Volume 28, Issue 5, May 1987
Index of content:

Branching rules for sp(2N ) algebra reduction on the chain sp(2N− 2)×sp(2)
View Description Hide DescriptionIn this paper branching rules for the reduction sp(2N) ■ sp(2N−2)×sp(2) are found, and a new pattern for labeling vectors belonging to the base for unitary irreducible representations BUIR’s is found.

Enlargeable graded Lie algebras of supersymmetry
View Description Hide DescriptionA criterion to enlarge infinite‐dimensional Lie algebras to analytic Lie groups is used here for the extension of graded Lie algebras of physical interest to super‐Lie groups.

Generalized Burgers equations and Euler–Painlevé transcendents. II
View Description Hide DescriptionIt was proposed earlier [P. L. Sachdev, K. R. C. Nair, and V. G. Tikekar, J. Math. Phys. 2 7, 1506 (1986)] that the Euler–Painlevé equationy y‘+a y’^{2}+ f(x)y y’+g(x) y ^{2}+b y’+c=0 represents the generalized Burgers equations (GBE’s) in the same manner as Painlevé equations do the KdV type. The GBE was treated with a damping term in some detail. In this paper another GBE u _{ t }+u ^{ a } u _{ x }+J u/2t =(gd/2)u _{ x x } (the nonplanar Burgers equation) is considered. It is found that its self‐similar form is again governed by the Euler–Painlevé equation. The ranges of the parameter α for which solutions of the connection problem to the self‐similar equation exist are obtained numerically and confirmed via some integral relations derived from the ODE’s. Special exact analytic solutions for the nonplanar Burgers equation are also obtained. These generalize the well‐known single hump solutions for the Burgers equation to other geometries J=1,2; the nonlinear convection term, however, is not quadratic in these cases. This study fortifies the conjecture regarding the importance of the Euler–Painlevé equation with respect to GBE’s.

R‐separation of variables for the time‐dependent Hamilton–Jacobi and Schrödinger equations
View Description Hide DescriptionThe theory of R‐separation of variables is developed for the time‐dependent Hamilton–Jacobi and Schrödinger equations on a Riemannian manifoldV ^{ n } where time‐dependent vector and scalar potentials are permitted. As an application it is shown how to obtain all R‐separable coordinates for the n‐sphere and Euclidean n‐space.

Maximum entropy summation of divergent perturbation series
View Description Hide DescriptionIn this paper the principle of maximum entropy is used to predict the sum of a divergent perturbation series from the first few expansion coefficients. The perturbation expansion for the ground‐state energy E(g) of the octic oscillator defined by H=p ^{2}/2+x ^{2}/2+g x ^{8} is a series of the form E(g)∼ 1/2 +∑(−1)^{ n+1} A _{ n } g ^{ n }. This series is terribly divergent because for large n the perturbation coefficients A _{ n } grow like (3n)!. This growth is so rapid that the solution to the moment problem is not unique and ordinary Padé summation of the divergent series fails. A completely different kind of procedure based on the principle of maximum entropy for reconstructing the function E(g) from its perturbation coefficients is presented. Very good numerical results are obtained.

Symmetries of static, spherically symmetric space‐times
View Description Hide DescriptionIn this paper it is shown that reduction from maximal to minimal static, spherical symmetry of a space‐time occurs in only one step reducing the number of independent Killing vector fields from 10 to 4. Maximal symmetry corresponds only to the de Sitter, anti‐de Sitter, and Minkowski metrics, without reference to the Einstein field equations.

Propagators for massive vector fields in anti‐de Sitter space‐time using Stueckelberg’s Lagrangian
View Description Hide DescriptionExpressions are found for homogeneous and inhomogeneous propagators for vector fields of arbitrary mass in anti‐de Sitter space‐time using a generalization of Stueckelberg’s Lagrangian for a massive vector field. The massless case (quantum electrodynamics) is also considered by taking the appropriate zero‐mass limit.

