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Nov 2003

Volume 44, Issue 11, pp. 4875-5456

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Recovery of a potential from the ratio of reflection and transmission coefficients

Tuncay Aktosun and Vassilis G. Papanicolaou

J. Math. Phys. 44, 4875 (2003); http://dx.doi.org/10.1063/1.1614871 (9 pages) | Cited 4 times

Online Publication Date: 21 October 2003

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For the one-dimensional 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, bound-state energies, bound-state 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 half-line, 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. © 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states

The Hopf algebra of identical, fermionic particle systems—Fundamental concepts and properties

Patrick Cassam-Chenaï and Frédéric Patras

J. Math. Phys. 44, 4884 (2003); http://dx.doi.org/10.1063/1.1611266 (23 pages) | Cited 7 times

Online Publication Date: 21 October 2003

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The 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 re-expand a wave function with the least possible number of spin orbitals is described.© 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods
05.30.Fk Fermion systems and electron gas

Scale calculus and the Schrödinger equation

Jacky Cresson

J. Math. Phys. 44, 4907 (2003); http://dx.doi.org/10.1063/1.1618923 (32 pages) | Cited 16 times

Online Publication Date: 21 October 2003

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This 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 f(t,ϵ) such that f(t,ϵ)→f(t) when ϵ goes to zero. This led to several new notions as representations: fractal functions and ϵ-differentiability. 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 relativity theory 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. © 2003 American Institute of Physics.
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03.65.Fd Algebraic methods
03.65.Ge Solutions of wave equations: bound states
02.20.Uw Quantum groups
05.45.Df Fractals
02.30.Tb Operator theory

On decoherence

Gianfausto Dell’Antonio

J. Math. Phys. 44, 4939 (2003); http://dx.doi.org/10.1063/1.1616202 (18 pages) | Cited 2 times

Online Publication Date: 21 October 2003

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Using a quantum particle in R3 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. © 2003 American Institute of Physics.
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03.65.Ta Foundations of quantum mechanics; measurement theory
42.50.-p Quantum optics

Some results on the eigenfunctions of the quantum trigonometric Calogero–Sutherland model related to the Lie algebra D4

J. Fernández Núñez, W. García Fuertes, and A. M. Perelomov

J. Math. Phys. 44, 4957 (2003); http://dx.doi.org/10.1063/1.1618362 (18 pages) | Cited 4 times

Online Publication Date: 21 October 2003

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We express the Hamiltonian of the quantum trigonometric Calogero–Sutherland model related to the Lie algebra D4 in terms of a set of Weyl-invariant 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. © 2003 American Institute of Physics.
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03.65.Fd Algebraic methods
02.20.Sv Lie algebras of Lie groups
02.20.Uw Quantum groups

A note on Anderson localization for the random hopping model

Frédéric Klopp and Shu Nakamura

J. Math. Phys. 44, 4975 (2003); http://dx.doi.org/10.1063/1.1616998 (6 pages) | Cited 6 times

Online Publication Date: 21 October 2003

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This short note is devoted to the proof of Lifshitz tails and a Wegner estimate, and thus, band edge localization, for the random hopping model. © 2003 American Institute of Physics.
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71.10.-w Theories and models of many-electron systems

The generalized MIC-Kepler system

Levon Mardoyan

J. Math. Phys. 44, 4981 (2003); http://dx.doi.org/10.1063/1.1619205 (7 pages) | Cited 11 times

Online Publication Date: 21 October 2003

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This paper deals with the dynamical system that generalizes the MIC-Kepler system. It is shown that the Schrödinger equation for this generalized MIC-Kepler system can be separated in spherical and parabolic coordinates. The spectral problem in spherical and parabolic coordinates is solved. © 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
03.65.Fd Algebraic methods
02.20.Uw Quantum groups

Random magnetic fields on line graphs

Fumihiko Nakano and Yuji Nomura

J. Math. Phys. 44, 4988 (2003); http://dx.doi.org/10.1063/1.1613377 (15 pages) | Cited 1 time

Online Publication Date: 21 October 2003

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We 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. © 2003 American Institute of Physics.
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73.43.Cd Theory and modeling
03.65.Ge Solutions of wave equations: bound states
02.10.Ox Combinatorics; graph theory

