Volume 23, Issue 5, May 1982
Index of content:

The character table for the corepresentations of magnetic groups
View Description Hide DescriptionA square character table is shown to exist for all finite magnetic groups. The table possesses row and column orthogonality properties similar to the character table for linear groups.

Complex orthogonal and symplectic matrices depending on parameters
View Description Hide DescriptionVersal deformations of elements of complex orthogonal and symplectic Lie algebras are studied. For a general element M of o(n,C) or sp(2n,C), a normal form M _{ A } is found which, unlike the Jordan normal form M _{ J }, depends holomorphically on M and on the similarity transformation M _{ A } = g M g ^{−1} from the corresponding group. Orthogonal and symplectic cases are treated simultaneously in order to underline their close relation. Bundles of matrices of low codimension are listed and bifurcation diagrams of two‐parameter families of orthogonal matrices are shown. Finally, versal deformations of all elements of o(6,C) are explicitly shown.

A theorem on A‐proper mappings and its application in scattering theory
View Description Hide DescriptionWe propose a projectionally complete scheme yielding an approximate solution of the functional equationB f = g in a Hilbert space. We prove that B is an A‐proper mapping. The result is applied to an integral equation with a kernel appearing in multichannel scatteringtheory.

Bargmann transform, Zak transform, and coherent states
View Description Hide DescriptionIt is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the Bargmann representation or by using the k q representation which was introduced by J. Zak. In this paper both methods are considered, in particular, in connection with expansions of generalized functions in what are called Gabor series. The setting consists of two spaces of generalized functions (tempered distributions and elements of the class S ^{*}) which appear in a natural way in the context of the Bargmann transform. Also, a thorough mathematical investigation of the Zak transform is given. This paper contains many comments and complements on existing literature; in particular, connections with the theory of interpolation of entire functions over the Gaussian integers are given.

Valleys and fall lines on a Riemannian manifold
View Description Hide DescriptionThe concepts of fall lines, ridges, and general stationary paths are defined for a potential energy function on a Riemannian manifold. Some theorems governing their properties and relationships are derived. These concepts are of interest in the classical mechanics of constrained systems and in the theory of collective motions in many‐body quantum mechanics.

A mechanical model with constraints
View Description Hide DescriptionWe analyze a dynamical system with a finite number of degrees of freedom. A complete analysis is presented both for first class constraints as well as for second class constraints. The results are applicable to Yang–Mills fields as well as higher spin fields.

Hamiltonian perturbation theory in noncanonical coordinates
View Description Hide DescriptionThe traditional methods of Hamiltonian perturbation theory in classical mechanics are first presented in a way which clearly displays their differential‐geometric foundations. These are then generalized to the case of noncanonical in phase space. In the new method the Hamiltonian H is treated, not as a scalar in phase space, but as one component of the fundamental form p d q−H d t. The perturbation analysis is applied to this entire form, in all of its components.

Electromagnetic multipole propagation in a homogeneous conducting wholespace
View Description Hide DescriptionThe electric and magnetic field components are expanded in terms of spin weighted spherical harmonics, thereby defining a multipole structure whose propagation through a homogeneous conducting wholespace is obtained. For current sources contained inside the unit sphere, the resultant exterior electromagnetic fields are uniquely associated with either currents from electrodes placed inside the unit sphere, or from insulated wires inside the unit sphere. The former fields have transverse magnetic fields, while the latter (with the exception of the magnetic monopole) have transverse electric fields. When displacement currents are ignored, the fields from step function current sources are described by the incomplete gamma functions, while the exact solutions containing the hyperbolic contributions are obtained from convolutions with integer order modified Bessel functions and functions dependent on the temporal behavior of the current source.

Nonrecurrence of the stochastic process for the hydrogen atom problem in stochastic electrodynamics
View Description Hide DescriptionIt is shown, following a criterion borrowed from Khas’minskii, that the stochastic process associated with the (approximate) Fokker–Planck equation of the hydrogen atom problem in stochastic electrodynamics (SED) is nonrecurrent and therefore also nonergodic. The demonstration of this nonrecurrence property does not use any explicit solution. The property implies, among other things, that all the invariant measures of the process will be nonfinite. Some remarks concerning the consequences for SED are made.

Pseudoscalar interaction of coupled quantum‐mechanical oscillators with independent Fermi systems
View Description Hide DescriptionUsing the techniques of constructive quantum field theory we analyze the dynamics of a cubic lattice of quantum mechanical oscillators with nearest‐neighbor coupling that interact with a corresponding lattice of finite‐volume truncations of independent relativistic Fermi fields. Since the model is nonrelativistic, we rely on a nonrelativistic version of the Osterwalder–Schrader (OS) reconstruction theorem. Also, the absence of Nelson’s symmetry in the (un)Euclidean picture is not serious because the transfer matrix in a given space direction is simple enough to make the verification of spatial OS positivity easy. After establishing for our model many of the basic results that hold for the more standard models, we give a proof of the Fortuin–Kasteleyn–Ginibre (FKG) inequality that is essentially independent of the dimension of the Fermi systems.

