Volume 54, Issue 5, May 2013
Index of content:

In this paper we propose a geometrization of the nonrelativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamicdependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincaré quantum recurrence in a physical system with finite energy levels.
 ARTICLES


Partial Differential Equations

Exact wave solutions for Bose–Einstein condensates with timedependent scattering length and spatiotemporal complicated potential
View Description Hide DescriptionWe consider a cubicquintic Gross–Pitaevskii equation which governs the dynamics of Bose–Einstein condensate matter waves with timedependent scattering length and spatiotemporal complex potential. By introducing phaseimprint parameters in the system, we present the integrable condition for the equation and obtain the exact analytical solutions, which describe the propagation of a solitary wave. By applying specific timemodulated feeding/loss functional parameter, various types of magnetic trap strengths, and phaseimprint parameters, the dynamics of the solutions can be controlled. Solitary wave solutions with breathing and snaking behaviors are reported.

On the strong solutions of onedimensional NavierStokesPoisson equations for compressible nonNewtonian fluids
View Description Hide DescriptionWe study the strong solutions of 1D NavierStokesPoisson equations for compressible nonNewtonian fluids in bounded intervals. The model is raised from the viscous isentropic gas flow under considering an external force and the nonNewtonian gravitational force term. By using the iterative method we prove the local existence and uniqueness of strong solutions based on some compatibility condition. The main condition is that the initial density vacuum is allowed.

Weak solutions for the NavierStokes equations with initial data
View Description Hide DescriptionOne of the major topics in the study of nonlinear partial differential equations of the evolutionary type is to look for as large as possible initial value spaces so that as many as possible solutions of such equations can be obtained. In the book “Recent Developments in the NavierStokes Problems,” LemariéRieusset proved that the NavierStokes equations have global weak solutions for initial data in the space (0 < r < 1), where X r is the space of functions whose pointwise products with H r functions belong to L 2, denotes the closure of in X r , and is the Besov space over . In this paper we partially extend this result of LemariéRieusset to the larger initial value space (0 < r < 1), where is a logarithmically modified version of the usual Besov space .

Bäcklund transformation and smooth multisoliton solutions for a modified CamassaHolm equation with cubic nonlinearity
View Description Hide DescriptionWe present a compact parametric representation of the smooth bright multisoliton solutions for the modified CamassaHolm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a reciprocal transformation and then find a novel Bäcklund transformation between solutions of the associated mCH equation and a model equation for shallowwater waves (SWW) introduced by Ablowitz et al. We combine this result with the expressions of the multisoliton solutions for the SWW and modified Kortewegde Vries equations to obtain the multisoliton solutions of the mCH equation. Subsequently, we investigate the properties of the one and twosoliton solutions as well as the general multisoliton solutions. We show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions. We also find that at a critical value of the parameter beyond which the solution becomes singular, the soliton solution exhibits a different feature from that of the peakon solution of the CH equation. Then, by performing an asymptotic analysis for large time, we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions. Finally, we use the Bäcklund transformation to derive an infinite number of conservation laws of the mCH equation.

Representation Theory and Algebraic Methods

Real second order freeness and Haar orthogonal matrices
View Description Hide DescriptionWe demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier.

Meromorphic openstring vertex algebras
View Description Hide DescriptionA notion of meromorphic openstring vertex algebra is introduced. A meromorphic openstring vertex algebra is an openstring vertex algebra in the sense of Kong and the author satisfying additional rationality (or meromorphicity) conditions for vertex operators. The vertex operator map for a meromorphic openstring vertex algebra satisfies rationality and associativity but in general does not satisfy the Jacobi identity, commutativity, the commutator formula, the skewsymmetry or even the associator formula. Given a vector space , we construct a meromorphic openstring vertex algebra structure on the tensor algebra of the negative part of the affinization of such that the vertex algebra structure on the symmetric algebra of the negative part of the Heisenberg algebra associated to is a quotient of this meromorphic openstring vertex algebra. We also introduce the notion of left module for a meromorphic openstring vertex algebra and construct left modules for the meromorphic openstring vertex algebra above.

Representations of the twisted affine NappiWitten algebras
View Description Hide DescriptionIn this paper, we study Verma modules for the twisted affine NappiWitten algebras and . The vertex operator representations of the affine NappiWitten algebras , , and are also constructed. Furthermore, the irreducible nonzero level quasifinite modules over the affine NappiWitten algebras are classified.

