Volume 27, Issue 6, June 1986
Index of content:

Soliton structure of the Drinfel’d–Sokolov–Wilson equation
View Description Hide DescriptionAn integrable equation due to Drinfel’d and Sokolov [Sov. Math. Dokl. 2 3, 457 (1981)] and Wilson [Phys. Lett. A 8 9, 332 (1982)] (DSW) is studied in detail. It is shown how this system can be obtained as a six‐reduction of the Kadomtsev–Petviashvili hierarchy. This equation presents a novel type of solutions called static solitons: they are static solutions that interact with moving solitons without deformations. Examples of such solutions are given, together with a general procedure for their construction. Finally the Painlevé analysis of the DSW equation is performed directly on the bilinear form, which constitutes a new application of the singularity analysis method.

Generalized Burgers equations and Euler–Painlevé transcendents. I
View Description Hide DescriptionInitial‐value problems for the generalized Burgers equation (GBE) u _{ t }+u ^{ β} u _{ x }+λu ^{α} =(δ/2)u _{ x x } are discussed for the single hump type of initial data—both continuous and discontinuous. The numerical solution is carried to the self‐similar ‘‘intermediate asymptotic’’ regime when the solution is given analytically by the self‐similar form. The nonlinear (transformed) ordinary differential equations (ODE’s) describing the self‐similar form are generalizations of a class discussed by Euler and Painlevé and quoted by Kamke. These ODE’s are new, and it is postulated that they characterize GBE’s in the same manner as the Painlevé equations categorize the Kortweg–de Vries (KdV) type. A connection problem for some related ODE’s satisfying proper asymptotic conditions at x=±∞, is solved. The range of amplitude parameter is found for which the solution of the connection problem exists. The other solutions of the above GBE, which display several interesting features such as peaking, breaking, and a long shelf on the left for negative values of the damping coefficient λ, are also discussed. The results are compared with those holding for the modified KdV equation with damping.

The four‐dimensional conformal Kepler problem reduces to the three‐dimensional Kepler problem with a centrifugal potential and Dirac’s monopole field. Classical theory
View Description Hide DescriptionThe four‐dimensional conformal Kepler problem is reduced by an S ^{1} action, when the associated momentum mapping takes nonzero fixed values. The reduced Hamiltonian system proves to be the three‐dimensional Kepler problem along with a centrifugal potential and Dirac’s monopole field. The negative‐energy surface turns out to be diffeomorphic to S ^{3}×S ^{2}, on which the symmetry group SO(4) acts. Constants of motion of the reduced system are also obtained, which include the total angular momentum vector and a Runge–Lenz‐like vector. The Kepler problem is thus generalized so as to admit the same symmetry group.

On an expression for the average resolvent using Grassmann integration
View Description Hide DescriptionThe integral representation of the inverse of a determinant and the Grassmann representation of a determinant are used to derive an expression for the average resolvent for a Gaussian orthogonal ensemble. The expression is compared with the one obtained using Lagrangian formalism.

Asymptotics of the maximum number of repulsive particles on a spherical surface
View Description Hide DescriptionThere are N equal particles interacting through a repulsive potential V(r)=A/r ^{β} ( β>0) placed insided the sphere. For β=1 (Coulomb case) and for all β<1, the minimum energy configuration will have all N particles on the inner surface of the sphere for any integer N. It is shown that starting from N=13 and for β>β_{crit} the minimum energy configuration will have only a fraction of particles on the inner surface while the rest of the particles will hang in the equilibrium inside the volume of the sphere. As far as β→1+, the maximum number of charges that can be held on the surface has an asymptotic N _{0}∼const/( β−1)^{2}. For const, the gap 6<C<8 was found. The theory may be applicable to, e.g., ‘‘magic numbers’’ in small atomic clusters. Some problems that are not yet solved are listed.

Logarithmic corrections to the uncertainty principle and infinitude of the number of bound states of N‐particle systems
View Description Hide DescriptionIt is shown that critical long‐distance behavior for a two‐body potential, defining the finiteness or infinitude of the number of negative eigenvalues of Schrödinger operators in ν dimensions, is given by v _{ k }(r)=−((ν−2)/2r ) ^{2}−1/(2r ln r)^{2} +⋅⋅⋅−1/(2r ln r⋅ln ln r ⋅⋅⋅ ln_{(k)} r)^{2}, where k=0,1,... for ν≠2 and k=1,2,... if ν=2. This result is a consequence of logarithmic corrections to an inequality known as the uncertainty principle. If the continuum threshold in the N‐body problem is defined by a two‐cluster breakup the results presented generate corrections to the existing sufficient conditions for the existence of infinitely many bound states.

