Volume 35, Issue 4, April 1994
Index of content:

The m‐function for Hamiltonians with Wigner–von Neumann potentials
View Description Hide DescriptionHamiltonians with strong Wigner–von Neumann potentials are known to possess bound states in the continuous spectrum. For smaller couplings these bound states become resonances, which appear as singularities of the m‐function. These singularities are intimately connected with the decay of the corresponding subordinate eigenfunction. For Coulomb‐type potentials with Schrödinger operators this has recently be shown by Atkinson, the author, and Hinton, Klaus and Shaw. Here these results are extended to a general class of Wigner–von Neumann potentials for Dirac and Schrödinger operators.

The relation between the Berry and the Anandan–Aharonov connections for U(N) bundles
View Description Hide DescriptionThe geometric phase has been defined as the holonomy of two bundles with different base spaces and different connections. In this article we establish the relation between the Berry–Simon and the Anandan–Aharonov definitions for a large class of cyclic Hamiltonians (with N‐fold degeneracy) using the classifying theorem for principal fiber bundles.

Ground state energy for the Hartree–Fock equations with Dirichlet boundary conditions
View Description Hide DescriptionThe Hartree–Fock equations for 2 N fermions in a one‐dimensional box are solved in terms of a power series in the coupling constant for various values of N and the quantum numbers. A general formula is found for the total energy and wave function for N=1, which is accurate to sixth order. Another formula for the total energy is conjectured for any N, which is accurate to second order. Using these results, the ground state energy, energy of interaction, and boundary energy are found to second‐order accuracy for arbitrary N.

N‐body quantum scattering theory in two Hilbert spaces. VII. Real‐energy limits
View Description Hide DescriptionA study is made of the real‐energy limits of approximate solutions of the Chandler–Gibson equations, as well as the real‐energy limits of the approximate equations themselves. It is proved that (1) the approximate time‐independent transition operator T ^{π}(z) and an auxiliary operator M ^{π}(z), when restricted to finite energy intervals, are trace class operators and have limits in trace norm for almost all values of the real energy; (2) the basic dynamical equation that determines the operator M ^{π}(z), when restricted to the space of trace class operators, has a real‐energy limit in trace norm for almost all values of the real energy; (3) the real‐energy limit of M ^{π}(z) is a solution of the real‐energy limit equation; (4) the diagonal (on‐shell) elements of the kernels of the real‐energy limit of T ^{π}(z) and of all solutions of the real‐energy limit equation exactly equal the on‐shell transition operator, implying that the real‐energy limit equation uniquely determines the physical transition amplitude; and (5) a sequence of approximate on‐shell transition operators converges strongly to the exact on‐shell transition operator. These mathematically rigorous results are believed to be the most general of their type for nonrelativistic N‐body quantum scattering theories.

Dissipation in Wigner–Poisson systems
View Description Hide DescriptionThe Wigner–Poisson (WP) system (or quantum Vlasov–Poisson system) is modified to include dissipative terms in the Hamiltonian. By utilizing the equivalence of the WP system to the Schrödinger–Poisson system, global existence and uniqueness are proved and regularity properties are deduced. The proof differs somewhat from that for the nondissipative case treated previously by Brezzi–Markowich and Illner et al.; in particular the Hille–Yosida Theorem is used since the linear evolution is not unitary, and a Liapunov function is introduced to replace the energy, which is not conserved.

Quantum scattering on a Cantor bar
View Description Hide DescriptionA solvable model of the scattering of a one‐dimensional particle by the familiar ‘‘middle‐third’’ Cantor set is considered. It is shown that the presence of positive energy levels in such a model is typical. From the high‐energy behavior of the scattering, data computation of the Hausdorff dimension of the scatterer is suggested.

Geometric quantization of Neumann‐type completely integrable Hamiltonian systems
View Description Hide DescriptionThe procedure of geometric quantization is developed for completely integrable dynamical systems on manifolds with exact symplectic structure. These results are applied to Neumann’s nonharmonic oscillatory system on two‐dimensional sphere S ^{2}.

The relativistic Hermite polynomial is a Gegenbauer polynomial
View Description Hide DescriptionIt is shown that the polynomials introduced recently by Aldaya, Bisquert, and Navarro‐Salas [Phys. Lett. A 156, 381 (1991)] in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of Gegenbauer polynomials. This fact is useful in the investigation of the properties of the corresponding wave function. Some examples are given, in particular, related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers.

Quantum measurements and quantum logics
View Description Hide DescriptionA quantum logic formulation of quantum measurements is introduced using Boolean powers of orthomodular σ‐posets to the description of the coupled system. A comparison with the traditional Hilbert space approach to quantum measurements and a discussion of some important classes of measurements is given.

Cosmological CP violation
View Description Hide DescriptionSpinor fields are studied in infinite, topologically multiply connected Robertson–Walker cosmologies. Unitary spinor representations for the discrete covering groups of the spacelike slices are constructed. The spectral resolution of Dirac’s equation is given in terms of horospherical elementary waves, on which the treatment of spin and energy is based in these cosmologies. The meaning of the energy and the particle–antiparticle concept is explained in the context of this varying cosmic background. Discrete symmetries, in particular inversions of the multiply connected spacelike slices, are studied. The violation of the unitarity of the parity operator, due to self‐interference of P‐reflected wave packets, is discussed. The violation of the CP and CPT invariance—already on the level of the free Dirac equation on this cosmological background—is pointed out.

