Volume 44, Issue 11, November 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Recovery of a potential from the ratio of reflection and transmission coefficients
View Description Hide DescriptionFor the onedimensional Schrödinger equation, the analysis is provided to recover the potential from the data consisting of the ratio of a reflection coefficient to the transmission coefficient. It is investigated whether such data uniquely constructs a reflection coefficient, the number of bound states, boundstate energies, boundstate norming constants, and a corresponding potential. In all three cases when there is no knowledge of the support of the potential, the support of the potential is confined to a halfline, and the support is confined to a finite interval, various uniqueness and nonuniqueness results are established, the precise criteria are provided for the uniqueness and the nonuniqueness and the degree of nonuniqueness, and the recovery is illustrated with some explicit examples.

The Hopf algebra of identical, fermionic particle systems—Fundamental concepts and properties
View Description Hide DescriptionThe Hopf algebra structure of the fermionic Fock space is unravelled. The tools provided by the Hopf algebra formalism are used to rederive in a more straightforward fashion some known theorems and to open the way to natural generalizations of these results. The algebraic concepts of rank, depth and length of a wave function are given. They allow one to cast a wave function into a canonical form that is simpler and more appropriate to a physical interpretation or a numerical treatment. An original algorithm to reexpand a wave function with the least possible number of spin orbitals is described.

Scale calculus and the Schrödinger equation
View Description Hide DescriptionThis paper is twofold. In a first part, we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus. The scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions such that when ε goes to zero. This led to several new notions as representations: fractal functions and εdifferentiability. The basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative. We then discuss some algebraic properties of these operators. We define a natural bialgebra, called quantum bialgebra, associated with them. Finally, we discuss a convenient geometric object associated with our study. In a second part, we define a first quantization procedure of classical mechanics following the scale relativitytheory developed by Nottale. We obtain a nonlinear Schrödinger equation via the classical Newton’s equation of dynamics using the scale operator. Under special assumptions we recover the classical Schrödinger equation and we discuss the relevance of these assumptions.

On decoherence
View Description Hide DescriptionUsing a quantum particle in as a toy model, and following the rules of Schrödinger’s quantum mechanics, we discuss to which extent one may be able to use “decoherence” to view the quantum particle as a “classical” measuring apparatus to measure position. We discuss also very briefly the measurement of momentum and the case of quantum optics.

Some results on the eigenfunctions of the quantum trigonometric Calogero–Sutherland model related to the Lie algebra
View Description Hide DescriptionWe express the Hamiltonian of the quantum trigonometric Calogero–Sutherland model related to the Lie algebra in terms of a set of Weylinvariant variables, namely, the characters of the fundamental representations of the Lie algebra. This parametrization allows us to solve for the energy eigenfunctions of the theory and to study properties of the system of orthogonal polynomials associated with them such as recurrence relations and generating functions.

A note on Anderson localization for the random hopping model
View Description Hide DescriptionThis short note is devoted to the proof of Lifshitz tails and a Wegner estimate, and thus, band edge localization, for the random hopping model.

The generalized MICKepler system
View Description Hide DescriptionThis paper deals with the dynamical system that generalizes the MICKepler system. It is shown that the Schrödinger equation for this generalized MICKepler system can be separated in spherical and parabolic coordinates. The spectral problem in spherical and parabolic coordinates is solved.

Random magnetic fields on line graphs
View Description Hide DescriptionWe study the spectral and transport properties of Schrödinger operators on line graphs with random magnetic fields. We show that it has a pure point spectrum with exponentially decaying eigenfunctions on spectral edges, whereas there appears an eigenvalue with infinite multiplicity due to the structure of line graphs. We compute the electrical conductivity which is zero on spectral edges, but is nonzero and finite on the isolated eigenvalue mentioned above. Some related problems are also discussed.

Radon–Nikodym derivatives of quantum operations
View Description Hide DescriptionGiven a completely positive (CP) map T, there is a theorem of the Radon–Nikodym type [W. B. Arveson, Acta Math. 123, 141 (1969); V. P. Belavkin and P. Staszewski, Rep. Math. Phys. 24, 49 (1986)] that completely characterizes all CP maps S such that is also a CP map. This theorem is reviewed, and several alternative formulations are given along the way. We then use the Radon–Nikodym formalism to study the structure of order intervals of quantum operations, as well as a certain onetoone correspondence between CP maps and positive operators, already fruitfully exploited in many quantum informationtheoretic treatments. We also comment on how the Radon–Nikodym theorem can be used to derive norm estimates for differences of CP maps in general, and of quantum operations in particular.

Perturbation expansions for a class of singular potentials
View Description Hide DescriptionHarrell’s modified perturbation theory [Ann. Phys. (N.Y.) 105, 379 (1977)] is applied and extended to obtain nonpower perturbation expansions for a class of singular Hamiltonians known as generalized spiked harmonic oscillators. The perturbation expansions developed here are valid for small values of the coupling λ>0, and they extend the results which Harrell obtained for the spiked harmonic oscillator Formulas for the excited states are also developed.

An extension of Fourier analysis for the torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian
View Description Hide DescriptionWe solved the Schrödinger equation for a particle in a uniform magnetic field in the dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus is defined as a quotient of the Euclidean space by an arbitrary dimensional lattice Λ. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potentialfree problems; the Schrödinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.