On the hydrogen‐oscillator connection: Passage formulas between wave functions
View Description Hide DescriptionRecent works on the hydrogen‐oscillator connection are extended to cover in a systematic (and easily computarizable) way the problem of the expansion of an R^{3} hydrogen wave function in terms of R^{4}oscillatorwave functions. Passage formulas from oscillator to hydrogen wave functions are obtained in six cases resulting from the combination of the following coordinate systems: spherical and parabolic coordinate systems for the hydrogen atom in three dimensions, and Cartesian, double polar, and hyperspherical coordinate systems for the isotropic harmonic oscillator in four dimensions. These coordinate systems are particularly useful in physical applications (e.g., Zeeman and Stark effects for hydrogenlike ions and coherent state approaches to the Coulomb problem).

Comment on momentum in stochastic mechanics
View Description Hide DescriptionStochastic mechanics is a probabilistic description of quantum systems in terms of stochastic differential equations. Davidson [M. Davidson, Lett. Math. Phys. 5, 523 (1981)] and de Falco, De Martino, and De Siena [D. de Falco, S. De Martino, and S. De Siena, Lett. Nuovo Cimento 3 6, 457 (1983)] have introduced momentum variables into this scheme. In this paper a discussion of this attempt is presented and some difficulties concerning the physical interpretation are pointed out.

The range of quantum probability
View Description Hide DescriptionThe set of all pair (and in fact higher‐order) distributions that are representable in quantum mechanics is characterized and compared with the classical range. Various interference phenomena yield pair distributions that are not classical; a few examples are discussed. These results shed light on some fundamental problems concerning the interpretation of quantum mechanics, in particular it is demonstrated how the ‘‘quantum logic’’ of Birkhoff and Von Neumann can be naturally interpreted in terms of truth values. Finally, the possibility of interpreting quantum probability in a realistic ‘‘quasiclassical’’ way is explored.

Non‐Abelian Aharonov–Bohm effects, Feynman paths, and topology
View Description Hide DescriptionThe Aharonov–Bohm effect in general gauge theories, for particles in gauge‐curvature‐free regions, is studied using the quantum mechanical propagator in the form of a Feynman sum over paths. Following Schulman [L. S. Schulman, T e c h n i q u e s a n d A p p l i c a t i o n s o f P a t h I n t e g r a t i o n (Wiley, New York, 1981)], such paths are divided into their homotopy equivalence classes, and the contributions from each class of paths of the Feynman sum are identified with propagators of a wave equation in the universal covering manifold of M, resulting in a simple form for the propagator on M. A group homomorphism from H, the fundamental homotopy group of M, to the gauge group G is shown to characterize possible Aharonov–Bohm effects, which can be divided into two types, Abelian and non‐Abelian, according to whether H*, the image of this homomorphism, is Abelian or non‐Abelian. For a non‐Abelian Aharonov–Bohm effect, it is necessary that both H and G be non‐Abelian. Simple examples illustrate the theory.

Reeh–Schlieder‐type density results in one‐ and n‐body Schrödinger theory and the ‘‘unique continuation problem’’
View Description Hide DescriptionA couple of Reeh–Schlieder‐type density results are proved to hold in one‐ and n‐body Schrödinger theory, that is, it is proved that states localized at time zero in an arbitrarily small open set of R^{ n } are already total after an arbitrarily small time (which implies much more than the well‐known acausal behavior of nonrelativistic theories). It is shown that there exists a close connection to the so‐called ‘‘unique continuation property’’ of elliptic partial differential operators. Furthermore, a certain machinery of analytic continuation is developed and the notion of generalized propagation kernels is introduced, which also might be of use elsewhere (e.g., in scattering theory).

Cosmologies with a noninteracting mixture of dust and radiation
View Description Hide DescriptionIn this paper a particular class of anisotropiccosmologies, the Kantowski–Sachs models, is considered. It is assumed that the matter content of the models consists of a noninteracting mixture of ordinary matter (‘‘dust’’) and thermal radiation. A qualitative study by means of a three‐dimensional autonomous system is carried out, giving us the global behavior of the ‘‘dust’’ density, the radiation density, and the shear anisotropy during the models’ evolution. All the models have past and future cosmological singularities where both the dust density and the radiation density diverge. A particular interesting result is a set of solutions of three‐measure zero, which is radiation dominated at one (past) singularity (of the ‘‘point’’ type) and evolving to a (future) singularity, where ordinary matter and thermal radiation become negligible.