A limit theorem for basic states of disordered structures
View Description Hide DescriptionLet a(t,ω) be a stationary process such that 0<E[1/a(t,ω)]<∞. It is shown that the random boundary‐value problem H y=−(d/d t)a(t,ω)(d y/d t)=λy, y(0)=y’(L)=0, has a unique solution (λ_{ i }(ω,L), y _{ i }(t,ω,L)) for i≥0 and λ_{ i }(ω,L)/λ_{ o i }(L)→1 almost surely as L→∞, where λ_{ o i }(L) is the ith eigenvalue of the averaged Hamiltonian H _{ o } y=−[1/E (1/a(t,ω))] (d ^{2} y/d t ^{2}) =λy, y(0)=y’(L)=0 .

Hypervirial theorem and parameter differentiation: Closed formulation for harmonic oscillator integrals
View Description Hide DescriptionA simple method has been developed to generate a closed formula for the calculation of matrix elements of arbitrary functions f(x) in the representation of the harmonic oscillator. The proposed algebraic procedure is based on the combined use of the hypervirial theorem with and without the second quantization formalism along with the parameter differentiation technique. The closed formula thus obtained is given in terms of a sum involving the jth derivative of f(x) evaluated at zero.

Linear stability of symplectic maps
View Description Hide DescriptionA general method is presented for analytically calculating linear stability limits for symplectic maps of arbitrary dimension in terms of the coefficients of the characteristic polynomial and the Krein signatures. Explicit results are given for dimensions 4, 6, and 8. The codimension and unfolding are calculated for all cases having a double eigenvalue on the unit circle. The results are applicable to many physical problems, including the restricted three‐body problem and orbital stability in particle accelerators.

Gauge and dual symmetries and linearization of Hirota’s bilinear equations
View Description Hide DescriptionIn Hirota’s [Hiroshima University Technical Report Nos. A 6, A 9, 1981; J. Phys. Soc. Jpn. 5 0, 3785 (1981)] bilinear difference equation which is satisfied by solutions to the Kadomtsev–Petviashvili (KP) hierarchy, gauge and dual symmetries are found, which enable one to reduce the problem of solving the nonlinear equation to solving a single linear equation.

Noether‐type conservation laws for perfect fluid motions
View Description Hide DescriptionA general approach is developed for the derivation of conservation laws in continuum physics. A Noether‐type theorem is applied in connection with transformations which leave the action functional invariant to within the integral of a divergence. Specific results are derived in the case of fluid dynamics: the pertinent equations are considered within the Lagrangian (material) description and are associated with a genuine variation formulation. The physical meaning of the conservation laws is emphasized and the greater generality of the approach is commented upon.

Factorization of the wave equation in higher dimensions
View Description Hide DescriptionThe factorization of the wave equation into a coupled system involving up‐and down‐going wave components is obtained for the case where the field quantities are multivariate functions of spatial variables, but the velocity c is a function of the z variable only. The form of the reflection operator is derived and the quadratic differential‐integral equation satisfied by its kernel is obtained.

Confinement and redistribution of charges and currents on a surface by external fields
View Description Hide DescriptionThe old problem of light scattering from a perfectly conducting surface is addressed. An electromagnetic field is incident upon the boundary, where it induces a charge and current distribution. These charges and currents emit the reflected fields. A set of equations for the charges and currents on the surface is derived by eliminating the E and B fields from Maxwell’sequations with the aid of the appropriate boundary conditions. An explicit and general solution is achieved, which reveals the confinement and redistribution of the charge and the current on the surface by the external field. Expressions are obtained for the surface resolvents, or the redistribution matrices, which represent the surface geometry. Action of a surface resolvent on the incident field, evaluated at the surface, then yields the charge and current distributions. The Faraday induction appears as an additional contribution to the charge density. Subsequently, the reflected fields are expanded in spherical waves, which have the surface‐multipole moments as a source. Explicit expressions are presented for the surface‐multipole moments, and it is pointed out that charge conservation on the surface sets constraints on these moments. The results apply to arbitrarily shaped surfaces and to any incident field. For a specific choice of the surface structure and the external field, the solutions for the charge, the current, and the reflected fields are amenable to numerical evaluation.

