Volume 47, Issue 10, October 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Explicit solutions for dimensional Schrödinger equations with positiondependent mass
View Description Hide DescriptionWith the consideration of spherical symmetry for the potential and mass function, onedimensional solutions of nonrelativistic Schrödinger equations with spatially varying effective mass are successfully extended to arbitrary dimensions within the frame of recently developed elegant nonperturbative technique, where the BenDanielDuke effective Hamiltonian in one dimension is assumed like the unperturbed piece, leading to wellknown solutions, whereas the modification term due to possible use of other effective Hamiltonians in one dimension and, together with the corrections coming from the treatments in higher dimensions, are considered as an additional term like the perturbation. Application of the model and its generalization for the completeness are discussed.

Unified theory of annihilationcreation operators for solvable (“discrete”) quantum mechanics
View Description Hide DescriptionThe annihilationcreation operators are defined as the positive/negative frequency parts of the exact Heisenberg operator solution for the “sinusoidal coordinate”. Thus are hermitian conjugate to each other and the relative weights of various terms in them are solely determined by the energy spectrum. This unified method applies to most of the solvable quantum mechanics of single degree of freedom including those belonging to the “discrete” quantum mechanics.

A note on the infimum problem of Hilbert space effects
View Description Hide DescriptionThe quantum effects for a physical system can be described by the set of positive operators on a complex Hilbert space that are bounded above by the identity operator . The infimum problem of Hilbert spaceeffects is to find under what condition the infimum exists for two quantum effects and . The problem has been studied in different contexts by Kadison, Gudder, Moreland, and Ando. In this Note, using the method of the spectral theory of operators, we give a affirmative answer of a conjecture of [S. Gudder, J. Math. Phys.37, 2637–2642 (1996)]. In addition, some properties of generalized infimum were considered.

Quaternionic diffusion by a potential step
View Description Hide DescriptionIn looking for qualitative differences between quaternionic and complex formulations of quantum physical theories, we provide a detailed discussion of the behavior of a wave packet in the presence of a quaternionic timeindependent potential step. In this paper, we restrict our attention to diffusion phenomena. For the group velocity of the wave packet moving in the potential region and for the reflection and transmission times, the study shows a striking difference between the complex and quaternionic formulations which could be matter of further theoretical discussions and could represent the starting point for a possible experimental investigation.

Resonance counting function in black box scattering
View Description Hide DescriptionSjostrand and Zworski [J. Am. Math. Soc.4, 729–769 (1991)] proved a universal upper bound for the resonance counting function in black box scattering. Examples are presented which show that there is no corresponding general lower bound.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Quantum invariants, modular forms, and lattice points II
View Description Hide DescriptionWe study the SU(2) WittenReshetikhinTuraev (WRT) invariant for the Seifert fibered homology spheres with exceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight and . By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the large limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in , is related to the number of integral lattice points inside the dimensional tetrahedron.

Canonical quantization of lattice HiggsMaxwellChernSimons fields: Krein Selfadjointness
View Description Hide DescriptionIt is shown how techniques from constructive quantum field theory may be applied to indefinite metric gauge theories in Hilbert space for the case of a HiggsMaxwellChernSimons theory on a lattice. The Hamiltonian operator is shown to be Krein essentially selfadjoint by means of unbounded but Krein unitary transformations relating the Hamiltonian to an essentially maximal accretive operator.

Large chiral diffeomorphisms on Riemann surfaces and algebras
View Description Hide DescriptionThe diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a BecchiRouetStora (BRS) formulation (for a given order of truncation) leading to a more algebraic setup. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the socalled algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between Kortewegde Vries (KdV) flows and diffeomorphims.
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 GENERAL RELATIVITY AND GRAVITATION


Threedimensional loop quantum gravity: Particles and the quantum double
View Description Hide DescriptionIt is well known that the quantum double structure plays an important role in threedimensional quantum gravity coupled to matter field. In this paper, we show how this algebraic structure emerges in the context of threedimensional Riemannian loop quantum gravity (LQG) coupled to a finite number of massive spinless point particles. In LQG, physical states are usually constructed from the notion of SU(2) cylindrical functions on a Riemann surfaced and the Hilbert structure is defined by the AshtekarLewandowski measure. In the case where is the sphere , we show that the physical Hilbert space is in fact isomorphic to a tensor product of simple unitary representations of the Drinfeld double DSU(2): the masses of the particles label the simple representations, the physical states are tensor products of vectors of simple representations, and the physical scalar product is given by intertwining coefficients between simple representations. This result is generalized to the case of any Riemann surface .

Automorphisms and a cartography of the solution space for vacuum Bianchi cosmologies: The Type III case
View Description Hide DescriptionThe theory of symmetries of systems of coupled, ordinary differential equations (ODEs) is used to develop a concise algorithm for cartographing the space of solutions to vacuum Bianchi Einstein’s Field Equations (EFE). The symmetries used are the well known automorphisms of the Lie algebra for the corresponding isometry group of each Bianchi Type, as well as the scaling and the time reparametrization symmetry. The application of the method to Type III results in (a) the recovery of all known solutions without a prior assumption of any extra symmetry; (b) the enclosure of the entire unknown part of the solution space into a single, second order ODE in terms of one dependent variable; and (c) a partial solution to this ODE. It is also worth mentioning that the solution space is seen to be naturally partitioned into three distinct, disconnected pieces: one consisting of the known Siklos (ppwave) solution, another occupied by the Type III member of the known EllisMacCallum family and the third described by the aforementioned ODE in which a one parameter subfamily of the known Kinnersley geometries resides. Lastly, preliminary results reported show that the unknown part of the solution space for other Bianchi Types is described by a strikingly similar ODE, pointing to a natural operational unification as far as the problem of solving the cosmological EFE’s is concerned.