Radon–Nikodym derivatives of quantum operations

Maxim Raginsky

J. Math. Phys. 44, 5003 (2003); http://dx.doi.org/10.1063/1.1615697 (18 pages) | Cited 5 times

Online Publication Date: 21 October 2003

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Given 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 T−S 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 one-to-one correspondence between CP maps and positive operators, already fruitfully exploited in many quantum information-theoretic 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. © 2003 American Institute of Physics.
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03.65.Fd Algebraic methods

Perturbation expansions for a class of singular potentials

Nasser Saad, Richard L. Hall, and Attila B. von Keviczky

J. Math. Phys. 44, 5021 (2003); http://dx.doi.org/10.1063/1.1616996 (21 pages) | Cited 1 time

Online Publication Date: 21 October 2003

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Harrell’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 H = −(d2/dx2)+x2+(A/x2)+(λ/xα) (A ≥ 0,α>2), 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 A = 0. Formulas for the excited states are also developed. © 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states

An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian

Makoto Sakamoto and Shogo Tanimura

J. Math. Phys. 44, 5042 (2003); http://dx.doi.org/10.1063/1.1616203 (28 pages) | Cited 3 times

Online Publication Date: 21 October 2003

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We solved the Schrödinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus Tn = Rn/Λ is defined as a quotient of the Euclidean space Rn by an arbitrary n-dimensional lattice Λ. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free 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. © 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
02.30.Nw Fourier analysis

A class of vector coherent states defined over matrix domains

K. Thirulogasanthar and S. Twareque Ali

J. Math. Phys. 44, 5070 (2003); http://dx.doi.org/10.1063/1.1617366 (14 pages) | Cited 18 times

Online Publication Date: 21 October 2003

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A general scheme is proposed for constructing vector coherent states, in analogy with the well-known canonical coherent states, and their deformed versions, when these latter are expressed as infinite series in powers of a complex variable z. In the present scheme, the variable z 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. © 2003 American Institute of Physics.
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03.65.Fd Algebraic methods
02.10.Yn Matrix theory
11.30.Hv Flavor symmetries

Phase space methods for particles on a circle

S. Zhang and A. Vourdas

J. Math. Phys. 44, 5084 (2003); http://dx.doi.org/10.1063/1.1616997 (11 pages) | Cited 4 times

Online Publication Date: 21 October 2003

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The phase space S×Z 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 R×R phase space. © 2003 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
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Mathematical structure of the temporal gauge in quantum electrodynamics

J. Löffelholz, G. Morchio, and F. Strocchi

J. Math. Phys. 44, 5095 (2003); http://dx.doi.org/10.1063/1.1603957 (13 pages) | Cited 1 time

Online Publication Date: 21 October 2003

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The 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 Ai satisfying locality and invariance under space–time translations, rotations and parity is derived in terms of the two point function of Fμν; 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, θ-vacua 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.© 2003 American Institute of Physics.
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12.20.Ds Specific calculations
11.10.Lm Nonlinear or nonlocal theories and models
11.15.Me Strong-coupling expansions
11.15.Tk Other nonperturbative techniques
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Embeddings in space–times sourced by scalar fields

E. Anderson, F. Dahia, J. E. Lidsey, and C. Romero

J. Math. Phys. 44, 5108 (2003); http://dx.doi.org/10.1063/1.1610237 (12 pages) | Cited 10 times

Online Publication Date: 21 October 2003

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The 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 self-interacting 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. © 2003 American Institute of Physics.
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04.20.Ex Initial value problem, existence and uniqueness of solutions
04.20.Gz Spacetime topology, causal structure, spinor structure

Parametric phenomena of the particle dynamics in a periodic gravitational wave field

Alexander B. Balakin, Veronika R. Kurbanova, and Winfried Zimdahl

J. Math. Phys. 44, 5120 (2003); http://dx.doi.org/10.1063/1.1617364 (21 pages) | Cited 7 times

Online Publication Date: 21 October 2003

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We 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. © 2003 American Institute of Physics.
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11.15.-q Gauge field theories
04.30.Nk Wave propagation and interactions
11.30.Hv Flavor symmetries

