Volume 22, Issue 1, January 1981
Index of content:

SU(m n⊇SU(m)×SU(n) isoscalar factors and S(f _{1}+f _{2})⊇S( f _{1})×S( f _{2}) isoscalar factors
View Description Hide DescriptionA simple relation is found between the isoscalar factor (ISF) of the unitary group and those of the permutation group, i.e. the SU(m n)⊆SU(m)×SU(n)ISF is equal to the S( f _{1}+f _{2})⊆S( f _{1})×S( f _{2}) ISF. Since the values of S( f _{1}+f _{2})⊆S( f _{1})×S( f _{2})ISF are independent of m and n, one arrives at an important conclusion that the values of SU(m n)⊆SU(m)×SU(n) ISF are also independent of m and n. Therefore they can be calculated for all m and n by a single stroke instead of one m and one n at a time. An eigenfunction metho for evaluating the SU(m n)⊆SU(m)×SU(n)ISF is given which can be easily translated into a computer program.

On the existence of real Clebsch–Gordan coefficients
View Description Hide DescriptionThe question of the possible general existence of real Clebsch–Gordan coefficients (coupling coefficients) for compact groups is considered. Criteria are established for a group to be susceptible to the classical rotation group approach in which a choice of standard irreducible matrix representations is made such that there is a fixed inner automorphism of the group carrying all standard representations into their complex conjugate. In connection with a generalization of this approach the concept of quasi‐ambivalence of a group is shown to be relevant.

On the matrix elements of the U(n) generators
View Description Hide DescriptionA straightforward derivation of the matrix elements of the U(n) generators is presented using algebraic infinitesimal techniques. An expression for the general fundamental Wigner coefficients of the group is obtained as a polynomial in the group generators. This enables generalized matrix elements to be defined without explicit reference to basis states. Such considerations are important for treating groups such as Sp(2n) whose basis states are not known.

Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform
View Description Hide DescriptionThe main purpose of this paper is to describe a technique of reduction, whereby from the class of evolution equations for matrices of order N solvable via the spectral transform associated to the (matrix) linear Schrödinger eigenvalue problem, one derives subclasses of nonlinear evolution equations involving less than N ^{2} fields. To illustrate the method, from the equations for matrices of order 2 two subclasses of equations for 2 fields (rather than 4) are obtained. The first class coincides, or rather includes, that solvable via the spectral transform associated to the generalized Zakharov–Shabat spectral problem; further reduction to nonlinear evolution equations for a single field reproduces a number of well‐known equations, but also yields a novel one (highly nonlinear). The second class also yields highly nonlinear equations; some examples are given, including another novel evolution equation for a single field.

On one method of solving the Helmholtz equation. I
View Description Hide DescriptionA method is suggested to solve the Helmholtz equation in an arbitrary domain with general form boundary conditions. The method permits reducing this equation to that of Poisson and an infinite set of simultaneous linear algebraic equations. Convergence of the method is proved for any wave number. Thus, it becomes possible to solve the Helmholtz equation by using any known method developed to solve the Poisson equation. As an illustration, an effective algorithm is constructed to solve the two dimensional diffraction problem on the arbitrary periodic boundary for any wavelength by using the conformal mapping techniques. If the boundary contains irregular points, then the field in all approximations has in all these points singularities of the needed type.

A Bessel–Watson transform pair
View Description Hide DescriptionA Bessel–Watson type of transform pair is developed. Since this result differs from a previously published result, a comparison between these two results is also presented.

