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

We 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 → ∞.
 ARTICLES


Partial Differential Equations

Global existence of the threedimensional viscous quantum magnetohydrodynamic model
View Description Hide DescriptionThe globalintime existence of weak solutions to the viscous quantum Magnetohydrodynamic equations in a threedimensional torus with large data is proved. The global existence of weak solutions to the viscous quantum Magnetohydrodynamic equations is shown by using the FaedoGalerkin method and weak compactness techniques.

On the mathematical and geometrical structure of the determining equations for shear waves in nonlinear isotropic incompressible elastodynamics
View Description Hide DescriptionUsing the theory of 1 + 1 hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll [Acta Mechanica3, 167–181 (1967)]. We show that a natural generalization of this class of solutions yields an infinite family of linear solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.

Some wellposedness and general stability results in Timoshenko systems with infinite memory and distributed time delay
View Description Hide DescriptionIn this paper, we consider a Timoshenko system in onedimensional bounded domain with infinite memory and distributed time delay both acting on the equation of the rotation angle. Without any restriction on the speeds of wave propagation and under appropriate assumptions on the infinite memory and distributed time delay convolution kernels, we prove, first, the wellposedness and, second, the stability of the system, where we present some decay estimates depending on the equalspeed propagation case and the opposite one. The obtained decay rates depend on the growths of the memory and delay kernels at infinity. In the nonequalspeed case, the decay rate depends also on the regularity of initial data. Our stability results show that the only dissipation resulting from the infinite memory guarantees the asymptotic stability of the system regardless to the speeds of wave propagation and in spite of the presence of a distributed time delay. Applications of our approach to specific coupled Timoshenkoheat and Timoshenkowave systems as well as the discrete time delay case are also presented.

The exponential decay of solutions and traveling wave solutions for a modified Camassa–Holm equation with cubic nonlinearity
View Description Hide DescriptionThe present paper is devoted to the study of persistence properties, infinite propagation, and the traveling wave solutions for a modified Camassa–Holm equation with cubic nonlinearity. We first show that persistence properties of the solution to the equation provided the initial datum is exponential decay and the initial potential satisfies a certain sign condition. Next, we get the infinite propagation if the initial datum satisfies certain compact conditions, while the solution to Eq. (1.1) instantly loses compactly supported, the solution has exponential decay as x goes to infinity. Finally, we prove Eq. (1.1) has a family traveling wave solutions.

Flows in a tube structure: Equation on the graph
View Description Hide DescriptionThe steadystate NavierStokes equations in thin structures lead to some elliptic second order equation for the macroscopic pressure on a graph. At the nodes of the graph the pressure satisfies Kirchofftype junction conditions. In the nonsteady case the problem for the macroscopic pressure on the graph becomes nonlocal in time. In the paper we study the existence and uniqueness of a solution to such onedimensional model on the graph for a pipewise network. We also prove the exponential decay of the solution with respect to the time variable in the case when the data decay exponentially with respect to time.

Generalized metaplectic operators and the Schrödinger equation with a potential in the Sjöstrand class
View Description Hide DescriptionIt is well known that the matrix of a metaplectic operator with respect to phasespace shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential operators in a Sjöstrand class enjoys the same decay properties. We study the behavior of these generalized metaplectic operators and represent them by Fourier integral operators. Our main result shows that the oneparameter group generated by a Hamiltonian operator with a potential in the Sjöstrand class consists of generalized metaplectic operators. As a consequence, the Schrödinger equation preserves the phasespace concentration, as measured by modulation space norms.

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.

Extensions of inhomogeneous polynomial representations for
View Description Hide DescriptionWe extend an inhomogeneous polynomial representation of special linear Lie superalgebras to an inhomogeneous representation on the tensor space of any irreducible tensor representation of the general linear Lie superalgebras with this polynomial space via Shen's mixed product. We find a sufficient and necessary condition for irreducibility of these inhomogeneous representations, and get the JordanH lder series of those which are not irreducible. Furthermore, we point out that all these irreducible modules are lowest weight modules. Their lowest weights and character formulas are obtained.

