Volume 52, Issue 9, September 2011
Index of content:
 ARTICLES


Quantum Mechanics (General and Nonrelativistic)

Effectivemass KleinGordonYukawa problem for bound and scattering states
View Description Hide DescriptionBound and scattering state solutions of the effectivemass KleinGordon equation are obtained for the Yukawa potential with any angular momentum ℓ. Energy eigenvalues, normalized wave functions, and scattering phase shifts are calculated as well as for the constant mass case. Bound statesolutions of the Coulomb potential are also studied as a limiting case. Analytical and numerical results are compared with the ones obtained before.

On the spectrum of D = 2 supersymmetric YangMills quantum mechanics
View Description Hide DescriptionWe investigate the structure of the spectrum of states in D = 2 SU(N) supersymmetric YangMills matrix quantum mechanics, which is a simplified model of Matrix theory. We compute the thermal partition function of this system and give evidence for the correctness of naively conjectured structure of the spectrum. It also suggests that ClaudsonHalpernSamuel solution is the unique eigenfunction of simultaneously diagonalizable hermitian operators, and we show that it is true in N = 3 and N = 4 cases.

Discrete spectra for confined and unconfined −a/r + br ^{2} potentials in ddimensions
View Description Hide DescriptionExact solutions to the ddimensional Schrödinger equation,d ⩾ 2, for Coulomb plus harmonic oscillator potentials V(r) = −a/r + br ^{2}, b > 0, and a ≠ 0 are obtained. The potential V(r) is considered both in all space, and under the condition of spherical confinement inside an impenetrable spherical box of radius R. With the aid of the asymptotic iteration method, the exact analytic solutions under certain constraints, and general approximate solutions, are obtained. These exhibit the parametric dependence of the eigenenergies on a, b, and R. The wave functions have the simple form of a product of a power function, an exponential function, and a polynomial.Polynomialsolutions are found for differential equations of the form py″ + qy′ − ry = 0, where p, q, and r are given polynomials withdegrees 4, 3, and 2, respectively.

Quantum Information and Computation

Entanglement evolution via quantum resonances
View Description Hide DescriptionWe consider two qubits interacting with local and collective thermal reservoirs. Each spinreservoir interaction consists of an energy exchange and an energy conserving channel. We prove a resonance representation of the reduced dynamics of the spins, valid for all times t ⩾ 0, with errors (small interaction) estimated rigorously, uniformly in time. Subspaces associated to noninteracting energy differences evolve independently, partitioning the reduced density matrix into dynamically decoupled clusters of jointly evolving matrix elements. Within each subspace the dynamics is Markovian with a generator determined entirely by the resonance data of the full Hamiltonian. Based on the resonance representation we examine the evolution of entanglement (concurrence). We show that, whenever thermalization takes place, entanglement of any initial state dies out in a finite time and will not return. For a concrete class of initially entangled spin states we find explicit bounds on entanglement survival and death times in terms of the initial state and the resonance data.

Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Distinguished selfadjoint extensions of Dirac operators via HardyDirac inequalities
View Description Hide DescriptionWe prove some HardyDirac inequalities with two different weights including measure valued and Coulombic ones. Those inequalities are used to construct distinguished selfadjoint extensions of Dirac operators for a class of diagonal potentials related to the weights in the mentioned inequalities.

A few remarks on the zero modes of the FaddeevPopov operator in the Landau and maximal Abelian gauges
View Description Hide DescriptionThe construction outlined by Henyey [Phys. Rev. D20, 1460 (1979)10.1103/PhysRevD.20.1460] is employed to provide examples of normalizable zero modes of the FaddeevPopov operator in the Landau and maximal Abelian gauges in SU(2) Euclidean YangMillstheories in d = 3 dimensions. The corresponding gauge configurations have all finite norm A^{2} < ∞. In particular, in the case of the Landau gauge, the explicit construction of an infinite class of normalizable zero modes with finite norm A^{2} is provided.

A symmetric approach to the massive nonlinear sigma model
View Description Hide DescriptionIn the present paper we extend the procedure of divergences subtraction to the massive case, previously introduced for the massless nonlinear sigma model (D = 4). Perturbative expansion in the number of loops is successfully constructed. The resulting theory depends on the spontaneous symmetry breaking parameter v, on the mass m, and on the radiative correction parameter Λ. Fermions are not considered in the present work. SU(2)⊗SU(2) is the group used.

