Volume 55, Issue 11, November 2014
Index of content:

Highly localized explicit solutions to the wave and Klein–Gordon–Fock multidimensional linear equations are presented. Their Fourier transforms are also found explicitly. Solutions depend on a set of parameters and demonstrate astigmatic properties. Asymptotic analysis for large and moderate time shows that constructed particlelike solutions have Gaussian localization near a point moving with the group speed.
 ARTICLES

 Partial Differential Equations

Mass concentration phenomenon for inhomogeneous fractional Hartree equations
View Description Hide DescriptionIn this paper, we consider the inhomogeneous fractional Hartree equation in masscritical case. First, we prove the existence of the ground states by studying the related constrained minimization problem. When the mass of initial data is larger than the mass of the ground states, we construct a sequence of blowup solutions whose initial data converge to the ground states. Moreover, we show the mass concentration phenomenon of the blowup solutions via applying the concentrationcompactness principle.
 Representation Theory and Algebraic Methods

Constructing new braided Tcategories over monoidal HomHopf algebras
View Description Hide DescriptionLet denote the set of all automorphisms of a monoidal Hopf algebra H with bijective antipode in the sense of Caenepeel and Goyvaerts [“Monoidal HomHopf algebras,” Commun. Algebra39, 2216–2240 (2011)] and let G be a crossed product group . The main aim of this paper is to provide new examples of braided Tcategory in the sense of Turaev [“Crossed groupcategories,” Arabian J. Sci. Eng., Sect. C33(2C), 483–503 (2008)]. For this purpose, we first introduce a class of new categories of (A, B)YetterDrinfeld Hommodules with . Then we construct a category and show that such category forms a new braided Tcategory, generalizing the main constructions by Panaite and Staic [“Generalized (anti) YetterDrinfel'd modules as components of a braided Tcategory,” Isr. J. Math.158, 349–366 (2007)]. Finally, we compute an explicit new example of such braided Tcategories.

New Turaev braided group categories and weak (co)quasiTuraev group coalgebras
View Description Hide DescriptionIn order to construct a class of new braided crossed Gcategories with nontrivial associativity and unit constraints, we study the Ggraded monoidal category over a family of algebras {H α}α∈G and introduce the notion of a weak (co)quasiTuraev G(co)algebra. Then we prove that the category of (co)representations of (co)quasitriangular weak (co)quasiTuraev π(co)algebras is exactly a braided crossed Gcategory. In fact, this (co)quasitriangular structure provides a solution to a generalized quantum YangBaxter type equation.

Deformations of momentum maps and Gsystems
View Description Hide DescriptionIn this note we give an explicit formula for a family of deformation quantizations for the momentum map associated with the cotangent lift of a Lie group action on . This family of deformations is parametrized by the formal Gsystems introduced in B. Dherin and I. Mencattini [“Gsystems and deformations of Gactions on ,” J. Math. Phys.55, 011702 (2014)] (see also B. Dherin and I. Mencattini [“Quantization of (volumepreserving) actions on ,” eprint arXiv:1202.0886]) and allows us to obtain classical invariant Hamiltonians that quantize without anomalies with respect to the quantizations of the action prescribed by the formal Gsystems.

Bosonfermion correspondence of type DA and multilocal Virasoro representations on the Fock space
View Description Hide DescriptionWe construct the bosonization of the Fock space of a single neutral fermion by using a 2point local Heisenberg field. We decompose as a direct sum of irreducible highest weight modules for the Heisenberg algebra , and thus we show that under the Heisenberg action the Fock space of the single neutral fermion is isomorphic to the Fock space of a pair of charged free fermions, thereby constructing the bosonfermion correspondence of type DA. As a corollary we obtain the Jacobi identity equating the graded dimension formulas utilizing both the Heisenberg and the Virasoro gradings on . We construct a family of 2pointlocal Virasoro fields with central charge , on . We construct a W 1 + ∞ representation on and show that under this W 1 + ∞ action is again isomorphic to .
 Quantum Mechanics

Modified wave operators for discrete Schrödinger operators with longrange perturbations
View Description Hide DescriptionWe consider the scattering theory for discrete Schrödinger operators on with longrange potentials. We prove the existence of modified wave operators constructed in terms of solutions of a HamiltonJacobi equation on the torus .

Generalized Kortewegde Vries equation induced from positiondependent effective mass quantum models and massdeformed soliton solution through inverse scattering transform
View Description Hide DescriptionWe consider onedimensional stationary positiondependent effective mass quantum model and derive a generalized Kortewegde Vries (KdV) equation in (1+1) dimension through Lax pair formulation, one being the effective mass Schrödinger operator and the other being the timeevolution of wave functions. We obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized KdV equation is an integrable system. Inverse scattering transform method is applied to obtain general solution of the nonlinear equation, and then Nsoliton solution is derived for reflectionless potentials. Finally, a special choice has been made for the variable mass function to get massdeformed soliton solution. The influence of position and timedependence of mass and also of the different representations of kinetic energy operator on the nature of such solitons is investigated in detail. The remarkable features of such solitons are demonstrated in several interesting figures and are contrasted with the conventional KdVsoliton associated with constantmass quantum model.

