Volume 55, Issue 8, August 2014
Index of content:
 ARTICLES


Representation Theory and Algebraic Methods

Affine KacMoody symmetric spaces related with
View Description Hide DescriptionSymmetric spaces associated with Lie algebras and Lie groups which are Riemannian manifolds have recently got a lot of attention in various branches of Physics for their role in classical/quantum integrable systems, transport phenomena, etc. Their infinite dimensional counter parts have recently been discovered which are affine KacMoody symmetric spaces. In this paper we have (algebraically) explicitly computed the affine KacMoody symmetric spaces associated with affine KacMoody algebras . We hope these types of spaces will play similar roles as that of symmetric spaces in many physical systems.

Generalized qdeformed TammDancoff oscillator algebra and associated coherent states
View Description Hide DescriptionIn this paper, we propose a full characterization of a generalized qdeformed TammDancoff oscillator algebra and investigate its main mathematical and physical properties. Specifically, we study its various representations and find the condition satisfied by the deformed qnumber to define the algebra structure function. Particular Fock spaces involving finite and infinite dimensions are examined. A deformed calculus is performed as well as a coordinate realization for this algebra. A relevant example is exhibited. Associated coherent states are constructed. Finally, some thermodynamics aspects are computed and discussed.

Classification of quadratic Lie algebras of low dimension
View Description Hide DescriptionIn this paper, we give the classification of the irreducible nonsolvable Lie algebras of dimensions with nondegenerate, symmetric, and invariant bilinear forms.

An uncertainty principle for unimodular quantum groups
View Description Hide DescriptionWe present a generalization of Hirschman's entropic uncertainty principle for locally compact Abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a wellknown uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finitedimensional quantum group algebras. In the nonunimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to qtraces of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the state.

ManyBody and Condensed Matter Physics

Joint moments of proper delay times
View Description Hide DescriptionWe calculate negative moments of the Ndimensional Laguerre distribution for the orthogonal, unitary, and symplectic symmetries. These moments correspond to those of the proper delay times, which are needed to determine the statistical fluctuations of several transport properties through classically chaotic cavities, like quantum dots and microwave cavities with ideal coupling.

Quantum Mechanics

On inverse scattering problem for the Schrödinger equation with repulsive potentials
View Description Hide DescriptionWe study multidimensional inverse scattering for Hamiltonians with repulsive potentials. Assuming shortrange conditions for the interaction potential, the high velocity limit of the scattering operator determines uniquely the shortrange part, using the EnssWeder timedependent method (1995). This work improves on a previous result obtained by Nicoleau (2006). We can allow interaction potentials to have not only slower decays but also Coulomblike singularities.

Spectra generated by a confined softcore Coulomb potential
View Description Hide DescriptionAnalytic and approximate solutions for the energy eigenvalues generated by a confined softcore Coulomb potentials of the form a/(r + β) in d > 1 dimensions are constructed. The confinement is effected by linear and harmonicoscillator potential terms, and also through “hard confinement” by means of an impenetrable spherical box. A byproduct of this work is the construction of polynomial solutions for a number of linear differential equations with polynomial coefficients, along with the necessary and sufficient conditions for the existence of such solutions. Very accurate approximate solutions for the general problem with arbitrary potential parameters are found by use of the asymptotic iteration method.

A squeezelike operator approach to positiondependent mass in quantum mechanics
View Description Hide DescriptionWe provide a squeezelike transformation that allows one to remove a position dependent mass from the Hamiltonian. Methods to solve the Schrödinger equation may then be applied to find the respective eigenvalues and eigenfunctions. As an example, we consider a positiondependentmass that leads to the integrable Morse potential and therefore to wellknown solutions.

Analytic matrix elements for the twoelectron atomic basis with logarithmic terms
View Description Hide DescriptionThe twoelectron problem for the heliumlike atoms in Sstate is considered. The basis containing the integer powers of ln r, where r is a radial variable of the Fock expansion, is studied. In this basis, the analytic expressions for the matrix elements of the corresponding Hamiltonian are presented. These expressions include only elementary and special functions, what enables very fast and accurate computation of the matrix elements. The decisive contribution of the correct logarithmic terms to the behavior of the twoelectron wave function in the vicinity of the triplecoalescence point is reaffirmed.