General Relativity and Gravitation

On holographic realization of logarithmic Galilean conformal algebra
View Description Hide DescriptionWe study twodimensional logarithmic Galilean conformal algebra (LGCA) by making use of a contraction of topologically massive gravity at critical point. We observe that using a naive contraction at the critical point fails to give a well defined theory, though contracting the theory while we are approaching the critical point leads to a well behaved expression for two point functions of the energymomentum tensors of LGCA.

Lensing by Kerr black holes. I. General lens equation and magnification formula
View Description Hide DescriptionWe develop a unified, analytic framework for gravitational lensing by Kerrblack holes. In this first paper, we present a new, general lens equation and magnification formula governing lensing by a compact object. Our lens equation assumes that the source and observer are in the asymptotically flat region and does not require a small angle approximation. Furthermore, it takes into account the displacement that occurs when the light ray's tangent lines at the source and observer do not meet on the lens plane. We then explore our lens equation in the case when the compact object is a Kerrblack hole. Specifically, we give an explicit expression for the displacement when the observer is in the equatorial plane of the Kerrblack hole as well as for the case of spherical symmetry.

Dynamical Systems

Stationary states of a nonlinear Schrödinger lattice with a harmonic trap
View Description Hide DescriptionWe study a discrete nonlinear Schrödinger lattice with a parabolic trapping potential. The model, describing, e.g., an array of repulsive BoseEinstein condensate droplets confined in the wells of an optical lattice, is analytically and numerically investigated. Starting from the linear limit of the problem, we use global bifurcation theory to rigorously prove that – in the discrete regime – all linear states lead to nonlinear generalizations thereof, which assume the form of a chain of discrete dark solitons (as the density increases). The stability of the ensuing nonlinear states is studied and it is found that the ground state is stable, while the excited states feature a chain of stability/instability bands. We illustrate the mechanisms under which discreteness destabilizes the darksoliton configurations, which become stable only in the continuum regime. Continuation from the anticontinuum limit is also considered, and a rich bifurcation structure is revealed.

Exponential stability of stochastic differential delay systems with delayed impulse effects
View Description Hide DescriptionIn this paper, we investigate the stability of stochastic delay differential systems with delayed impulses by Razumikhin methods. Some criteria on the pth moment exponential stability are obtained. It is shown that if a stochastic delay differential system is exponentially stable, then under some conditions, its stability is robust or weaken with respect to delayed impulses. Moreover, it is also shown that an unstable stochastic delay system can be successfully stabilized by delayed impulses. The effectiveness of the proposed results is illustrated by three examples.

The GouldHopper polynomials in the NovikovVeselov equation
View Description Hide DescriptionWe use the GouldHopper (GH) polynomials to investigate the NovikovVeselov (NV) equation. The root dynamics of the σflow in the NV equation is studied using the GH polynomials and then the Lax pair is found. In particular, when N = 3, 4, 5, one can get the Goldfish model. The smooth rational solutions of the NV equation are also constructed via the extended Moutard transformation and the GH polynomials. The asymptotic behavior is discussed and then the smooth rational solution of the Liouville equation is obtained.

Classical Mechanics and Classical Fields

Super central configurations of the threebody problem under the inverse integer power law
View Description Hide DescriptionIn this paper, we consider the problem of central configurations of the nbody problem with the general homogenous potential 1/r ^{α}, where α is a positive integer. A configuration q = (q _{1}, q _{2}, ⋅⋅⋅, q _{ n }) is called a super central configuration if there exists a positive mass vector m = (m _{1}, ⋅⋅⋅, m _{ n }) such that q is a central configuration for m with m _{ i } attached to q _{ i } and q is also a central configuration for m′, where is a permutation of m. The main result in this paper is the existence and classifications of super central configurations in the rectilinear threebody problem with general homogenous potential. Our results extend the previous work [Xie, Z., J. Math. Phys.51, 042902 (2010)]10.1063/1.3345125 from the case in which α = 2 to the case in which α is a positive integer. Descartes’ rule of sign is extensively used in the proof of the main theorem.