Symmetries of the energy–momentum tensor of spherically symmetric Lorentzian manifolds

M. Sharif

J. Math. Phys. 44, 5141 (2003); http://dx.doi.org/10.1063/1.1610779 (18 pages) | Cited 8 times

Online Publication Date: 21 October 2003

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Matter 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 finite-dimensional, i.e., four or ten. © 2003 American Institute of Physics.
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04.20.Gz Spacetime topology, causal structure, spinor structure
04.20.Ex Initial value problem, existence and uniqueness of solutions
02.40.Gh Noncommutative geometry
02.10.Ud Linear algebra
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A further solvable three-body problem in the plane

F. Calogero, J.-P. Françoise, and A. Guillot

J. Math. Phys. 44, 5159 (2003); http://dx.doi.org/10.1063/1.1610778 (7 pages) | Cited 4 times

Online Publication Date: 21 October 2003

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The solution is provided of a three-body problem in the plane, which is the third of a trio recently identified as likely to display a particularly simple time-evolution 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 super-Painlevé character of certain nonlinear ordinary differential equations, namely, the fact that their general solution is an entire function of the independent variable. © 2003 American Institute of Physics.
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45.50.Jf Few- and many-body systems
02.30.Hq Ordinary differential equations
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Gauge principle revisited: Towards a unification of space–time and internal gauge interactions

V. Aldaya, J. L. Jaramillo, and J. Guerrero

J. Math. Phys. 44, 5166 (2003); http://dx.doi.org/10.1063/1.1604183 (19 pages) | Cited 2 times

Online Publication Date: 21 October 2003

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The 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 infinite-dimensional 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. © 2003 American Institute of Physics.
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11.15.Me Strong-coupling expansions
11.10.Ef Lagrangian and Hamiltonian approach
11.30.Ly Other internal and higher symmetries
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Existence of suitable weak solutions of complex Ginzburg–Landau equations and properties of the set of singular points

Xiaofeng Liu and Houyu Jia

J. Math. Phys. 44, 5185 (2003); http://dx.doi.org/10.1063/1.1618360 (9 pages)

Online Publication Date: 21 October 2003

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In 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 mathn or the whole space. We also discuss the properties of the set of the singular points of the suitable weak solution in mathn, 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. © 2003 American Institute of Physics.
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74.20.De Phenomenological theories (two-fluid, Ginzburg-Landau, etc.)
02.30.Jr Partial differential equations
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Multiplicative noise: A mechanism leading to nonextensive statistical mechanics

Celia Anteneodo and Constantino Tsallis

J. Math. Phys. 44, 5194 (2003); http://dx.doi.org/10.1063/1.1617365 (10 pages) | Cited 31 times

Online Publication Date: 21 October 2003

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A large variety of microscopic or mesoscopic models lead to generic results that accommodate naturally within Boltzmann–Gibbs statistical mechanics [based on S1 ≡ −k ∫ du p(u)ln p(u)]. Similarly, other classes of models point toward nonextensive statistical mechanics [based on Sq ≡ k[1− ∫ du[p(u)]q]/[q−1], where the value of the entropic index q ∊ math depends on the specific model]. We show here a family of models, with multiplicative noise, which belongs to the nonextensive class. More specifically, we consider Langevin equations of the type math = f(u)+g(u)ξ(t)+η(t), where ξ(t) and η(t) are independent zero-mean Gaussian white noises with respective amplitudes M and A. This leads to the Fokker–Planck equation ∂tP(u,t) = −∂u[f(u)P(u,t)]+M∂u{g(u)∂u[g(u)P(u,t)]}+A∂uuP(u,t). Whenever the deterministic drift is proportional to the noise induced one, i.e., f(u) = −τg(u)g′(u), the stationary solution is shown to be P(u,∞)∝{1−(1−q)β[g(u)]2}1/(1−q) [with q ≡ (τ+3M)/(τ+M) and β = (τ+M/2A)]. This distribution is precisely the one optimizing Sq with the constraint 〈[g(u)]2〉q ≡ { ∫ du [g(u)]2[P(u)]q}/{ ∫ du [P(u)]q} = const. We also introduce and discuss various characterizations of the width of the distributions.© 2003 American Institute of Physics.
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05.20.-y Classical statistical mechanics
05.40.Ca Noise