Spectral analysis of radial Dirac operators in the KerrNewman metric and its applications to timeperiodic solutions
View Description Hide DescriptionWe investigate the existence of timeperiodic solutions of the Dirac equation in the KerrNewman background metric. To this end, the solutions are expanded in a Fourier series with respect to the time variable , and the Chandrasekhar separation ansatz is applied so that the question of existence of a timeperiodic solution is reduced to the solvability of a certain coupled system of ordinary differential equations. First, we prove the already known result that there are no timeperiodic solutions in the nonextreme case. Then, it is shown that in the extreme case for fixed black hole data there is a sequence of particle masses for which a timeperiodic solution of the Dirac equation does exist. The period of the solution depends only on the data of the black hole described by the KerrNewman metric.

Symmetries of the RobinsonTrautman equation
View Description Hide DescriptionWe study point symmetries of the RobinsonTrautman equation. The cases of one and twodimensional algebras of infinitesimal symmetries are discussed in detail. The corresponding symmetry reductions of the equation are given. Higher dimensional symmetries are shortly discussed. It turns out that all known exact solutions of the RobinsonTrautman equation are symmetric.
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 DYNAMICAL SYSTEMS


Goldfishing: A new solvable manybody problem
View Description Hide DescriptionA recent technique allows one to identify and investigate solvable dynamical systems naturally interpretable as classical manybody problems, being characterized by equations of motion of Newtonian type (generally in twodimensional space). In this paper we tersely review results previously obtained in this manner and present novel findings of this kind: mainly solvable variants of the goldfish manybody model, including models that feature isochronous classes of completely periodic solutions. Different formulations of these models are presented. The behavior of one of these isochronous dynamical systems in the neighborhood of its equilibrium configuration is investigated, and in this manner some remarkable Diophantine findings are obtained.

On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy
View Description Hide DescriptionWe derive dispersionless Hirota equations of the extended dispersionless BKP(EdBKP) hierarchy proposed by Takasaki from the method of kernel formula provided by Carroll and Kodama. Moreover, we verify associativity equations (WDVV equations) in the EdBKP hierarchy from dispersionless Hirota equations and give a realization of associative algebra with structure constants expressed in terms of the residue formula.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Polarizability of the dielectric doublesphere
View Description Hide DescriptionAn explicit solution for the longitudinal and transverse polarizability of the symmetric dielectric intersecting double sphere is obtained as a rapidly converging series of integral operators, which is fast enough for real time calculation in Java Applet.
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 STATISTICAL PHYSICS


On characterization of reversible Markov processes by monotonicity of the fluctuation spectral density
View Description Hide DescriptionFor the system driven by a stationary Markov process, if it is in nonequilibrium steady state (i.e., irreversible), then there exists a function such that the fluctuation spectrum (or say: power spectrum density) of is nonmonotonic in under mild conditions, which means that there exist nonzero spectrum peaks of the fluctuation spectrum. For the system driven by a Markov chain with discrete time , even if it is in equilibrium state (i.e., reversible), one cannot distinguish the equilibrium and nonequilibrium steady state in terms of the monotonicity of the fluctuation spectrum any more.

Nonlinear diffusion equation and nonlinear external force: Exact solution
View Description Hide DescriptionThe solutions of the nonlinear diffusionequation are investigated by considering the presence of an external force which exhibits an explicit dependence on the distribution. First, the stationary case is considered; after that the dynamical case, i.e., the case dependent on time. The stationary solution is obtained by considering the external force and the result found is related to the distributions which emerge from the Tsallis statistics or the BoltzmannGibbs statistics. The dynamical solution is investigated by considering the external force and related to the Levy distributions in the asymptotic limit. In both cases, the solutions are expressed in terms of the exponentials and the logarithmics functions which emerge from the Tsallis formalism.

Limiting laws of linear eigenvalue statistics for Hermitian matrix models
View Description Hide DescriptionWe study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of Hermitian matrices as . Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al. [ Commun. Pure Appl. Math.52, 1325–1425 (1999)] for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of intervals, then in the global regime the variance of statistics is a quasiperiodic function of as generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not, in general, variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.

Energy correlations for a random matrix model of disordered bosons
View Description Hide DescriptionLinearizing the Heisenberg equations of motion around the ground state of an interacting quantum manybody system, one gets a timeevolution generator in the positive cone of a real symplectic Lie algebra. The presence of disorder in the physical system determines a probability measure with support on this cone. The present paper analyzes a discrete family of such measures of exponential type, and does so in an attempt to capture, by a simple random matrix model, some generic statistical features of the characteristic frequencies of disordered bosonic quasiparticle systems. The level correlation functions of the said measures are shown to be those of a determinantal process, and the kernel of the process is expressed as a sum of biorthogonal polynomials. While the correlations in the bulk scaling limit are in accord with sinekernel or Gaussian Unitary Ensemble universality, at the lowfrequency end of the spectrum an unusual type of scaling behavior is found.

Overlap fluctuations from the Boltzmann random overlap structure
View Description Hide DescriptionWe investigate overlap fluctuations of the SherringtonKirkpatrick mean field spin glass model in the framework of the Random Overlap Structure (ROSt). The concept of ROSt has been introduced recently by Aizenman and coworkers, who developed a variational approach to the SherringtonKirkpatrick model. Here we propose an iterative procedure to show that, in the socalled Boltzmann ROSt, AizenmanContucci polynomials naturally arise for almost all values of the inverse temperature (not in average over some interval only). These polynomials impose restrictions on the overlap fluctuations in agreement with Parisi theory.