Quantum Information and Computation

Modulus of convexity for operator convex functions
View Description Hide DescriptionGiven an operator convex function f(x), we obtain an operatorvalued lower bound for cf(x) + (1 − c)f(y) − f(cx + (1 − c)y), c ∈ [0, 1]. The lower bound is expressed in terms of the matrix Bregman divergence. A similar inequality is shown to be false for functions that are convex but not operator convex.

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

Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states
View Description Hide DescriptionWe discuss the quantization of linearized gravity on globally hyperbolic, asymptotically flat, vacuum spacetimes, and the construction of distinguished states which are both of Hadamard form and invariant under the action of all bulk isometries. The procedure, we follow, consists of looking for a realization of the observables of the theory as a subalgebra of an auxiliary, nondynamical algebra constructed on future null infinity ℑ^{+}. The applicability of this scheme is tantamount to proving that a solution of the equations of motion for linearized gravity can be extended smoothly to ℑ^{+}. This has been claimed to be possible provided that a suitable gauge fixing condition, first written by Geroch and Xanthopoulos [“Asymptotic simplicity is stable,” J. Math. Phys.19, 714 (1978)], is imposed. We review its definition critically, showing that there exists a previously unnoticed obstruction in its implementation leading us to introducing the concept of radiative observables. These constitute an algebra for which a Hadamard state induced from null infinity and invariant under the action of all spacetime isometries exists and it is explicitly constructed.

General Relativity and Gravitation

Firstorder equivalent to EinsteinHilbert Lagrangian
View Description Hide DescriptionA firstorder Lagrangian L ^{∇} variationally equivalent to the secondorder EinsteinHilbert Lagrangian is introduced. Such a Lagrangian depends on a symmetric linear connection, but the dependence is covariant under diffeomorphisms. The variational problem defined by L ^{∇} is proved to be regular and its Hamiltonian formulation is studied, including its covariant Hamiltonian attached to ∇.

Dynamical Systems

Spectra of discrete Schrödinger operators with primitive invertible substitution potentials
View Description Hide DescriptionWe study the spectral properties of discrete Schrödinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction expansion, a strictly larger class than primitive invertible substitution sequences). It is known that operators from this family have spectra which are Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of this set tends to 1 as coupling constant λ tends to 0. Moreover, we also show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly with respect to λ. Additionally, we show that, in the small coupling regime, the density of states measure for an operator in this family is exact dimensional. The dimension of the density of states measure is strictly smaller than the Hausdorff dimension of the spectrum and tends to 1 as λ tends to 0.

Invariant tori with prescribed frequency for nearly integrable hamiltonian systems
View Description Hide DescriptionIn this paper, we consider real analytic nearly integrable hamiltonian systems under Bruno nondegeneracy condition, and prove that if the rank of the Jacobian of the ndimensional frequency vectors with respect to action variables is n − 1, then there exists a oneparameter analytic family of invariant tori, whose frequencies are the small dilation of a prescribed Diophantine frequency vector.

