Volume 46, Issue 11, November 2005
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Analytic and contour representations in the unit disk based on SU(1,1) coherent states
View Description Hide DescriptionA contour representation in the unit disk based on SU(1,1) coherent states is introduced. The scalar product is given by a contour integral. The regions of convergence of the functions representing ket and bra states are studied. An analytic representation in the unit disk is also considered, where the scalar product is represented by an integral over the unit disk, with the Lobachevsky measure. Various relations which connect these analytic functions with other phasespace quantities are derived.

A general treatment of deformation effects in Hamiltonians for inhomogeneous crystalline materials
View Description Hide DescriptionIn this paper, a general method of treating Hamiltonians of deformed nanoscale systems is proposed. This method is used to derive a secondorder approximation both for the strong and weak formulations of the eigenvalue problem. The weak formulation is needed in order to allow deformations that have discontinuous first derivatives at interfaces between different materials. It is shown that, as long as the deformation is twice differentiable away from interfaces, the weak formulation is equivalent to the strong formulation with appropriate interfaceboundary conditions. It is also shown that, because the Jacobian of the deformation appears in the weak formulation, the approximations of the weak formulation is not equivalent to the approximations of the strong formulation with interfaceboundary conditions. The method is applied to two onedimensional examples (a sinusoidal and a quantumwell potential) and one twodimensional example (a freestanding quantum wire), where it is shown that the energy eigenvalues of the secondorder approximations lie within 1% of the exact energy eigenvalues for a linear strain of up to 9.8%, whereas the firstorder approximation has an error of less than 1% for a linear strain of up to 5.5%.

Remarks on the repulsive WignerPoisson system
View Description Hide DescriptionWe present an estimate for the spacetime integral of classical solutions to the repulsive WignerPoisson system. We use Markowich’s formalism between WignerPossion and SchrödingerPoisson systems. Through this formalism and Morawetz interaction potentials, we derive the same a priori estimate given by Chae and Ha for the repulsive VlasovPoisson system.

Smoothness of wave functions in thermal equilibrium
View Description Hide DescriptionWe consider the thermal equilibrium distribution at inverse temperature , or canonical ensemble, of the wave function of a quantum system. Since spaces contain more nondifferentiable than differentiable functions, and since the thermal equilibrium distribution is very spread out, one might expect that has probability zero to be differentiable. However, we show that for relevant Hamiltonians the contrary is the case: with probability 1, is infinitely often differentiable and even analytic. We also show that with probability 1, lies in the domain of the Hamiltonian.

Large expansion from variational perturbation theory
View Description Hide DescriptionWe derive recursively the perturbation series for the groundstateenergy of the dimensional anharmonic oscillator and resum it using variational perturbation theory (VPT). From the exponentially fast converging approximants, we extract the coefficients of the largeexpansion to higher orders. The calculation effort is much smaller than in the standard fieldtheoretic approach based on the HubbardStratonovich transformation.

Existence of twocluster threshold resonances and the body Efimov effect
View Description Hide DescriptionWe prove the existence of twocluster threshold resonances in body problems and study their perturbation by intercluster interactions. As application, we construct concrete examples based on Yukawa potentials for which the body Efimov effect happens with .

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Divergences in quantum electrodynamics on a graph
View Description Hide DescriptionWe consider a model of quantum electrodynamics (QED) on a graph as the generalization of dimensional deconstruction with the Abelian symmetry. Arbitrary structures of the theory space correspond to the graphs consisting of vertices and edges. The mass spectrum of the model is expressed in terms of eigenvalues of the Laplacian for the graph. We also find that physical massless scalar modes are associated with the fundamental tie set matrix on the graph. We further investigate the oneloop divergences in the model by use of the background field method.

 GENERAL RELATIVITY AND GRAVITATION


Pure gravitational radiation with twisting rays in the linear approximation
View Description Hide DescriptionSolutions of types N and III with twisting rays are derived in the linear approximation by means of complex coordinate transformations. Some solutions are shown to have Riemann tensors which vanish asymptotically and are everywhere regular.

Volume elements and torsion
View Description Hide DescriptionWe reexamine here the issue of consistency of minimal action formulation with the minimal coupling procedure (MCP) in spaces with torsion. In RiemannCartan spaces, it is known that a proper use of the MCP requires that the trace of the torsion tensor be a gradient, , and that the modified volume element be used in the action formulation of a physical model. We rederive this result here under considerably weaker assumptions, reinforcing some recent results about the inadequacy of propagating torsion theories of gravity to explain the available observational data. The results presented here also open the door to possible applications of the modified volume element in the geometric theory of crystalline defects.

Soldered bundle background for the de Sitter top
View Description Hide DescriptionWe prove that the mathematical framework for the de Sitter top system is the de Sitter fiber bundle. In this context, the concept of soldering associated with a fiber bundle plays a central role. We comment on the possibility that our formalism may be of particular interest in different contexts including MacDowellMansouri theory, two time physics, and oriented matroid theory.

