Volume 41, Issue 12, December 2000
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


The onset of superconductivity in semiinfinite strips
View Description Hide DescriptionThe existence of a superconducting branch bifurcating from the normal state is proved in semiinfinite strips. It is proved that the critical magnetic field at which bifurcation takes place, or the onset field, for a semiinfinite strip is greater than the onset field for an infinite strip with the same width. In addition we find the loci of the vortices far away from the corners and show convergence of the bifurcating modes in long rectangles to those in the semiinfinite strip with the same width.

Inverse atomic densities and inequalities among density functionals
View Description Hide DescriptionRigorous relationships among physically relevant quantities of atomic systems (e.g., kinetic, exchange, and electron–nucleus attraction energies, information entropy) are obtained and numerically analyzed. They are based on the properties of inverse functions associated to the oneparticle density of the system. Some of the new inequalities are of great accuracy and/or improve similar ones previously known, and their validity extends to other manyfermion systems and to arbitrary dimensionality.

Eigenvalues in spectral gaps of the twodimensional Pauli operator
View Description Hide DescriptionWe consider purely magnetic twodimensional Pauli operators with a spectral gap, perturbed by a magnetic field Assuming that and vanish at infinity, we ask whether eigenvalues will cross the gap as Furthermore, we give an example of a twodimensional Pauli operator with periodic magnetic field of zero flux which has at least one spectral gap.

Multipartite generalization of the Schmidt decomposition
View Description Hide DescriptionWe find a canonical form for pure states of a general multipartite system, in which the constraints on the coordinates (with respect to a factorizable orthonormal basis) are simply that certain ones vanish and certain others are real. For identical particles they are invariant under permutations of the particles. As an application, we find the dimension of the generic local equivalence class.

Group theoretical quantum tomography
View Description Hide DescriptionThe paper is devoted to the mathematical foundation of quantum tomography using the theory of squareintegrable representations of unimodular Lie groups.

Phaseintegral formulas for quantal matrix elements
View Description Hide DescriptionSimple phaseintegral formulas, not involving wave functions, are derived for quantal matrix elements associated with bound states of a quantal particle in a smooth singlewell potential. In these formulas one uses an arbitrary order of the phaseintegral approximation generated from an unspecified base function.

Confluent hypergeometric equations and related solvable potentials in quantum mechanics
View Description Hide DescriptionThe connection between the Schrödinger and confluent hypergeometric equations is discussed. It is shown that the factorization of the confluent hypergeometric equation gives a unifying powerful algebraic tool in order to study some quantum mechanical eigenvalue problems. That description includes the linear and Ndimensional harmonic oscillators, as well as the Coulomb and Morse potentials.

Matroid theory and Chern–Simons
View Description Hide DescriptionIt is shown that matroid theory may provide a natural mathematical framework for a duality symmetries not only for quantum Yang–Mills physics, but also for Mtheory. Our discussion is focused in an action consisting purely of the Chern–Simons term, but in principle the main ideas can be applied beyond such an action. In our treatment the theorem due to Thistlethwaite, which gives a relationship between the Tutte polynomial for graphs and Jones polynomial for alternating knots and links, plays a central role. Before addressing this question we briefly mention some important aspects of matroid theory and we point out a connection between the Fano matroid and supergravity. Our approach also seems to be related to loop solutions of quantum gravity based in an Ashtekar formalism.

On the absolutely continuous spectrum of Stark Hamiltonians
View Description Hide DescriptionWe study the spectral properties of the Schrödinger operator with a constant electric field perturbed by a bounded potential. It is shown that if the derivative of the potential in the direction of the electric field is smaller at infinity than the electric field, then the spectrum of the corresponding Stark operator is purely absolutely continuous. In one dimension, the absolute continuity of the spectrum is implied by just the boundedness of the derivative of the potential. The sharpness of our criterion for higher dimensions is illustrated by constructing smooth potentials with bounded partial derivatives for which the corresponding Stark operators have a dense point spectrum.