On the oscillator realization of conformal U(2, 2) quantum particles and their particlehole coherent states
View Description Hide DescriptionWe revise the unireps. of U(2, 2) describing conformal particles with continuous mass spectrum from a manybody perspective, which shows massive conformal particles as compounds of two correlated massless particles. The statistics of the compound (boson/fermion) depends on the helicity h of the massless components (integer/halfinteger). Coherent states (CS) of particlehole pairs (“excitons”) are also explicitly constructed as the exponential action of exciton (noncanonical) creation operators on the ground state of unpaired particles. These CS are labeled by points Z (2 × 2 complex matrices) on the CartanBergman domain , and constitute a generalized (matrix) version of Perelomov U(1, 1) coherent states labeled by points z on the unit disk . First, we follow a geometric approach to the construction of CS, orthonormal basis, U(2, 2) generators and their matrix elements and symbols in the reproducing kernel Hilbert space of analytic squareintegrable holomorphic functions on , which carries a unitary irreducible representation of U(2, 2) with index (the conformal or scale dimension). Then we introduce a manybody representation of the previous construction through an oscillator realization of the U(2, 2) Lie algebra generators in terms of eight boson operators with constraints. This particle picture allows us for a physical interpretation of our abstract mathematical construction in the manybody jargon. In particular, the index λ is related to the number 2(λ − 2) of unpaired quanta and to the helicity h = (λ − 2)/2 of each massless particle forming the massive compound.

Analogues of Lusztig's higher order relations for the qOnsager algebra
View Description Hide DescriptionLet A, A ^{*} be the generators of the qOnsager algebra. Analogues of Lusztig's r−th higher order relations are proposed. In a first part, based on the properties of tridiagonal pairs of qRacah type which satisfy the defining relations of the qOnsager algebra, higher order relations are derived for r generic. The coefficients entering in the relations are determined from a twovariable polynomial generating function. In a second part, it is conjectured that A, A ^{*} satisfy the higher order relations previously obtained. The conjecture is proven for r = 2, 3. For r generic, using an inductive argument recursive formulae for the coefficients are derived. The conjecture is checked for several values of r ≥ 4. Consequences for coideal subalgebras and integrable systems with boundaries at q a root of unity are pointed out.

Symmetries and the ucondition in HomYetterDrinfeld categories
View Description Hide DescriptionLet (H, S, α) be a monoidal HomHopf algebra and the HomYetterDrinfeld category over (H, α). Then in this paper, we first find sufficient and necessary conditions for to be symmetric and pseudosymmetric, respectively. Second, we study the ucondition in and show that the HomYetterDrinfeld module (H, adjoint, Δ, α) (resp., (H, m, coadjoint, α)) satisfies the ucondition if and only if S ^{2} = id. Finally, we prove that over a triangular (resp., cotriangular) HomHopf algebra contains a rich symmetric subcategory.

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.

SO(4) algebraic approach to the threebody bound state problem in two dimensions
View Description Hide DescriptionWe use the permutation symmetric hyperspherical threebody variables to cast the nonrelativistic threebody Schrödinger equation in two dimensions into a set of (possibly decoupled) differential equations that define an eigenvalue problem for the hyperradial wave function depending on an SO(4) hyperangular matrix element. We express this hyperangular matrix element in terms of SO(3) group ClebschGordan coefficients and use the latter's properties to derive selection rules for potentials with different dynamical/permutation symmetries. Threebody potentials acting on three identical particles may have different dynamical symmetries, in order of increasing symmetry, as follows: (1) S 3 ⊗ O L (2), the permutation times rotational symmetry, that holds in sums of pairwise potentials, (2) O(2) ⊗ O L (2), the socalled “kinematic rotations” or “democracy symmetry” times rotational symmetry, that holds in areadependent potentials, and (3) O(4) dynamical hyperangular symmetry, that holds in hyperradial threebody potentials. We show how the different residual dynamical symmetries of the nonrelativistic threebody Hamiltonian lead to different degeneracies of certain states within O(4) multiplets.