Fluids

BiotSavart helicity versus physical helicity: A topological description of ideal flows
View Description Hide DescriptionFor an isentropic (thus compressible) flow, fluid trajectories are considered as orbits of a family of one parameter, smooth, orientationpreserving, and nonsingular diffeomorphisms on a compact and smoothboundary domain in the Euclidian 3space which necessarily preserve a finite measure, later interpreted as the fluid mass. Under such diffeomorphisms the BiotSavart helicity of the pushforward of a divergencefree and tangent to the boundary vector field is proved to be conserved and since these circumstances present an isentropic flow, the conservation of the “BiotSavart helicity” is established for such flows. On the other hand, the well known helicity conservation in ideal flows which here we call it “physical helicity” is found to be an independent constant with respect to the BiotSavart helicity. The difference between these two helicities reflects some topological features of the domain as well as the velocity and vorticity fields which is discussed and is shown for simply connected domains the two helicities coincide. The energy variation of the vorticity field is shown to be formally the same as for the incompressible flow obtained before. For fluid domains consisting of several disjoint solid tori, at each time, the harmonic knot subspace of smooth vector fields on the fluid domain is found to have two independent base sets with a special type of orthogonality between these two bases by which a topological description of the vortex and velocity fields depending on the helicity difference is achieved since this difference is shown to depend only on the harmonic knot parts of velocity, vorticity, and its BiotSavart vector field. For an ideal magnetohydrodynamics (MHD) flow three independent constant helicities are reviewed while the helicity of magnetic potential is generalized for nonsimply connected domains by inserting a special harmonic knot field in the dynamics of the magnetic potential. It is proved that the harmonic knot part of the vorticity in hydrodynamics and the magnetic field in MHD is presented by constant coefficients (fluxes) when expanded in terms of one of the time dependent base functions.

Statistical Physics

Classical nonMarkovian Boltzmann equation
View Description Hide DescriptionThe modeling of particle transport involves anomalous diffusion, ⟨x ^{2}(t) ⟩ ∝ t ^{α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advectiondispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the threeparticle distribution function are neglected. We show that the inclusion of higherorder distribution functions give rise to an exact, nonMarkovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two and the threeparticle distribution functions are considered under the assumption that the two and the threeparticle correlation functions are translationally invariant that allows us to obtain advectiondispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.

Moments of the Gaussian β ensembles and the largeN expansion of the densities
View Description Hide DescriptionThe loop equation formalism is used to compute the 1/N expansion of the resolvent for the Gaussian β ensemble up to and including the term at O(N ^{−6}). This allows the moments of the eigenvalue density to be computed up to and including the 12th power and the smoothed density to be expanded up to and including the term at O(N ^{−6}). The latter contain nonintegrable singularities at the endpoints of the support—we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings β = 1, 2, and 4 there are characterisations of both the resolvent and the moments which allows for the corresponding expansions to be extended, in some recursive form at least, to arbitrary order. In this regard, we give fifth order linear differential equations for the density and resolvent at β = 1 and 4, which complements the known third order linear differential equations for these quantities at β = 2.

Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle
View Description Hide DescriptionThe maximum entropy principle is applied to the formal derivation of isothermal, Eulerlike equations for semiclassical fermions (electrons and holes) in graphene. After proving general mathematical properties of the equations so obtained, their asymptotic form corresponding to significant physical regimes is investigated. In particular, the diffusive regime, the MaxwellBoltzmann regime (high temperature), the collimation regime and the degenerate gas limit (vanishing temperature) are considered.

Ising model observables and nonbacktracking walks
View Description Hide DescriptionThis paper presents an alternative proof of the connection between the partition function of the Ising model on a finite graph G and the set of nonbacktracking walks on G. The techniques used also give formulas for spinspin correlation functions in terms of nonbacktracking walks. The main tools used are Viennot's theory of heaps of pieces and turning numbers on surfaces.

Longrun growth rate in a random multiplicative model
View Description Hide DescriptionWe consider the longrun growth rate of the average value of a random multiplicative process x i + 1 = a i x i where the multipliers have Markovian dependence given by the exponential of a standard Brownian motion W i . The average value ⟨x n ⟩ is given by the grand partition function of a onedimensional lattice gas with twobody linear attractive interactions placed in a uniform field. We study the Lyapunov exponent , at fixed , and show that it is given by the equation of state of the lattice gas in thermodynamical equilibrium. The Lyapunov exponent has discontinuous partial derivatives along a curve in the (ρ, β) plane ending at a critical point (ρ C , β C ) which is related to a phase transition in the equivalent lattice gas. Using the equivalence of the lattice gas with a bosonic system, we obtain the exact solution for the equation of state in the thermodynamical limit n → ∞.