The origin of the Schott term in the electromagnetic self force of a classical point charge
View Description Hide DescriptionThe Schott term is the third order term in the electromagnetic self force of a charged point particle. The self force may be obtained by integrating the electromagnetic stressenergymomentum tensor over the side of a narrow hypertube enclosing a section of worldline. This calculation has been repeated many times using two different hypertubes known as the Dirac tube and the Bhabha tube; however, in previous calculations using a Bhabha tube the Schott term does not arise as a result of this integration. In order to regain the LorentzAbrahamDirac equation, many authors have added an ad hoccompensatory term to the nonelectromagnetic contribution to the total momentum. In this article the Schott term is obtained by direct integration of the electromagnetic stressenergymomentum tensor.

Fractal structure of ferromagnets: The singularity structure analysis
View Description Hide DescriptionFollowing the WeissTaborCarnevale approach [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys.24, 522 (1983)10.1063/1.525721; J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys.25, 13 (1984).]10.1063/1.526009 designed for studying the integrability properties of nonlinear partial differential equations, we investigate the singularitystructure of a (2+1)dimensional waveequation describing the propagation of polariton solitary waves in a ferromagnetic slab. As a result, we show that, out of any damping instability, the system above is integrable. Looking forward to unveiling its complete integrability, we derive its Bäcklund transformation and Hirota's bilinearization useful in generating a set of soliton solutions. In the wake of such results, using the arbitrary functions to enter into the Laurent series of solutions to the above system, we discuss some typical class of excitations generated from the previous solutions in account of a classification based upon the different expressions of a generic lower dimensional function. Accordingly, we unearth the nonlocal excitations of lowest amplitudes, the dromion and lump patterns of higher amplitudes, and finally the stochastic pattern formations of highest amplitudes, which arguably endow the aforementioned system with the fractalproperties.

Fluids

Internal heating driven convection at infinite Prandtl number
View Description Hide DescriptionWe derive an improved rigorous lower bound on the space and time averaged temperature ⟨T⟩ of an infinite Prandtl number Boussinesq fluid contained between isothermal noslip boundaries driven by uniform internal heating. A singular stable stratification is introduced as a perturbation to a nonsingular background profile yielding ⟨T⟩ ⩾ 0.419[R log R]^{−1/4} where R is the heat Rayleigh number. The analysis relies on a generalized HardyRellich inequality that is proved in the Appendix.

Global strong solution for a threedimensional viscous liquidgas twophase flow model with vacuum
View Description Hide DescriptionThis paper is concerned with the Cauchy problem of the 3D viscousliquidgas twophase flowmodel, where the initial vacuum is allowed. The existence of the global strong solution is proved when the energy of the initial data is small enough.

Statistical Physics

On the area of excursion sets of spherical Gaussian eigenfunctions
View Description Hide DescriptionThe high frequency behaviour for random eigenfunctions of the spherical Laplacian has been recently the object of considerable interest, also because of strong motivation arising from physics and cosmology. In this paper, we are concerned with the high frequency behaviour of excursion sets; in particular, we establish a uniform central limit theorem for the empirical measure, i.e., the proportion of spherical surface, where spherical Gaussian eigenfunctions lie below a level z. Our proofs borrow some techniques from the literature on stationary long memory processes; in particular, we expand the empirical measure into Hermite polynomials, and establish a uniform weak reduction principle, entailing that the asymptotic behaviour is asymptotically dominated by a single term in the expansion. As a result, we establish a functional central limit theorem; the limiting process is fully degenerate.

Universality for eigenvalue correlations from the unitary ensemble associated with a family of singular weights
View Description Hide DescriptionWe study the asymptotic behavior of the eigenvalue correlations for the unitary ensemble associated with a family of singular weights w(x; μ) = exp { − (1 − x ^{2})^{−μ}}, x ∈ ( − 1, 1) for μ > 0. When μ ∈ (0, 1/2) these are Szegö class weights, and are nonSzegö when μ ⩾ 1/2. It is proved that the behavior in the bulk of the spectrum is described in terms of the sine kernel, which persists the socalled universality results. While the local behavior at the edge of the spectrum is described in terms of the Airy kernel. A specific scaling of the limit reflects the singular behavior of orthogonal polynomials on [ − 1, 1], with respect to the weight w(x; μ).

Weak coupling limits in a stochastic model of heat conduction
View Description Hide DescriptionWe study the Brownian momentum process, a model of heat conduction, weakly coupled to heat baths. In two different settings of weak coupling to the heat baths, we study the nonequilibrium steady state and its proximity to the local equilibrium measure in terms of the strength of coupling. For three and four site systems, we obtain the twopoint correlation function and show it is generically not multilinear.