Nonorthogonal R‐separable coordinates for four‐dimensional complex Riemannian spaces
View Description Hide DescriptionWe classify all R‐separable coordinate systems for the equations Δ_{4}Ψ=J^{4} _{ i, j=1} g ^{−1/2} ∂_{ j }(g ^{1/2} g ^{ i j }∂_{ i }Ψ) =0 and J^{4} _{ i, j=1} g ^{ i j }∂_{ i } W∂_{ j } W =0 with special emphasis on nonorthogonal coordinates, and give a group theoretic interpretation of the results. For flat space we show that the two equations separate in exactly the same coordinate systems and present a detailed list of the possibilities. We demonstrate that every R‐separable system for the Laplace equation Δ_{4}Ψ=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Δ_{4}Φ=λΦ on one of the manifoldsE _{4}, S _{1}×S _{3}, S _{2}×S _{2}, and S _{4}.

Stability of constant‐amplitude motions in slow‐fluctuation approximation
View Description Hide DescriptionOscillatory motions at constant amplitude admit of an especially simple description of their stability in phase space: orbital stability is equivalent to stability of the amplitudes alone, regardless of phase behavior, while Liapunov stability can subsequently be inferred from the phases alone. Stability arguments simplify further in slow‐fluctuation approximation because of the availability of explicit quadratures for the amplitudes and phases depending ultimately on a single polynomial. Thus, all orbital stability information about near‐resonant constant‐amplitude motions in conservative, autonomous systems can be extracted solely from that one polynomial. Explicit analytic criteria for orbital stability are derived, and auxiliary methods for the construction of stability charts are developed. Liapunov stability is shown to be a rare exception, but Liapunov instability is encountered in distinctly varying degrees; a fairly wide class of motions in a fairly wide class of systems is shown to be Liapunov–unstable only in the third order of a certain approximation. Five examples are given at some length; they differ starkly in detail. Owing to their tractable stability properties the abundant constant‐amplitude motions play in slow‐fluctuation approximation the role of the often nonexistent, purely periodic solutions of the traditional theories.

Independent variables in quantum mechanics
View Description Hide DescriptionIt is conjectured that the particle states of quantum mechanics are represented by functions of independent variables. These functions obey a linear differential equation which has an invariance group homomorphic to the inhomogeneous Lorentz group, thus giving a linear, Lorentz‐invariant theory. Simple one‐particle examples of equations which lead to a discrete particle spectrum are given, using both space–time variable, x _{μ}, and sets of spinlike variables (pairs of complex numbers). Some of the examples have internal symmetry. No examples of realistic ’’many‐body’’ particle theories are given, but we can deduce general characteristics. The differential equation must be of second or higher order to give an interaction. Products of single‐particle states will be solutions of the equation and will form a complete set for widely separated particles. But products of one‐particle states are not solutions of the equation for strongly interacting particles, and this permits the creation of particles. The origin of antisymmetry in such a theory is not clear.

Representation and properties of para‐Bose oscillator operators. II. Coherent states and the minimum uncertainty states
View Description Hide DescriptionThe energy, position, and momentum eigenstates of a para‐Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para‐Bose system. This brings in, in a natural way, the coherent states ‖z;α〉 defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ‖z;α〉〈z;α‖ and the other containing the pseudodiagonal basis ‖z;α〉〈−z;α‖. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible α dependence of this minimum value is considered.

Ermakov systems, velocity dependent potentials, and nonlinear superposition
View Description Hide DescriptionWe derive new Ermakov systems with velocity‐dependent potentials. The extended Ermakov system presented contains all known one‐dimensional cases and many new systems. These Ermakov systems lead to a nonlinear superposition law for the solutions.

Random walks and quantum currents in networks
View Description Hide DescriptionThe free electron network model of a metal is reformulated in terms of restricted random walks; this allows direct calculation of the propagator. The reformulation gives more freedom in the choice of boundary conditions and is suitable for the investigation of topologically disordered networks.