On the generalized problem of the Boltzmann equation and the moment method in kinetic theory of gases

M. Chen

J. Math. Phys. 44, 5204 (2003); http://dx.doi.org/10.1063/1.1615696 (8 pages) | Cited 1 time

Online Publication Date: 21 October 2003

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In this paper we formulate the generalized problem of the Boltzmann equation based on the kinetic entropy balance equation in conjunction with the maximum entropy principle. First we prove that the solution of this generalized problem is unique. We then prove that the entropy balance equation obtained by Eu in extended irreversible thermodynamics is valid, if, and only if, the one-particle distribution function fa is the solution of this generalized problem. As a by-product of this result, we also obtain a statistical expression of the thermodynamic entropy balance equation that shares the same formula as the kinetic entropy balance equation. © 2003 American Institute of Physics.
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05.20.Dd Kinetic theory
51.10.+y Kinetic and transport theory of gases
05.70.Ln Nonequilibrium and irreversible thermodynamics
05.60.-k Transport processes
05.70.Ce Thermodynamic functions and equations of state

Stochastic dynamics of the scattering amplitude generating K-distributed noise

Timothy R. Field and Robert J. A. Tough

J. Math. Phys. 44, 5212 (2003); http://dx.doi.org/10.1063/1.1611264 (12 pages) | Cited 10 times

Online Publication Date: 21 October 2003

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We derive the stochastic dynamics of the complex valued amplitude resulting from coherent scattering from a random population of scatterers when this becomes asymptotically large. Considerations of a random walk model, introduced by Jakeman, are used to derive stochastic differential equations for the amplitude and corresponding intensity and phase stochastic processes. An analysis of the correlation structure in the fluctuations is provided and interpreted geometrically in terms of the gauge invariant properties of the field and the Markov property. A Fokker–Planck description for the evolution of the probability density is given and the equilibrium and detailed balance conditions shown to hold. Expressions for the intensity autocorrelation function and power spectral density are provided in closed form. The practical implications of the stochastic theory are discussed. © 2003 American Institute of Physics.
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05.40.Fb Random walks and Levy flights
02.50.Cw Probability theory
02.50.Ey Stochastic processes
02.50.Ga Markov processes

Interacting squares in arbitrary external field

Christian Tutschka

J. Math. Phys. 44, 5224 (2003); http://dx.doi.org/10.1063/1.1613042 (19 pages)

Online Publication Date: 21 October 2003

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A model of a many-body system composed of squares with contact pair interactions in an arbitrary external field is presented. The formulation uses a mapping of the system onto polydisperse hard core mixtures. On the polydisperse level then, a simplified Hamiltonian function is specified. This assumption together with a further one about the global free energy functional for the pure hard core part of the idealized mixture make the model solvable. It is expected to hold for high temperatures, low densities, or low temperatures. The validity of the method of construction in the latter case is illustrated by a further application to a corresponding lattice system, for which exact results to compare with are readily available when the temperature is sufficiently low.© 2003 American Institute of Physics.
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05.20.-y Classical statistical mechanics
05.70.Ce Thermodynamic functions and equations of state
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.30.Xx Calculus of variations
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Ring-shaped exact Hopf solitons

C. Adam and J. Sánchez-Guillén

J. Math. Phys. 44, 5243 (2003); http://dx.doi.org/10.1063/1.1612897 (7 pages) | Cited 9 times

Online Publication Date: 21 October 2003

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The existence of ring-like structures in exact Hopfion solutions is shown.© 2003 American Institute of Physics.
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05.45.Yv Solitons
02.40.Pc General topology
02.30.Hq Ordinary differential equations
02.30.Cj Measure and integration
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