Ideal quantum gases in dimensional space and powerlaw potentials
View Description Hide DescriptionWe investigate ideal quantum gases in dimensional space and confined in a generic external potential by using the semiclassical approximation. In particular, we derive density of states, density profiles and critical temperatures for Fermions and Bosons trapped in isotropic powerlaw potentials. From such results, one can easily obtain those of quantum gases in a rigid box and in a harmonic trap. Finally, we show that the Bose–Einstein condensation can set up in a confining powerlaw potential if and only if where is the space dimension and is the powerlaw exponent.

The semiclassical propagator for spin coherent states
View Description Hide DescriptionWe use a continuoustime path integral to obtain the semiclassical propagator for minimalspread spin coherent states. We pay particular attention to the “extra phase” discovered by Solari and Kochetov, and show that this correction is related to an anomaly in the fluctuation determinant. We show that, once this extra factor is included, the semiclassical propagator has the correct short time behavior to and demonstrate its consistency under dissection of the path.

Representation of quantum mechanical resonances in the Lax–Phillips Hilbert space
View Description Hide DescriptionWe discuss the quantum Lax–Phillips theory of scattering and unstable systems. In this framework, the decay of an unstable system is described by a semigroup. The spectrum of the generator of the semigroup corresponds to the singularities of the Lax–Phillips Smatrix. In the case of discrete (complex) spectrum of the generator of the semigroup associated with resonances, the decay law is exactly exponential. We explain how this profound difference between the quantum Lax–Phillips theory and the description of unstable systems in the framework of the standard quantum theory emerges. The states corresponding to these resonances (eigenfunctions of the generator of the semigroup) lie in the Lax–Phillips Hilbert space, and therefore all physical properties of the resonant states can be computed. In the special case of a timeindependent potential problem lifted trivially to the quantum Lax–Phillips theory, we show that the Lax–Phillips Smatrix is unitarily related to the Smatrix of standard scattering theory by a unitary transformation parametrized by the spectral variable σ of the Lax–Phillips theory. Analytic continuation in σ has some of the properties of a method developed some time ago for application to dilation analytic potentials. We work out an illustrative example of the theory using a Lee–Friedrichs model, which is generalized to a rank one potential in the Lax–Phillips Hilbert space.

Generalized affine coherent states: A natural framework for the quantization of metriclike variables
View Description Hide DescriptionAffine variables, which have the virtue of preserving the positivedefinite character of matrixlike objects, have been suggested as replacements for the canonical variables of standard quantization schemes, especially in the context of quantum gravity. We develop the kinematics of such variables, discussing suitable coherent states, their associated resolution of unity, polarizations, and finally the realization of the coherentstate overlap function in terms of suitable pathintegral formulations.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


A new superconformal mechanics
View Description Hide DescriptionIn this paper we propose a new supersymmetric extension of conformal mechanics. The Grassmannian variables that we introduce are the basis of the forms and of the vector fields built over the symplectic space of the original system. Our supersymmetric Hamiltonian itself turns out to have a clear geometrical meaning being the Lie derivative of the Hamiltonian flow of conformal mechanics. Using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original nonsupersymmetric model. Besides the supersymmetric extension of the original Hamiltonian, we also provide the extension of the other conformal generators present in the original system. These extensions also have a supersymmetric character being the square of some Grassmannian charge. We build the whole superalgebra of these charges and analyze their closure. The representation of the even part of this superalgebra on the odd part turns out to be integer and not spinorial in character.

