Volume 51, Issue 6, June 2010
 ARTICLES

 Quantum Mechanics (General and Nonrelativistic)

Localized eigenfunctions in Šeba billiards
View Description Hide DescriptionWe describe some new families of quasimodes for the Laplacian perturbed by the addition of a potential formally described by a Dirac delta function. As an application, we find, under some additional hypotheses on the spectrum, subsequences of eigenfunctions of Šeba billiards that localize around a pair of unperturbed eigenfunctions.

On the nonlocality of the fractional Schrödinger equation
View Description Hide DescriptionA number of papers over the past eight years have claimed to solve the fractional Schrödinger equation for systems ranging from the onedimensional infinite square well to the Coulomb potential to onedimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schrödinger equation from being solved in the usual piecewise fashion. We focus on the onedimensional infinite square well and show that the purported ground state, which is based on a piecewise approach, is definitely not a solution of the fractional Schrödinger equation for the general fractional parameter . On a more positive note, we present a solution to the fractional Schrödinger equation for the onedimensional harmonic oscillator with .

Riccati equation and the problem of decoherence
View Description Hide DescriptionThe block operator matrix theory is used to investigate the problem of a single qubit. We establish a connection between the Riccati operator equation and the possibility of obtaining an exact reduced dynamics for the qubit in question. The model of the half spin particle in the rotating magnetic field coupling with the external environment is discussed. We show that the model defined in such a way can be reduced to a time independent problem.

Grouptheoretical approach to Bloch electron in magnetic field problem
View Description Hide DescriptionIn this paper magnetictranslation group theory is extended to include full rotational symmetry of Hamiltonian. Proper generalization of small representation and star of the representation concepts are derived. Irreducible representations of magnetictranslation group and magneticspace group are presented. Correct form of symmetrized basis function is derived, reflecting symmetry of the magneticpoint group. From viewpoint of group theory reduction of Hamiltonian symmetry group caused by magnetic field and splitting of energy levels is investigated.

Cranking approach for a complex quantum system
View Description Hide DescriptionWithin random matrix model, the response of a complex quantum system on time variations in an external parameter . In the limit of weak coupling of the external parameter to the quantum system, the different dynamical regimes of the diffusion in a space of the occupancies of eigenstates of are considered. The main focus is made on measuring the role of memory effects in the quantum diffusive dynamics. Possible macroscopic manifestations of the quantum mechanical diffusion are discussed in the context of the cranking approach, where the time variations in the classical parameter provides the constancy of energy of the total system.
 Quantum Information and Computation

Fast universal quantum computation with railroadswitch local Hamiltonians
View Description Hide DescriptionWe present two universal models of quantum computation with a timeindependent, frustrationfree Hamiltonian. The first construction uses 3local (qubit) projectors and the second one requires only 2local qubitqutrit projectors. We build on Feynman’s Hamiltonian computer idea [R. Feynman, Optics News11, 11 (1985)] and use a railroadswitchtypeclock register. The resources required to simulate a quantum circuit with gates in this model are smalldimensional quantum systems (qubits or qutrits), a timeindependent Hamiltonian composed of local, constant norm, projector terms, the possibility to prepare computational basis product states, a running time , and the possibility to measure a few qubits in the computational basis. Our models also give a simplified proof of the universality of 3local adiabatic quantum computation.

Separability of particle fermionic states for arbitrary partitions
View Description Hide DescriptionWe present a criterion of separability for arbitrary partitions of particle fermionic pure states. We show that, despite the superficial nonfactorizability due to the antisymmetry required by the indistinguishability of the particles, the states which meet our criterion have factorizable correlations for a class of observables which are specified consistently with the states. The separable states and the associated class of observables share an orthogonal structure, whose nonuniqueness is found to be intrinsic to the multipartite separability and leads to the nontransitivity in the factorizability, in general. Our result generalizes the previous result obtained by Ghirardi et al. [J. Stat. Phys.108, 49 (2002)] for the and case.
 Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Abelian link invariants and homology
View Description Hide DescriptionWe consider the link invariants defined by the quantum Chern–Simons field theory with compact gauge group U(1) in a closed oriented 3manifold . The relation of the Abelian link invariants with the homology group of the complement of the links is discussed. We prove that, when is a homology sphere or when a link—in a generic manifold—is homologically trivial, the associated observables coincide with the observables of the sphere . Finally, we show that the U(1) Reshetikhin–Turaev surgery invariant of the manifold is not a function of the homology group only, nor a function of the homotopy type of alone.