Gequivariant ϕcoordinated quasi modules for quantum vertex algebras
View Description Hide DescriptionThis is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras. In this paper, we introduce a notion of Ttype quantum vertex algebra and a notion of Gequivariant ϕcoordinated quasi module for a Ttype quantum vertex algebra with an automorphism group G. We refine and extend several previous results and we obtain a commutator formula for Gequivariant ϕcoordinated quasi modules. As an illustrating example, we study a special case of the deformed Virasoro algebra with q = −1, to which we associate a Clifford vertex superalgebra and its Gequivariant ϕcoordinated quasi modules.

ManyBody and Condensed Matter Physics

Bound states of the spinorbit coupled ultracold atom in a onedimensional shortrange potential
View Description Hide DescriptionWe solve the bound state problem for the Hamiltonian with the spinorbit and the Raman coupling included. The Hamiltonian is perturbed by a onedimensional shortrange potential V which describes the impurity scattering. In addition to the bound states obtained by considering weak solutions through the Fourier transform or by solving the eigenvalue equation on a suitable domain directly, it is shown that ordinary pointinteraction representations of V lead to spinorbit induced extra states.

Quantum Mechanics

Hartman effect and dissipative quantum systems
View Description Hide DescriptionThe dwell time for dissipative quantum system is shown to increase with barrier width. It clearly precludes Hartman effect for dissipative systems. Here calculation has been done for inverted parabolic potential barrier.

Coulomb problem in noncommutative quantum mechanics
View Description Hide DescriptionThe aim of this paper is to find out how it would be possible for space noncommutativity (NC) to alter the quantum mechanics (QM) solution of the Coulomb problem. The NC parameter λ is to be regarded as a measure of the noncommutativity – setting λ = 0 which means a return to the standard quantum mechanics. As the very first step a rotationally invariant NC space , an analog of the Coulomb problem configuration space (R 3 with the origin excluded) is introduced. is generated by NC coordinates realized as operators acting in an auxiliary (Fock) space . The properly weighted HilbertSchmidt operators in form , a NC analog of the Hilbert space of the wave functions. We will refer to them as “wave functions” also in the NC case. The definition of a NC analog of the hamiltonian as a hermitian operator in is one of the key parts of this paper. The resulting problem is exactly solvable. The full solution is provided, including formulas for the bound states for E < 0 and lowenergy scattering for E > 0 (both containing NC corrections analytic in λ) and also formulas for highenergy scattering and unexpected bound states at ultrahigh energy (both containing NC corrections singular in λ). All the NC contributions to the known QM solutions either vanish or disappear in the limit λ → 0.

Spherical Schrödinger operators with δtype interactions
View Description Hide DescriptionWe investigate spectral properties of spherical Schrödinger operators (also known as Bessel operators) with δpoint interactions concentrated on a discrete set. We obtain necessary and sufficient conditions for these Hamiltonians to be selfadjoint, lowersemibounded and also we investigate their spectra. We also extend the classical Bargmann estimate to such Hamiltonians. In certain cases we express the number of negative eigenvalues explicitly by means of point interactions and the corresponding intensities. We apply our results to Schrödinger operators in with a singular interaction supported by an infinite family of concentric spheres.

Thermal nonlinear coherent states on a flat space and on a sphere
View Description Hide DescriptionIn this paper, we first define thermal nonlinear coherent states on a sphere and show that these states are essentially twomode squeezed nonlinear coherent states of the sphere at zero temperature. Then we consider quantum statistical properties of the thermal sphere nonlinear coherent states. In particular, we investigate temperature effects on transition of the constructed states from nonclassical states to classical ones. By using the Mandel parameter, we obtain a transition temperature and show that this transition temperature increases by increasing the curvature of the physical space. It turns out that, increasing curvature of the space provides nonlinear coherent states with nonclassical properties in higher temperature ranges.

Analytical solutions of the Schrödinger equation for a twodimensional exciton in magnetic field of arbitrary strength
View Description Hide DescriptionThe FeranchukKomarov operator method is developed by combining with the LeviCivita transformation in order to construct analytical solutions of the Schrödinger equation for a twodimensional exciton in a uniform magnetic field of arbitrary strength. As a result, analytical expressions for the energy of the ground and excited states are obtained with a very high precision of up to four decimal places. Especially, the precision is uniformly stable for the whole range of the magnetic field. This advantage appears due to the consideration of the asymptotic behaviour of the wavefunctions in strong magnetic field. The results could be used for various physical analyses and the method used here could also be applied to other atomic systems.

