Volume 55, Issue 9, September 2014
Index of content:

The partition function of the ChernSimons theory on the threesphere with the unitary group U(N) provides a onematrix model. The corresponding Nparticle system can be mapped to the determinantal point process whose correlation kernel is expressed by using the StieltjesWigert orthogonal polynomials. The matrix model and the point process are regarded as qextensions of the random matrix model in the Gaussian unitary ensemble and its eigenvalue point process, respectively. We prove the convergence of the Nparticle system to an infinitedimensional determinantal point process in N → ∞, in which the correlation kernel is expressed by Jacobi's theta functions. We show that the matrix model obtained by this limit realizes the oscillatory matrix model in ChernSimons theory discussed by de Haro and Tierz.
 ARTICLES


Partial Differential Equations

Traveling wave solutions of degenerate coupled Kortewegde Vries equation
View Description Hide DescriptionWe give a detailed study of the traveling wave solutions of (ℓ = 2) KaupBoussinesq type of coupled KdV equations. Depending upon the zeros of a fourth degree polynomial, we have cases where there exist no nontrivial real solutions, cases where asymptotically decaying to a constant solitary wave solutions, and cases where there are periodic solutions. All such possible solutions are given explicitly in the form of Jacobi elliptic functions. Graphs of some exact solutions in solitary wave and periodic shapes are exhibited. Extension of our study to the cases ℓ = 3 and ℓ = 4 are also mentioned.

The decay rate of solution for the bipolar NavierStokesPoisson system
View Description Hide DescriptionIn this paper, the Cauchy problem for the bipolar NavierStokesPoisson system is studied. We obtain the global existence and the H ^{ s } decay rate of classical solutions for this problem. Our approach is mainly based on the analysis of the Green's function of the linearized system and some elaborate energy estimates. It should be mentioned that with the help of long waveshort wave decomposition, the decay rate of the higher order derivatives of the solutions is obtained. Furthermore, based on the H ^{ s } decay rate of the solutions, we also give the L ^{ p } estimate of the solution.

Persistence properties and unique continuation for a generalized CamassaHolm equation
View Description Hide DescriptionIn this paper, persistence properties of solutions are investigated for a generalized CamassaHolm equation (gk bCH) having (k+1)degree nonlinearities and containing as its integrable members the CamassaHolm, the DegasperisProcesi, and the Novikov equations. The persistence properties will imply that strong solutions of the gk bCH equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters b and k. Finally, existence of global solutions is established when b = k+1.

Representation Theory and Algebraic Methods

Simple HarishChandra modules over super Schrödinger algebra in (1+1) dimensional spacetime
View Description Hide DescriptionThe N = 1 super Schrödinger algebra in (1+1) dimensional spacetime contains a subalgebra isomorphic to module. Let V be a simple weight module for the N = 1 super Schrödinger algebra but not a simple module. Let ω ∈ supp(V). If V is neither a highest weight module nor a lowest weight module for , we prove that , and all nontrivial weight spaces of V have the same dimension. We prove that if V is a HarishChandra module, then it is a highest weight module, or lowest weight module, or a twisted localization of a highest weight module.

Quantum Mechanics

The large connectivity limit of the Anderson model on tree graphs
View Description Hide DescriptionWe consider the Anderson localization problem on the infinite regular tree. Within the localized phase, we derive a rigorous lower bound on the free energy function recently introduced by Aizenman and Warzel. Using a finite volume regularization, we also derive an upper bound on this free energy function. This yields upper and lower bounds on the critical disorder such that all states at a given energy become localized. These bounds are particularly useful in the large connectivity limit where they match, confirming the early predictions of AbouChacra, Anderson, and Thouless.

Energy eigenfunctions for positiondependent mass particles in a new class of molecular Hamiltonians
View Description Hide DescriptionBased on recent results on quasiexactly solvable Schrodinger equations, we review a new phenomenological potential class lately reported. In the present paper, we consider the quantum differential equations resulting from positiondependent mass (PDM) particles. We first focus on the PDM version of the hyperbolic potential V(x) = asech^{2} x + bsech^{4} x, which we address analytically with no restrictioon the parameters and the energies. This is the celebrated Manning potential, a doublewell widely used in molecular physics, until now not investigated for PDM. We also evaluate the PDM version of the sixth power hyperbolic potential V(x) = asech^{6} x + bsech^{4} x for which we could find exact expressions under some special settings. Finally, we address a triplewell case V(x) = asech^{6} x + bsech^{4} x + csech^{2} x of particular interest for its connection to the new trends in atomtronics. The PDM Schrodinger equations studied in the present paper yield analytical eigenfunctions in terms of local Heun functions in its confluents forms. In all the cases PDM particles are more likely tunneling than ordinary ones. In addition, it is observed a merging of eigenstates when the mass becomes nonuniform.