Killing spinors and gravitational perturbations
View Description Hide DescriptionIt is shown that in a vacuum space‐time, possibly with a nonzero cosmological constant, which admits a D(1,0) Killing spinor, one component of the perturbed Weyl spinor that satisfies a decoupled equation, when multiplied by an appropriate factor made out of the components of the Killing spinor, constitutes a Debye potential that generates metric perturbations of the considered background. It is also shown that in the case where the background is of type N, there is an operation that relates the gravitational perturbations and the zero‐rest‐mass fields of spin‐0, ‐ 1/2 , and ‐1.

Gravitational perturbations of algebraically special space‐times via the HH equation
View Description Hide DescriptionBy using the approach to the Einstein field equations based on the existence of a congruence of null two‐dimensional surfaces, it is shown that a scalar potential that satisfies a second‐order linear partial differential equation generates gravitational perturbations of a given algebraically special solution of the Einstein vacuum field equations with cosmological constant. Generalizations of this result to the case of simultaneous perturbations of the gravitational and the matter fields are also indicated.

Parallel‐propagated frame along the geodesics of the metrics admitting a Killing–Yano tensor
View Description Hide DescriptionIt is shown that the equations for a parallel‐propagated frame along geodesics can be solved explicitly by separation of variables assuming the existence of a valence‐2 Killing–Yano tensor that is indecomposable and such that the associated Killing tensor has no constant eigenvalue.

A method for generating exact Bianchi type II cosmological models
View Description Hide DescriptionA method for generating exact Bianchi type II cosmological models with a perfect fluid distribution of matter is presented. Two new classes of Bianchi type II solutions have been generated from Lorenz’s solution [D. Lorenz, Phys. Lett. A 7 9, 19 (1980)]. A detailed study of physical and kinematic properties of one of them has been carried out.

Invariant operators for the n‐dimensional super‐Poincaré algebra and the decomposition of the scalar superfield
View Description Hide DescriptionAn analysis of supersymmetric Kaluza–Klein theories is begun by obtaining the Casimir operators for the super‐Poincaré algebra in any number of dimensions. The knowledge of these operators is used to decompose the general scalar superfield in 11 dimensions into its irreducible parts. The irreducible superfields are expressed as products of Grassmann–Hermite functions and Grassmann–Bargmann–Wigner multispinor fields. Some Lagrangians for these superfields are written down. The formulation is off shell but global.

Superfield and irreducible superfield structure
View Description Hide DescriptionField contents for scalar superfields in different numbers of dimensions are tabulated. Tables of field contents for some irreducible superfields included in the scalar superfield in 11 dimensions are also given.

Local properties of quantum systems
View Description Hide DescriptionLocal properties of a quantum system are defined as the expectation values of its observables in a microstate of some complete set of commuting observables. An equation for the time evolution of local properties is obtained for any system whose statistical operator (density matrix) evolves by unitary transformation in accordance with the von Neumann equation. The formalism is applied to the example of a system of one particle. In this case the local properties are fields, the time‐evolution equation is an equation of continuity with source terms. For constants of motion the source terms vanish, giving equations of continuity for the fields. For each scalar field a flux vector for its transport current is defined. For momentum, a stress tensor is obtained. The effects on local properties of realization of a latent ensemble of the statistical operator (an entropy‐increasing mechanism recently proposed to explain approach to equilibrium) are also considered; a non‐negative local entropy production is identified, as well as a discontinuous redistribution of local properties among microstates.

Wave vector dependent susceptibility of a free electron gas in D dimensions and the singularity at 2k _{F}
View Description Hide DescriptionThe static susceptibility of a free electron gas in D dimensions at T=0 is obtained by techniques of dimensional regularization. Our solutions for the susceptibility χ(k,D) are given in terms of the hypergeometric function. For any integer dimensions analytic expressions are possible. The high‐ and low‐k series solutions are shown to be related by an analytic continuation if D is an odd integer, but not related if D is an even integer. The singularity at 2k _{F} is a branch point, whereupon the series solutions are absolutely convergent, yielding χ(k=2k _{F},D)=(D−1)^{−} ^{1}. The relationship of χk D has the appearance of a PVT diagram.