 DYNAMICAL SYSTEMS


Constrained reductions of twodimensional dispersionless Toda hierarchy, Hamiltonian structure, and interface dynamics
View Description Hide DescriptionFinitedimensional reductions of the twodimensional dispersionless Toda hierarchy constrained by the “string equation” are studied. These include solutions determined by polynomial, rational, or logarithmic functions, which are of interest in relation to the “Laplacian growth” or HeleShaw problem governing interface dynamics. The consistency of such reductions is proved, and the Hamiltonian structure of the reduced dynamics is derived. The Poisson structure of the rationally reduced dispersionless Toda hierarchies is also derived.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The effects of nonlocality on the evolution of higher order fluxes in nonequilibrium thermodynamics
View Description Hide DescriptionThe role of gradient dependent constitutive spaces is investigated on the example of Extended Thermodynamics of rigid heat conductors. Different levels of nonlocality are developed and the different versions of extended thermodynamics are classified. The local form of the entropy density plays a crucial role in the investigations. The entropyinequality is solved under suitable constitutive assumptions. Balance form of evolution equations is obtained in special cases. Closure relations are derived on a phenomenological level.

An invariant variational principle for canonical flows on Lie groups
View Description Hide DescriptionIn this paper we examine the existence of Lie groups, whose canonical geodesic flows are variational with respect to a leftinvariant regular—but not necessarily quadratic (i.e., metric)—Lagrange function. We give effective necessary and sufficient conditions for the existence of an invariant variational principle generating the canonical flow. With these results, taken in conjunction with the classification of Lie algebras, we solve the inverse problem of invariant Lagrangian dynamics in dimensions up to four.

Hamiltonian multivector fields and Poisson forms in multisymplectic field theory
View Description Hide DescriptionWe present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing explicit expressions for the Poisson bracket between two Poisson forms.

Ideality criterion for unilateral constraints in timedependent impulsive mechanics
View Description Hide DescriptionWe construct a new geometric framework based on the concepts of left and right jetbundles of a classical spacetime in order to analyze the impulsive behavior of a unilateral constraint . The setup allows deep insights into how one can choose an ideality criterion for the constraint when the hypothesis of conservation of kinetic energy is assumed. We show that the conservation of kinetic energy alone univocally determines the impulsive reaction when the codimension of is 1, and that it leaves the impulsive reaction partially undetermined when the codimension of is greater than 1. If the codimension of is greater than 1, we prove that an additional minimality requirement determines a physically meaningful constitutive characterization of . We show that both the Newtonlike and the Poissonlike approaches to the description of the reactive impulse are equivalent, in the sense that both give the same results about the ideality criterion. Moreover, we prove that the same results hold using the classical approach based on reflection operators, possible only in case of codimension 1. We present also several physically meaningful examples.

The GaussLandauHall problem on Riemannian surfaces
View Description Hide DescriptionWe introduce the notion of GaussLandauHall magnetic field on a Riemannian surface. The corresponding LandauHall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model corresponds with a limit case obtained when the force of the GaussLandauHall magnetic field increases arbitrarily. We also obtain properties related with the completeness of flowlines for general magnetic fields. The paper also contains results relative to the LandauHall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field.

 STATISTICAL PHYSICS


Fluiddynamic equations for granular particles in a host medium
View Description Hide DescriptionA kinetic model for a granular gas interacting with a given background by binary dissipative collisions is analyzed, with particular reference to the derivation of macroscopic equations for the fundamental observables. Particles are modelled as inelastic hard spheres under the assumption of collision dominated regime (small mean free path). Closure of the relevant moment equations is achieved by resorting to a maximum entropy principle, and two specific entropy functionals have been worked out in detail, in the class of the admissible ones for the relevant linear extended Boltzmann equation. Considered macroscopic fields include density, mass velocity, and granular temperature. In the hydrodynamic limit when the mean free path tends to zero, a single driftdiffusion equation of NavierStokes type is recovered for the only hydrodynamic variable of the physical problem.

Random multioverlap structures for optimization problems
View Description Hide DescriptionWe extend to the KSAT and XORSAT optimization problems the results recently achieved, by introducing the concept of random multioverlap structure, for the VianaBray model of diluted mean fieldspin glass. More precisely we can prove a generalized bound and an extended variational principle for the free energy per site in the thermodynamic limit. Moreover a trial function implementing ultrametric breaking of replica symmetry is exhibited. The ultrametric structure exhibits the same factorization property as the optimal structures for the VianaBray model and the SherringtonKirkpatrick nondiluted model.

 METHODS OF MATHEMATICAL PHYSICS


Harmonic fields on the extended projective disk and a problem in optics
View Description Hide DescriptionThe Hodge equations for 1forms are studied on Beltrami’s projective disk model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for weakly harmonic 1fields, changing type on the unit circle, is derived under Dirichlet conditions imposed on the noncharacteristic portion of the boundary. A similar system arises in the analysis of wave motion near a caustic. A class of elliptichyperbolic boundaryvalue problems is formulated for those equations as well. For both classes of boundaryvalue problems, an arbitrarily small lowerorder perturbation of the equations is shown to yield solutions which are strong in the sense of Friedrichs.

Integrable quasiclassical deformations of cubic curves
View Description Hide DescriptionA general scheme for determining and studying hydrodynamic type systems describing integrable deformations of algebraic curves is applied to cubic curves. Lagrange resolvents of the theory of cubic equations are used to derive and characterize these deformations.