BRST detour quantization: Generating gauge theories from constraints
View Description Hide DescriptionWe present the Becchi–Rouet–Stora–Tyutin (BRST) cohomologies of a class of constraint (super) Lie algebras as detour complexes. By interpreting the components of detour complexes as gauge invariances, Bianchi identities, and equations of motion, we obtain a large class of new gauge theories. The pivotal new machinery is a treatment of the ghost Hilbert space designed to manifest the detour structure. Along with general results, we give details for three of these theories which correspond to gauge invariant spinning particle models of totally symmetric, antisymmetric, and Kähler antisymmetric forms. In particular, we give details of our recent announcement of a form Kähler electromagnetism. We also discuss how our results generalize to other special geometries.

U(1)invariant membranes: The geometric formulation, Abel, and pendulum differential equations
View Description Hide DescriptionThe geometric approach to study the dynamics of U(1)invariant membranes is developed. The approach reveals an important role of the Abel nonlinear differential equation of the first type with variable coefficients depending on time and one of the membrane extendedness parameters. The general solution of the Abel equation is constructed. Exact solutions of the whole system of membrane equations in the Minkowski spacetime are found and classified. It is shown that if the radial component of the membrane world vector is only time dependent, then the dynamics is described by the pendulum equation.

Drinfel’d superdoubles and Poisson–Lie Tplurality in low dimensions
View Description Hide DescriptionDefining the real Lie superalgebra as real graded vector space we classify real Manin supertriples and Drinfel’d superdoubles of superdimensions (2,2), (4,2), and (2,4). The Drinfel’d doubles of the superdimension (2,2) are then used for construction of the simplest models related by Poisson–Lie Tplurality.
 General Relativity and Gravitation

A dynamic correspondence between Bose–Einstein condensates and Friedmann–Lemaître–Robertson–Walker and Bianchi I cosmology with a cosmological constant
View Description Hide DescriptionIn some interesting work of James Lidsey, the dynamics of Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology with positive curvature and a perfect fluid matter source is shown to be modeled in terms of a timedependent, harmonically trapped Bose–Einstein condensate. In the present work, we extend this dynamic correspondence to both FLRW and Bianchi I cosmologies in arbitrary dimension, especially when a cosmological constant is present.

On asymptotic structure at null infinity in five dimensions
View Description Hide DescriptionWe discuss the asymptotic structure of null infinity in five dimensional spacetimes. Since it is known that the conformal infinity is not useful for odd higher dimensions, we shall employ the coordinate based method such as the Bondi coordinate first introduced in four dimensions. Then we will define the null infinity and identify the asymptotic symmetry. We will also derive the Bondi mass expression and show its conservation law.
 Classical Mechanics and Classical Fields

Hysteresis and phase transitions for onedimensional and threedimensional models in shape memory alloys
View Description Hide DescriptionBy means of the Ginzburg–Landau theory of phase transitions, we study a nonisothermal model to characterize the austenitemartensite transition in shape memoryalloys. In the first part of this paper, the onedimensional model proposed by Berti et al. [“Phase transitions in shape memoryalloys: A nonisothermal GinzburgLandau model,” Physica D239, 95 (2010)] is modified by varying the expression of the free energy. In this way, the description of the phenomenon of hysteresis, typical of these materials, is improved and the related stressstrain curves are recovered. Then, a generalization of this model to the threedimensional case is proposed and its consistency with the principles of thermodynamics is proven. Unlike other threedimensional models, the transition is characterized by a scalar valued order parameter and the Ginzburg–Landau equation, ruling the evolution of , allows us to prove a maximum principle, ensuring the boundedness of itself.
 Statistical Physics

On the solvability of two dimensional semigroup gauge theories
View Description Hide DescriptionWe study the solvability of two dimensional semigroup gauge theories by Migdal’s link elimination method. We determine certain conditions that ensure that the partition sum corresponding to the join of two plaquettes depends only on the holonomy around the boundary of the joined plaquettes. These conditions are checked for a few types of semigroups: 0groups, cyclic, inverse symmetric, and Brandt semigroups.