A class of vector coherent states defined over matrix domains
View Description Hide DescriptionA general scheme is proposed for constructing vector coherent states, in analogy with the wellknown canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable In the present scheme, the variable is replaced by matrix valued functions over appropriate domains. As particular examples, we analyze the quaternionic extensions of the canonical coherent states and the Gilmore–Perelomov and Barut–Girardello coherent states arising from representations of SU(1,1). Possible physical applications are indicated.

Phase space methods for particles on a circle
View Description Hide DescriptionThe phase space for a particle on a circle is considered. Displacement operators in this phase space are introduced and their properties are studied. Wigner and Weyl functions in this context are also considered and their physical interpretation and properties are discussed. All results are compared and contrasted with the corresponding ones for the harmonic oscillator in the phase space.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Mathematical structure of the temporal gauge in quantum electrodynamics
View Description Hide DescriptionThe conflict between Gauss’ law constraint and the existence of the propagator of the gauge fields, at the basis of contradictory proposals in the literature, is shown to lead to only two alternatives, both with peculiar features with respect to standard quantum field theory. In the positive (interacting) case, the Gauss’ law holds in operator form, but only the correlations of exponentials of gauge fields exist (nonregularity) and the space translations are not strongly continuous, so that their generators do not exist. Alternatively, a Källen–Lehmann representation of the two point function of satisfying locality and invariance under space–time translations, rotations and parity is derived in terms of the two point function of positivity is violated, the Gauss’ law does not hold, the energy spectrum is positive, but the relativistic spectral condition does not hold. In the free case, θvacua exist on the observable fields, but they do not have time translationally invariant extensions to the gauge fields; the vacuum is faithful on the longitudinal field algebra and defines a modular structure (even if the energy is positive). Functional integral representations are derived in both cases, with the alternative between ergodic measures on real random fields or complex Gaussian random fields.
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 GENERAL RELATIVITY AND GRAVITATION


Embeddings in space–times sourced by scalar fields
View Description Hide DescriptionThe extension of the Campbell–Magaard embedding theorem to general relativity with minimally coupled scalar fields is formulated and proven. The result is applied to the case of a selfinteracting scalar field for which new embeddings are found, and to Brans–Dicke theory. The relationship between the Campbell–Magaard theorem and the General Relativity Cauchy and initial value problems is outlined.

Parametric phenomena of the particle dynamics in a periodic gravitational wave field
View Description Hide DescriptionWe establish exactly solvable models for the motion of neutral particles, electrically charged point and spin particles [U(1) symmetry], isospin particles [SU(2) symmetry], and particles with color charges [SU(3) symmetry] in a gravitational wave background. Special attention is devoted to parametric effects induced by the gravitational field. In particular, we discuss parametric instabilities of the particle motion and parametric oscillations of the vectors of spin, isospin, and color charge.

Symmetries of the energy–momentum tensor of spherically symmetric Lorentzian manifolds
View Description Hide DescriptionMatter collineations of spherically symmetric Lorentzian manifolds are considered. These are investigated when the energy–momentum tensor is nondegenerate and also when it is degenerate. We have classified space–times admitting higher symmetries and space–times admitting SO(3) as the maximal isometry group. For the nondegenerate case, we obtain either four, six, seven, or ten independent matter collineations in which four are isometries and the rest are proper. The results of the previous paper [Sharif and Sehar (Gen. Relativ. Gravit. 35, 1091 (2003)] are recovered as a special case. It is worth noting that we have also obtained two cases where the energy–momentum tensor is degenerate but the group of matter collineations is finitedimensional, i.e., four or ten.
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 DYNAMICAL SYSTEMS


A further solvable threebody problem in the plane
View Description Hide DescriptionThe solution is provided of a threebody problem in the plane, which is the third of a trio recently identified as likely to display a particularly simple timeevolution hence to be amenable to exact treatment. This conjecture, already validated by providing the solution of the first two of these three models, is now completely proven by exhibiting the solution of the third. This finding also demonstrates the conjectured superPainlevé character of certain nonlinear ordinary differential equations, namely, the fact that their general solution is an entire function of the independent variable.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Gauge principle revisited: Towards a unification of space–time and internal gauge interactions
View Description Hide DescriptionThe minimal coupling principle is revisited under the quantum perspectives of the space–time symmetry. This revision is better realized on a group approach to quantization (GAQ) where group cohomology and extensions of groups play a preponderant role. We first consider the case of the electromagnetic potential; the Galilei and/or Poincaré group is (noncentrally) extended by the “local” U(1) group. The resulting group can also be seen as a central extension, parametrized by both the mass and the electric charge, of an infinitedimensional group, on which GAQ leads to the dynamics of a particle moving in the presence of an electromagnetic field. Then we try the gravitational interaction of a particle by making the space–time translations “local.” However, promoting to “local” the space–time subgroup of the true symmetry of the quantum free relativistic particle, i.e., the centrally extended by U(1) Poincaré group, results in a new electromagneticlike force of pure gravitational origin. This is a consequence of the space–time translations not being an invariant subgroup of the extended Poincaré group and constitutes a preliminary attempt to a nontrivial mixing of space–time and internal gauge interactions.
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 FLUIDS


Existence of suitable weak solutions of complex Ginzburg–Landau equations and properties of the set of singular points
View Description Hide DescriptionIn this paper, we consider the supercritical complex Ginzburg–Landau equation. We discuss the existence of suitable weak solution in Ω, where Ω is a bounded domain in or the whole space. We also discuss the properties of the set of the singular points of the suitable weak solution in which means that the possible singular points are located in a bounded ball for any given time and there is no singular point on the whole space after limited time.