Solutions of the wave equation for superposed potentials with application to charmonium spectroscopy
View Description Hide DescriptionIn this paper complete solutions of the Schrödinger equation for three different superposed potentials have been obtained. In particular high‐energy asymptotic expansions of the bound‐state eigenfunctions and eigenvalues are derived. Various properties of these expansions have been examined including the behavior of Regge trajectories. Finally the relevance of these investigations to various aspects of the spectroscopy of heavy quark composites is discussed in detail.

Quantum mechanical scattering theory for potentials of the form V(r) = (sin r)/r ^{ β} , 1/3<β≤1/2
View Description Hide DescriptionA rigorous nonrelativistic time dependent quantum‐mechanical scattering theory for a single particle is developed for potentials of the form V(r) = (sin r)/r ^{ β} , 1/3<β⩽ 1/2 . The positive‐energy solutions of the radial Schrödinger equation are used to construct modified wave operators which converge as t→±∞ on a dense set of states to the familiar time‐independent formulas for the wave operators.

Generalized completeness relations in the theory of resonant scattering
View Description Hide DescriptionGeneralized completeness relations involving resonance states are constructed within the framework of analytically continued symmetrized scattering kernels into the unphysical sheet of the complex‐energy plane. The bases states utilized are identified with complex‐energy generalized eigenvectors over an extended or rigged Hilbert space. The resulting relations are uniquely defined and do not exhibit the usual divergence problems encountered with the regularization methods.

Eigenvalues of an anharmonic oscillator. II
View Description Hide DescriptionAn expression is derived for the sixth nonzero term in the WKBJ approximation. The six‐term WKBJ approximation is applied to calculate the eigenvalues for the potential V(x) = 1/2 k x ^{2}+a x ^{4}, k≳0 and a≳0. At low values of λ[ = ah//( μk ^{3})^{1/2}], the calculated results are in excellent agreement, to 15 significant figures, with those of Banerjee e t a l. for all quantum numbers. At medium and high values of λ, the calculated results are poor at low quantum numbers, but improve rapidly as n increases. A 15‐significant‐figure accuracy is achieved at n = 10 for λ = 0.05, and at about n = 15 for λ = 20 000. For λ = 0.5, eigenvalues are calculated to 20 significant figures and an argument is presented to show that by n = 50, the calculated value is correct to 19 significant figures. The accuracy further improves at higher quantum numbers.

Existence theorem for solutions of Witten’s equation and nonnegativity of total mass
View Description Hide DescriptionWe prove that given an asymptotically flat (in a very weak sense) initial data set, there always exists a spinor field that satisfies Witten’s equation and that becomes constant at infinity. Thus we fill a gap in Witten’s arguments on the nonnegativity of the total mass of an isolated system, when measured at spatial infinity. We also include a review of Witten’s argument.

Linear transport in nonhomogeneous media. II
View Description Hide DescriptionElementary continuum and discrete solutions are constructed for transport equations with scattering ratios which are bilinear functions of position. Numerical results are given for some albedo problems.

Unrenormalized Schwinger–Dyson equations and dynamical mass generation
View Description Hide DescriptionIn dynamical mass generation of fermion fields by fermion‐spin‐1‐boson interactions, anomalous magnetic moment‐like terms in three‐point functions play an essential role, so that the Schwinger–Dyson equations do not admit a trivial solution, i.e., a solution without dynamically generated mass. Applicability of Altman’s and Lika’s versions of nonlinear operator theory to unrenormalized Schwinger–Dyson equations is discussed and algorithms for construction of approximate solutions are proposed.

Remarks on spacetime symmetries and nonabelian gauge fields
View Description Hide DescriptionSome properties of nonabelian gauge fields invariant under a given group of spacetime transformations are studied and applied to the case of spatial homogeneity and isotropy. All the k = 0 and k = 1 Robertson–Walker models filled with a SO (3) gauge field are derived.

SU(3) wave solutions
View Description Hide DescriptionAn a n s a t z yielding propagating wave solutions for pure SU(3) gauge theories is exhibited. The solutions are self‐dual and have a superposition property like their SU(2) analogs. Possible generalizations of the a n s a t z which may be used to obtain additional irreducible SU(3) solutions are also suggested.

Supermanifolds and BRS transformations
View Description Hide DescriptionWe use a supermanifold formalism to support a geometrical interpretation of a recently proposed superfield formulation of BRS (Becchi–Ronet–Stora) and anti‐BRS transformations.