On the electromagnetic field and the Teukolsky–Press relations in arbitrary space‐times
View Description Hide DescriptionThe relations on the electromagnetic field obtained by Teukolsky and Press for type D vacuum space‐times are considered; these are four second‐order equations in two complex components of the field with respect to a principal null tetrad. A rigorous geometric interpretation of these relations is given, showing the essential role played by the Maxwellian character of the basic null tetrad. It appears that, generically, the Teukolsky–Press relations are incomplete. Once completed, their generalizations to the general Maxwell equations (with source term) with respect to non‐necessarily Maxwellian tetrads on arbitrary space‐times are given.

Existence and analyticity of many‐body scattering amplitudes at low energies
View Description Hide DescriptionTwo‐cluster–two‐cluster scattering amplitudes for N‐body quantum systems are studied. Our attention is restricted to energies below the lowest three‐cluster threshold. For potentials falling off like r ^{−1−δ} it is proved that in this energy range these amplitudes exist, are continuous, and that the asymptotic completeness holds. Moreover, if the potentials fall off exponentially it is proved that these amplitudes can be meromorphically continued in the energy, with square root or logarithmic branch points at the two‐cluster thresholds.

A new class of lattice identities
View Description Hide DescriptionSome identities involving two‐dimensional lattice sums of a class of integrals are derived. A simple application to theta functions is given.

Two comments on nonlinear Schrödinger equations
View Description Hide DescriptionIn a recent paper, Brüll and Lange [Expos. Math. 4, 279 (1986); Math. Meth. Appl. Sci. 8, 559 (1986)] have discussed a class of nonlinear Schrödinger equations with rather general nonlinearities which comprises various cases occurring in the literature. Although the ‘‘potentials’’ in these equations are quite complicated, the equations admit various invariance properties. The present paper has two aims. First several local and global conservation laws related to conservation of mass, impulse, and energy are exhibited. One of these laws seems to be new, though not surprising. Then it is shown that the equation defined by Brüll and Lange is just suitable to apply some transformations which reduce the problem of solitary waves to a relatively simple Hamiltonian system in the plane. This method of transforming the phase plane problem into normal form is, in some respects, similar to the transformations introduced by Hadeler [Proc. Math. Soc. Edinburgh, to be published; F r e e B o u n d a r y P r o b l e m s : T h e o r y a n d A p p l i c a t i o n s, Montecatini Conference, 1981, edited by A.Fasano and M. Primicerio (Pitman, New York, 1983), Vol. II, pp. 664–671] for parabolic and hyperbolic reaction diffusion equations.

The Hamiltonian structures of the nonlinear Schrödinger equation in the classical limit
View Description Hide DescriptionUsing Madelung’s hydrodynamical variables, it is shown that the bi‐Hamiltonian structure of the nonlinear Schrödinger equation goes over to a bi‐Hamiltonian structure of the shallow water wave equations in the classical limit.

Semiclassical treatment of spin system by means of coherent states
View Description Hide DescriptionThe semiclassical time‐dependent propagator is studied in terms of the SU(2) coherent states for spin systems. The first‐ and second‐order terms are obtained by means of a detailed calculation. While the first‐order term was established in the earlier days of coherent states the second‐order one is a subject of contradiction. The present approach is developed through a polygonal expansion of the discontinuous paths that enter the path integral. The results here presented are in agreement with only one of the previous approaches, i.e., the one developed on Glauber’s coherent states by means of a direct WKB approximation. It is shown that the present approach gives the exact result in a simple case where it is also possible to observe differences with previous works.