Formulas for joint probabilities for the asymmetric simple exclusion process
View Description Hide DescriptionIn earlier work, the authors [Tracy, C. A. and Widom, H., “Integral formulas for the asymmetric simple exclusion process,” Commun. Math. Phys.279, 815 (2008)] obtained integral formulas for probabilities for a single particle in the asymmetric simple exclusion process. Here, formulas are obtained for joint probabilities for several particles. In the case of a single particle, the derivation here is simpler than the one in the earlier work for one of its main results.

Multiscaling for systems with a broad continuum of characteristic lengths and times: Structural transitions in nanocomposites
View Description Hide DescriptionThe multiscale approach to body systems is generalized to address the broad continuum of long time and length scales associated with collective behaviors. A technique is developed based on the concept of an uncountable set of time variables and of order parameters (OPs) specifying major features of the system. We adopt this perspective as a natural extension of the commonly used discrete set of time scales and OPs which is practical when only a few, widely separated scales exist. The existence of a gap in the spectrum of time scales for such a system (under quasiequilibrium conditions) is used to introduce a continuous scaling and perform a multiscale analysis of the Liouville equation. A functionaldifferential Smoluchowski equation is derived for the stochastic dynamics of the continuum of Fourier component OPs. A continuum of spatially nonlocal Langevin equations for the OPs is also derived. The theory is demonstrated via the analysis of structural transitions in a composite material, as occurs for viral capsids and molecular circuits.

New representations of and Dirac delta using the nonextensivestatisticalmechanics exponential function
View Description Hide DescriptionWe present a generalization of the representation in plane waves of Dirac delta, , namely, , using the nonextensivestatisticalmechanics exponential function, with , being any real number, for real values of within the interval . Concomitantly, with the development of these new representations of Dirac delta, we also present two new families of representations of the transcendental number . Incidentally, we remark that the plane wave form which emerges, namely, , is normalizable for , in contrast to the standard one, , which is not.

Bifurcation of binary systems with the Onsager mobility
View Description Hide DescriptionThe main objective of this article is to study the effect of the (nonlinear) Onsager mobility to the phase separation of the binary system, using rigorous bifurcation analysis. In particular, a nondimensional parameter , depending on the molar density of the homogeneous state, and the critical temperature is derived; the sign of this parameter dictates the type of transition. Also, the analysis indicates that the type of the transition, the critical temperature , and the strength of the deviation of the transition solutions from the homogeneous state are all independent of the choices of the Onsager mobility.
 Methods of Mathematical Physics

Fate of the Julia set of higher dimensional maps in the integrable limit
View Description Hide DescriptionBy studying higher dimensional rational maps, we have shown, in our previous papers, that periodic points of integrable maps with sufficient number of invariants form invariant varieties of periodic points (IVPPs) different for each period. In this paper, we study the transition of a nonintegrable map to an integrable one. In particular, we investigate analytically where the Julia set goes and how it disappears when the map becomes integrable. We show that the behavior of the Julia set is different, depending on whether the map has an unstable variety of fixed point (UVFP), which becomes nonfixed in the integrable limit. If the map does not have an UVFP, all periodic points approach IVPP or fixed points of the integrable map. Otherwise, a large part of periodic points of all periods approach the UVFP and the UVFP itself becomes a variety of indeterminate points unless it disappears from the map. Moreover, we show that a map recovered by the singularity confinement generates the sequence of all IVPPs.