On the WKB approximation to the propagator for arbitrary Hamiltonians
View Description Hide DescriptionThis paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian H _{ c } to H is such that it associates aP_{ i }Q^{ j } +(1−a)Q^{ j }P_{ i } to p _{ i } q ^{ j }, then the formula gives K _{WKB}=K _{VV}exp {( 1/2 −a)F_{ T }(∂^{2} H _{ c }/∂q ^{ i }∂p _{ i } )(q _{ c }(t),p _{ c }(t),t) d t}, where K _{VV}≡(2πih/)^{−n/2} (det M)^{1/2}exp(i S _{ c }/h/) is Van Vleck’s well‐known formula, S _{ c } being the action functional evaluated at the classical path (q _{ c },p _{ c }) and M _{ i j } ≡−∂^{2} S _{ c }/∂q _{ a } ^{ i }∂q _{ b } ^{ j }. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].

Degenerate perturbations in nonrelativistic quantum mechanics
View Description Hide DescriptionWe investigate the effects on the discrete spectrum of an arbitrary quantum mechanical Schrödinger operator H, which are caused by the addition of a real rank N separable potential to H. For such a potential the bound state energies E _{ n }(λ) as function of the r e a l potential strength λ are in general confined to certain b o u n d e d intervals. This remarkable phenomenon can be seen as a particular case of the general situation of c o m p l e x potential strengths.

Aharonov–Bohm scattering and the velocity operator
View Description Hide DescriptionIt is shown that the existence of Aharonov–Bohm scattering depends upon the criteria used for establishing the stationary states. If one applies Pauli’s criterion, there is no scattering. It is shown further that applying the usual criteria that the wave functions be continuous and single valued, as was done by Aharonov and Bohm, leads to stationary state wave functions which, with two exceptions, are eigenstates of the acceleration operator corresponding to eigenvalue zero. The acceleration operator is undefined for the remaining two states. Thus, only the eigenfunctions satisfying the Pauli criterion lead to well‐defined, sensible physics.

Some models of anisotropic spheres in general relativity
View Description Hide DescriptionA heuristic procedure is developed to obtain interior solutions of Einstein’s equations for anisotropic matter from known solutions for isotropic matter. Five known solutions are generalized to give solutions with anisotropic sources.

Observer frame rotation rates and magnetic fields in spatially homogeneous universes
View Description Hide DescriptionWe discuss how the rotation of an observer’s Cartesian reference frame is related to the precession of the shear tensor’s principal axes and to the rotation vector of the fluid. An approximate solution of Einstein’s equations illuminates this relationship further. In the case with a magnetic field and fluid flow a vorticity is allowed in Bianchi I cosmologies.

On the Hoenselaers–Kinnersley–Xanthopoulos spinning mass fields
View Description Hide DescriptionThe metrics of the Hoenselaers–Kinnersley–Xanthopoulos family of spinning mass solutions with arbitrary positive number distortion parameter δ and with twin rotation‐reflection parameters besides mass parameter are studied. When values of two parameters are unequal or equal, the metrics are asymmetric or symmetric with respect to the reflection at the equatorial plane, respectively. The metrics for any distortion parameter δ contain no event horizon.

On stationary axially symmetric Einstein–Maxwell scalar and Brans–Dicke–Maxwell fields
View Description Hide DescriptionA particular type of exact solutions of Einstein–Maxwell massless scalar field equations corresponding to stationary axially symmetric fields is presented here. The solutions are linear combinations of static fields with constant coefficients. Further, by a proper choice of conformal transformation the solutions have been transformed to the Brans–Dicke fields coupled with source‐free electromagnetic fields. Finally these solutions have been transformed to a general form through unit transformations.

The quantum equivalence principle and finite particle creation in expanding universes
View Description Hide DescriptionA formulation of the equivalence principle in quantum field theory is introduced. The quantum equivalence principle yields implementable Bogolyubov transformations. In this way we find a theory for a scalar field in curved space–time where particle creation is finite for every value of the coupling constant. In the particular case of conformal coupling the initial conditions of positive and negative frequency wave functions coincide with the ones of a first order WKB approximation. The coefficients of the Bogolyubov transformations are exactly computed and the created energy density is also finite.