Volume 41, Issue 12, December 2000
 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.