Solving fractional Schrödingertype spectral problems: Cauchy oscillator and Cauchy well
View Description Hide DescriptionThis paper is a direct offspring of the work of Garbaczewski and Stephanovich [“Lévy flights and nonlocal quantum dynamics,” J. Math. Phys.54, 072103 (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions were made with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to socalled fractional quantum mechanics spectral problems. Presently, we give a decisive computerassisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computerassisted outcomes (potentially defective, in view of various cutoffs), i.e., detailed eigenvalues and shapes of eigenfunctions, has been analyzed.

Quantum evolution in the regime of quantum wells in a semiclassical island with artificial interface conditions
View Description Hide DescriptionWe introduce a modified Schrödinger operator where the semiclassical Laplacian is perturbed by artificial interface conditions occurring at the boundaries of the potential's support. The corresponding dynamics is analyzed in the regime of quantum wells in a semiclassical island. Under a suitable energy constraint for the initial states, we show that the time propagator is stable with respect to the nonselfadjont perturbation, provided that this is parametrized through infinitesimal functions of the semiclassical parameter “h.” It has been recently shown that hdependent artificial interface conditions allow a new approach to the adiabatic evolution problem for the shape resonances in models of resonant heterostructures. Our aim is to provide with a rigorous justification of this method.

Timedependent Schrödingerlike equation with nonlocal term
View Description Hide DescriptionWe investigate a timedependent Schrödingerlike equation in presence of a nonlocal term by using the method of variable separation and the Green function approach. We analyze the Green function for different forms of the memory kernel and the nonlocal term. Results for delta potential energy function are presented. Distributed order memory kernels are also considered, and the asymptotic behaviors of the Green function are derived by using Tauberian theorem. The obtained results for the Green function for the considered Schrödingerlike equation may be transformed to those for the probability distribution function of a diffusionlike equation with memory kernel and can be used to explain various anomalous diffusive behaviors.

Quantum dynamics of a particle constrained to lie on a surface
View Description Hide DescriptionWe consider the quantum dynamics of a charged particle in Euclidean space subjected to electric and magnetic fields under the presence of a potential that forces the particle to stay close to a compact surface. We prove that, as the strength of this constraining potential tends to infinity, the motion of this particle converges to a motion generated by a Hamiltonian over the surface superimposed by an oscillatory motion in the normal directions. Our result extends previous results by allowing magnetic potentials and more general constraining potentials.

Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Gauge theories in noncommutative homogeneous Kähler manifolds
View Description Hide DescriptionWe construct a gauge theory on a noncommutative homogeneous Kähler manifold, where we employ the deformation quantization with separation of variables for Kähler manifolds formulated by Karabegov. A key point in this construction is to obtaining vector fields which act as inner derivations for the deformation quantization. We give an explicit construction of this gauge theory on noncommutative and noncommutative .

Construction of the noncommutative complex ball
View Description Hide DescriptionWe describe the construction of the noncommutative complex ball whose commutative analog is the Hermitian symmetric space D = SU (m, 1)/U(m), with the method of coherent state quantization. In the commutative limit, we obtain the standard manifold. We also consider a quantum field theory model on the noncommutative manifold.

Phaseintegral method for the radial Dirac equation
View Description Hide DescriptionA phaseintegral (WKB) solution of the radial Dirac equation is calculated up to the third order of approximation, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the zerothorder transition points. The potential is allowed to be of scalar, vector, or tensor type, or any combination of these. The connection problem is investigated in detail. Explicit formulas are given for singleturningpoint phase shifts and singlewell energy levels.

Mean field limit for bosons with compact kernels interactions by Wigner measures transportation
View Description Hide DescriptionWe consider a class of manybody Hamiltonians composed of a free (kinetic) part and a multiparticle (potential) interaction with a compactness assumption on the latter part. We investigate the mean field limit of such quantum systems following the Wigner measures approach. We prove in particular the propagation of these measures along the flow of a nonlinear (Hartree) field equation. This enhances and complements some previous results of the same type shown in Z. Ammari and F. Nier and Fröhlich et al. [“Mean field limit for bosons and propagation of Wigner measures,” J. Math. Phys.50(4), 042107 (2009); Z. Ammari and F. Nier and Fröhlich et al. , “Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states,” J. Math. Pures Appl.95(6), 585–626 (2011); Z. Ammari and F. Nier and Fröhlich et al. , “Meanfield and classical limit of manybody Schrödinger dynamics for bosons,” Commun. Math. Phys.271(3), 681–697 (2007)].