Semiclassical approximation for bound states of the Schrödinger equation with a Coulomb‐like potential
View Description Hide DescriptionThe eigenvalue problem for the radial Schrödinger equation with an ‘‘almost’’ Coulomb potential is considered. This problem provides the simplest example of a system whose classical trajectories have singularities. Thus the standard semiclassical quantization procedure (in the sense of Einstein–Brillouin–Keller–Maslov) cannot be applied straightforwardly to this situation. The equation under consideration has two transition points on the half axis 0<x<∞: a regular singular point at the origin and a turning point for some x _{0}≳0; these points coalesce at low energy levels. The well‐known comparison equation method provides uniform asymptotics for equations of this kind, but has the following shortcomings: it does not appeal to the corresponding classical mechanical problem and the formulas for the eigenvalues and eigenfunctions contain phase integrals which are not analytic with respect to the parameter characterizing the closeness of the transition points. In the present article we derive new formulas for the eigenfunctions in the form of inverse Fourier transforms of rapidly oscillating exponentials multiplied by some powers of x, together with simple formulas for the eigenvalues. These formulas are directly connected with the classical trajectories and do not have the shortcomings of the comparison equation method, i.e., all the functions entering the asymptotic expansions are analytic in x and energy. As a by‐product, an asymptotic expansion of integrals having two stationary phase points which coalesce for some value of the parameter and tend to infinity when the parameter tends to zero is obtained.

Second harmonic generation in nonlinear optical films
View Description Hide DescriptionSecond harmonic generation, an important phenomenon in nonlinear optics, is modeled in this work. The model is derived from a nonlinear system of Maxwell’sequations, which overcomes the known shortcomings of some commonly used models in the literature. Existence and uniqueness of solutions are established by a combination of a variational approach and the contraction mapping principle. Some numerical results are also presented.

Nonlinear random wave equations
View Description Hide DescriptionNonlinear waveequations describing forced vibrations of a string subject to random perturbations of intensity ε are studied herein. Two kinds of random perturbations are considered; one is Gaussian in the form of initial white noise and the other is a non‐Gaussian random forcing which involves the formal derivative of a Brownian sheet. The solutions u ^{ε}(t,x) of the complete forced equation are a stochastic process in space and time which has a unique stochastic equilibrium. The time variable t varies in a finite interval I:=[0, T], where T is a fixed positive time, and the space variable x varies in an interval J. Assuming that the initial conditions are of sufficient regularity, the fluctuations u ^{ε}(t,x) from u ^{0}(t,x), the unperturbed solution, are analyzed as the intensity of perturbation ε↓0.

Nöther formalism for conserved quantities in classical gauge field theories
View Description Hide DescriptionGauge theories coupled with the gravitational field and external matter fields are considered and Nötherian techniques are used to build the canonical conserved quantities. Their superpotentials are found and the relation between the canonical stress tensor and the Hilbert tensor is determined. The case of Yang–Mills theories coupled with gravity and scalar matter fields is considered in detail and it is shown that in this case the canonical stress tensor and the Hilbert tensor coincide.

On the group theory of the polarization states of a massless field
View Description Hide DescriptionIt is shown that the theory of representations of the Poincaré group applied to the vector potentials of a massless field yields in a simple and direct way (without assuming that the field is a gauge field, and without detailed assumption on the form of the equation of motion), the structure of the polarizations, the Gupta–Bleuler condition, and gauge invariance of the theory. The method is shown to apply to the Maxwell field and to a five‐dimensional generalization of the Maxwell field (which properly contains the Maxwelltheory) associated with manifestly covariant dynamics.

Energy‐momentum conservation in gauge theories
View Description Hide DescriptionUsing the formulation in terms of principal fiber bundles, it is shown that the external symmetry implies the conservation of the energy‐momentum tensor for a general gauge theory. When applying this theory to the case of Dirac fields, one is naturally forced to replace the dependence on the metric by the dependence on the underlying spin structure in the variational problems.

Third‐order bounds on the conductivity of a random stacking of cubes
View Description Hide DescriptionThird‐order bounds on the conductivity of a random stacking of cubes are constructed. Since a random stacking of cubes is a symmetric cell material the bounds are determined by a Miller parameter G [J. Math. Phys. 10, 1988 (1969)] that is independent of volume fractions. This parameter is evaluated with a boundary integral technique to G=0.1375164293203.

A generalized Henon–Heiles system and related integrable Newton equations
View Description Hide DescriptionA detailed description is given of integrable cases of the generalized Henon–Heiles systems which differs from the standard H–H ones by the term α/q ^{2} _{2}. Their connection with fifth‐order one‐component soliton equations is discussed. Lax representations are constructed, and the bi‐Hamiltonian formulation of dynamics is given. It is also shown that the gH–H system can be mapped onto another system of Newton equations with a nonstandard Hamiltonian structure.

Quadrics on complex Riemannian spaces of constant curvature, separation of variables, and the Gaudin magnet
View Description Hide DescriptionIntegrable systems that are connected with orthogonal separation of variables in complex Riemannian spaces of constant curvature are considered herein. An isomorphism with the hyperbolic Gaudin magnet, previously pointed out by one of the authors, extends to coordinates of this type. The complete classification of these separable coordinate systems is provided by means of the corresponding L matrices for the Gaudin magnet. The limiting procedures (or ε calculus) which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems.

Local geometric invariants of integrable evolution equations
View Description Hide DescriptionThe integrable hierarchy of commuting vector fields for the localized induction equation of 3D hydrodynamics, and its associated recursion operator, are used to generate families of integrable evolution equations which preserve local geometric invariants of the evolving curve or swept‐out surface.