantum theory of lattice Bose systems
View Description Hide DescriptionWe give a model independent description of criteria under which expectation values of observables associated with a finite part of the lattice Bose system can be made to converve to those of the associated Fermi or finite spin system.

Nonlinear internal symmetry
View Description Hide DescriptionA theory of nscalar fields is outlined in which the equations of motion are invariant under all nonsingular global transformations of the fields amongst themselves, whether linear or nonlinear.

U(N) integral for the generating functional in lattice gauge theory
View Description Hide DescriptionThe one link integral or equivalently the generating functional of U(N) integrals in the lattice gauge theory is explicitly evaluated in terms of a character expansion.

Poincaré wave equations as Fourier transforms of Galilei wave equations
View Description Hide DescriptionThe relationship between the Poincaré and Galilei groups allows us to write the Poincaré wave equations for arbitrary spin as a Fourier transform of the Galilean ones. The relation between the Lagrangian formulation for both cases is also studied.

A solution of an inverse‐source problem in coherence theory
View Description Hide DescriptionAn explicit solution is presented for a special case of the problem of determining the cross spectral density of a radiation source from its far‐field intensity pattern. The case considered is that of a flat, circular source radiating into a half‐space, where the sourcecoherenceproperties exhibit statistical homogeneity and isotropy. The solution is in the form of a uniformly convergent series of hypergeometric functions that are straightforwardly related to Legendre polynomials.

Quantum statistics of multimode m‐photon absorption process
View Description Hide DescriptionA density matrix method is used to obtain an exact solution for the reduced density matrix of the field in an arbitrary multimode m‐photon absorption process. The results of some earlier specialized studies of photon statistics in multiphoton absorption process can be recovered from this solution.