Bogomolov multiplier, double classpreserving automorphisms, and modular invariants for orbifolds
View Description Hide DescriptionWe describe the group of braided tensor autoequivalences of the Drinfeld centre of a finite group G isomorphic to the identity functor (just as a functor). We prove that the semidirect product Out 2 − cl (G)⋉B(G) of the group of double class preserving automorphisms and the Bogomolov multiplier of G is a subgroup of . An automorphism of G is double class preserving if it preserves conjugacy classes of pairs of commuting elements in G. The Bogomolov multiplier B(G) is the subgroup of its Schur multiplier H ^{2}(G, k ^{*}) of classes vanishing on abelian subgroups of G. We show that elements of give rise to different realisations of the charge conjugation modular invariant for Gorbifolds of holomorphic conformal field theories.

General Relativity and Gravitation

A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case
View Description Hide DescriptionMotivated by the desire to understand the causal structure of physical signals produced in curved spacetimes – particularly around black holes – we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional Euclidean spaces. The general formula is applied to (d ≥ 2)dimensional de Sitter spacetime, and the scalar Green's function is demonstrated to be sourced by a line emanating infinitesimally close to the origin of the ambient (d + 1)dimensional Minkowski spacetime and piercing orthogonally through the de Sitter hyperboloids of all finite sizes. This method does not require solving the de Sitter wave equation directly. Only the zero mode solution to an ordinary differential equation, the “wave equation” perpendicular to the hyperboloid – followed by a onedimensional integral – needs to be evaluated. A topological obstruction to the general construction is also discussed by utilizing it to derive a generalized Green's function of the Laplacian on the (d ≥ 2)dimensional sphere.

Classical Mechanics and Classical Fields

Isotropic invariants of a completely symmetric thirdorder tensor
View Description Hide DescriptionIn both theoretical and applied mechanics, the modeling of nonlinear constitutive relations of materials is a topic of prime importance. To properly formulate consistent constitutive laws some restrictions need to be imposed on tensor functions. To that aim, representations theorems for both isotropic and anisotropic functions have been extensively investigated since the middle of the 20th century. Nevertheless, in threedimensional physical space, most of the results are restricted to sets of tensors up to secondorder. The purpose of the present paper is thus to get one step further and to provide an integrity basis for isotropic polynomial functions of a completely symmetric thirdorder tensor. We exploit the link between the O(3)action on harmonic tensors and the action on the space of binary forms to explicitly construct this basis. We believe that such an integrity basis may found interesting applications both in continuum mechanics and in other fields of theoretical physics.

Fluids

Wave breaking phenomenon for a modified twocomponent DullinGottwaldHolm equation
View Description Hide DescriptionThis paper is concerned with the wave breaking phenomenon for a modified twocomponent DullinGottwaldHolm shallow water system. We give sufficient conditions on the initial data to guarantee blowup of solutions in finite time.

On some classes of nonstationary axially symmetric solutions to the NavierStokes equations
View Description Hide DescriptionIn the paper, the NavierStokes equations are studied in axially symmetric cases of nonstationary motion with rotation of incompressible viscous fluids. The problem is reduced to a nonlinear system of three partial differential equations for three unknown functions of the cylindrical coordinates r and z and time t. The three functions are sought in the form of power series in r with coefficients depending on t and z. For the unknown coefficients recurrence relations are obtained which contain three arbitrary functions. The relations are examined in three particular cases in which they give analytical solutions to the NavierStokes equations for any values of the coordinates t, z, and r.

Statistical Physics

Large deviations in stochastic heatconduction processes provide a gradientflow structure for heat conduction
View Description Hide DescriptionWe consider three onedimensional continuoustime Markov processes on a lattice, each of which models the conduction of heat: the family of Brownian Energy Processes with parameter m (BEP(m)), a Generalized Brownian Energy Process, and the KipnisMarchioroPresutti (KMP) process. The hydrodynamic limit of each of these three processes is a parabolic equation, the linear heat equation in the case of the BEP(m) and the KMP, and a nonlinear heat equation for the Generalized Brownian Energy Process with parameter a (GBEP(a)). We prove the hydrodynamic limit rigorously for the BEP(m), and give a formal derivation for the GBEP(a). We then formally derive the pathwise largedeviation rate functional for the empirical measure of the three processes. These rate functionals imply gradientflow structures for the limiting linear and nonlinear heat equations. We contrast these gradientflow structures with those for processes describing the diffusion of mass, most importantly the class of Wasserstein gradientflow systems. The linear and nonlinear heatequation gradientflow structures are each driven by entropy terms of the form −log ρ; they involve dissipation or mobility terms of order ρ^{2} for the linear heat equation, and a nonlinear function of ρ for the nonlinear heat equation.