The classical Kepler problem and geodesic motion on spaces of constant curvature
View Description Hide DescriptionIn this paper we clarify and generalize previous work by Moser and Belbruno concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as Hamiltonian systems and the phase flow in each system is characterized by the value of the corresponding Hamiltonian and one other parameter (the mass parameter in the Kepler problem and the curvature parameter in the geodesic motion problem). Using a canonical transformation the Hamiltonian vector field for the geodesic motion problem is transformed into one which is proportional to that for the Kepler problem. Within this framework the energy of the Kepler problem is equal to (minus) the curvature parameter of the constant curvature space and the mass parameter is given by the value of the Hamiltonian for the geodesic motion problem. We work with the corresponding family of evolution spaces and present a unified treatment which is valid for all values of energy continuously. As a result, there is a correspondence between the constants of motion for both systems and the Runge–Lenz vector in the Kepler problem arises in a natural way from the isometries of a space of constant curvature. In addition, the canonical nature of the transformation guarantees that the Poisson bracket Lie algebra of constants of motion for the classical Kepler problem is identical to that associated with geodesic motion on spaces of constant curvature.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Poisson algebras associated with constrained dispersionless modified Kadomtsev–Petviashvili hierarchies
View Description Hide DescriptionWe investigate the biHamiltonian structures associated with constrained dispersionless modified Kadomtsev–Petviashvili (KP) hierarchies which are constructed from truncations of the Lax operator of the dmKP hierarchy. After transforming their second Hamiltonian structures to those of the Gelfand–Dickeytype, we obtain the Poisson algebras of the coefficient functions of the truncated Lax operators. Then we study the conformal property and freefield realizations of these Poisson algebras. Some examples are worked out explicitly to illustrate the obtained results.

Integrability of the and Ruijsenaars–Schneider models
View Description Hide DescriptionWe study the and Ruijsenaars–Schneider models with interaction potential of trigonometric and rational types. The Lax pairs for these models are constructed and the involutive Hamiltonians are also given. Taking a nonrelativistic limit, we also obtain the Lax pairs for the corresponding Calogero–Moser systems.

Geometric phases for corotating elliptical vortex patches
View Description Hide DescriptionWe describe a geometric phase that arises when two elliptical vortex patches corotate. Using the Hamiltonian moment model of Melander, Zabusky, and Styczek [J. Fluid Mech. 167, 95–115 (1986)] we consider two corotating uniform elliptical patches evolving according to the second order truncated equations of the model. The phase is computed in the adiabatic setting of a slowly varying Hamiltonian as in the work of Hannay [J. Phys. A 18, 221–230 (1985)] and Berry [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We also discuss the geometry of the symplectic phase space of the model in the context of nonadiabatic phases. The adiabatic phase appears in the orientation angle of each patch—it is similiar in form and is calculated using a multiscale perturbation procedure as in the point vortex configuration of Newton [Physica D 79, 416–423 (1994)] and Shashikanth and Newton [J. Nonlinear Sci. 8, 183–214 (1998)], however, an extra factor due to the internal stucture of the patch is present. The final result depends on the initial orientation of the patches unlike the phases in the works of Hannay and Berry [J. Phys. A 18, 221–230 (1985)]; [Proc. R. Soc. London, Ser. A 392, 45–57 (1984)]. We then show that the adiabatic phase can be interpreted as the holonomy of a connection on the trivial principal fiber bundle where is identified with the product of the momentum level sets of two Kirchhoff vortex patches and is diffeomorphic to the momentum level set of two point vortex motion. This two point vortex motion is the motion that the patch centroids approach in the adiabatic limit.

 RELATIVITY AND GRAVITATION


Multiply warped products with nonsmooth metrics
View Description Hide DescriptionIn this article we study manifolds with metrics and properties of Lorentzian multiply warped products. We represent the interior Schwarzschild space–time as a multiply warped product space–time with warping functions and we also investigate the curvature of a multiply warped product with warping functions. We give the Ricci curvature in terms of for the multiply warped products of the form

 MISCELLANEOUS TOPICS IN MATHEMATICAL METHODS


unification model building II. Clebsch–Gordan coefficients of
View Description Hide DescriptionWe have computed the Clebsch–Gordan coefficients for the product where (000 001) is the adjoint 78dimensional representation of The results are presented for the dominant weights of the irreducible representations in this product. As a simple application we express the singlet operator in in terms of multiplets of the Standard Model gauge group.