BakerAkhiezer functions and generalised MacdonaldMehta integrals
View Description Hide DescriptionFor the rational BakerAkhiezer functions associated with special arrangements of hyperplanes with multiplicities we establish an integral identity, which may be viewed as a generalisation of the selfduality property of the usual Gaussian function with respect to the Fourier transformation. We show that the value of properly normalised BakerAkhiezer function at the origin can be given by an integral of MacdonaldMehta type and explicitly compute these integrals for all known BakerAkhiezer arrangements. We use the DotsenkoFateev integrals to extend this calculation to all deformed root systems, related to the nonexceptional basic classical Lie superalgebras.

On infinitedimensional state spaces
View Description Hide DescriptionIt is well known that the canonical commutation relation [x, p] = i can be realized only on an infinitedimensional Hilbert space. While any finite set of experimental data can also be explained in terms of a finitedimensional Hilbert space by approximating the commutation relation, Occam's razor prefers the infinitedimensional model in which [x, p] = i holds on the nose. This reasoning one will necessarily have to make in any approach which tries to detect the infinitedimensionality. One drawback of using the canonical commutation relation for this purpose is that it has unclear operational meaning. Here, we identify an operationally welldefined context from which an analogous conclusion can be drawn: if two unitary transformations U, V on a quantum system satisfy the relation V −1 U 2 V = U 3, then finitedimensionality entails the relation UV −1 UV = V −1 UVU; this implication strongly fails in some infinitedimensional realizations. This is a result from combinatorial group theory for which we give a new proof. This proof adapts to the consideration of cases where the assumed relation V −1 U 2 V = U 3 holds only up to ε and then yields a lower bound on the dimension.

Geometric approach to nonrelativistic quantum dynamics of mixed states
View Description Hide DescriptionIn this paper we propose a geometrization of the nonrelativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamicdependent metric tensor on the principal manifold, such that the projection of the geodesic flow to the base manifold gives the temporal evolution predicted by the von Neumann equation. Using that approach we can describe every conserved quantum observable as a Killing vector field, and provide a geometric proof for the Poincaré quantum recurrence in a physical system with finite energy levels.

Entropy and complexity analysis of hydrogenic Rydberg atoms
View Description Hide DescriptionThe internal disorder of hydrogenic Rydberg atoms as contained in their position and momentum probability densities is examined by means of the following informationtheoretic spreading quantities: the radial and logarithmic expectation values, the Shannon entropy, and the Fisher information. As well, the complexity measures of CrámerRao, FisherShannon, and López RuizManciniCalvet types are investigated in both reciprocal spaces. The leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the Laguerre and Gegenbauer orthogonal polynomials which control the wavefunctions of the Rydberg states in both position and momentum spaces. The associated generalized Heisenberglike, logarithmic and entropic uncertainty relations are also given. Finally, application to linear (l = 0), circular (l = n − 1), and quasicircular (l = n − 2) states is explicitly done.

Quantum Information and Computation

Approximately clean quantum probability measures
View Description Hide DescriptionA quantum probability measure–or quantum measurement–is said to be clean if it cannot be irreversibly connected to any other quantum probability measure via a quantum channel. The notion of a clean quantum measure was introduced by Buscemi et al. [“Clean positive operator valued measures,” J. Math. Phys.46(8), 082109 (Year: 2005)10.1063/1.2008996] for finitedimensional Hilbert space, and was studied subsequently by Kahn [“Clean positive operatorvalued measures for qubits and similar cases,” J. Phys. A40(18), 4817–4832 (Year: 2007)10.1088/17518113/40/18/009] and Pellonpää [“Complete characterization of extreme quantum observables in finite dimensions,” J. Phys. A44(8), 085304 (Year: 2011)10.1088/17518113/44/8/085304]. The present paper provides new descriptions of clean quantum probability measures in the case of finitedimensional Hilbert space. For Hilbert spaces of infinite dimension, we introduce the notion of “approximately clean quantum probability measures” and characterise this property for measures whose range determines a finitedimensional operator system.

Quantum logarithmic Sobolev inequalities and rapid mixing
View Description Hide DescriptionA family of logarithmic Sobolev inequalities on finite dimensional quantum state spaces is introduced. The framework of noncommutative spaces is reviewed and the relationship between quantum logarithmic Sobolev inequalities and the hypercontractivity of quantum semigroups is discussed. This relationship is central for the derivation of lower bounds for the logarithmic Sobolev (LS) constants. Essential results for the family of inequalities are proved, and we show an upper bound to the generalized LS constant in terms of the spectral gap of the generator of the semigroup. These inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups, when the LS constant and the spectral gap are of the same order. Convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory. We provide a number of examples, where improved bounds on the mixing time of several semigroups are obtained, including the depolarizing semigroup and quantum expanders.