Combined stateadding and statedeleting approaches to type III multistep rationally extended potentials: Applications to ladder operators and superintegrability
View Description Hide DescriptionType III multistep rationally extended harmonic oscillator and radial harmonic oscillator potentials, characterized by a set of k integers m 1, m 2, ⋯, m k , such that m 1 < m 2 < ⋯ < m k with m i even (resp. odd) for i odd (resp. even), are considered. The stateadding and statedeleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators. The eigenstates of the Hamiltonians are shown to separate into m k + 1 infinitedimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebras. These ladder operators are then used to build a higherorder integral of motion for seven new infinite families of superintegrable twodimensional systems separable in cartesian coordinates. The finitedimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent onedimensional Hamiltonian eigenstates and provide an algebraic derivation of the superintegrable systems whole spectrum including the level total degeneracies.

Scattering through a straight quantum waveguide with combined boundary conditions
View Description Hide DescriptionScattering through a straight twodimensional quantum waveguide with Dirichlet boundary conditions on and Neumann boundary condition on is considered using stationary scattering theory. The existence of a matching conditions solution at x = 0 is proved. The use of stationary scattering theory is justified showing its relation to the wave packets motion. As an illustration, the matching conditions are also solved numerically and the transition probabilities are shown.

Atoms confined by very thin layers
View Description Hide DescriptionThe Hamiltonian of an atom with N electrons and a fixed nucleus of infinite mass between two parallel planes is considered in the limit when the distance a between the planes tends to zero. We show that this Hamiltonian converges in the norm resolvent sense to a Schrödinger operator acting effectively in whose potential part depends on a. Moreover, we prove that after an appropriate regularization this Schrödinger operator tends, again in the norm resolvent sense, to the Hamiltonian of a twodimensional atom (with the threedimensional Coulomb potentialone over distance) as a → 0. This makes possible to locate the discrete spectrum of the full Hamiltonian once we know the spectrum of the latter one. Our results also provide a mathematical justification for the interest in the twodimensional atoms with the threedimensional Coulomb potential.
 Quantum Information and Computation

Decoherence free subspaces of a quantum Markov semigroup
View Description Hide DescriptionWe give a full characterisation of decoherence free subspaces of a given quantum Markov semigroup with generator in a generalised Lindbald form which is valid also for infinitedimensional systems. Our results, extending those available in the literature concerning finitedimensional systems, are illustrated by some examples.

Quantum skew divergence
View Description Hide DescriptionIn this paper, we study the quantum generalisation of the skew divergence, which is a dissimilarity measure between distributions introduced by Lee in the context of natural language processing. We provide an indepth study of the quantum skew divergence, including its relation to other state distinguishability measures. Finally, we present a number of important applications: new continuity inequalities for the quantum JensenShannon divergence and the Holevo information, and a new and short proof of Bravyi's Small Incremental Mixing conjecture.

On the reduction criterion for random quantum states
View Description Hide DescriptionIn this paper, we study the reduction criterion for detecting entanglement of large dimensional bipartite quantum systems. We first obtain an explicit formula for the moments of a random quantum state to which the reduction criterion has been applied. We show that the empirical eigenvalue distribution of this random matrix converges strongly to a limit that we compute, in three different asymptotic regimes. We then employ tools from free probability theory to study the asymptotic positivity of the reduction operators. Finally, we compare the reduction criterion with other entanglement criteria, via thresholds.
 General Relativity and Gravitation

A potential foundation for emergent spacetime
View Description Hide DescriptionWe present a novel derivation of both the Minkowski metric and Lorentz transformations from the consistent quantification of a causally ordered set of events with respect to an embedded observer. Unlike past derivations, which have relied on assumptions such as the existence of a 4dimensional manifold, symmetries of spacetime, or the constant speed of light, we demonstrate that these now familiar mathematics can be derived as the unique means to consistently quantify a network of events. This suggests that spacetime need not be physical, but instead the mathematics of space and time emerges as the unique way in which an observer can consistently quantify events and their relationships to one another. The result is a potential foundation for emergent spacetime.

Greybody factors for Myers–Perry black holes
View Description Hide DescriptionThe Myers–Perry black holes are higherdimensional generalizations of the usual (3+1)dimensional rotating Kerr black hole. They are of considerable interest in Kaluza–Klein models, specifically within the context of braneworld versions thereof. In the present article, we shall consider the greybody factors associated with scalar field excitations of the Myers–Perry spacetimes, and develop some rigorous bounds on these greybody factors. These bounds are of relevance for characterizing both the higherdimensional Hawking radiation, and the superradiance, that is expected for these spacetimes.
 Dynamical Systems

All the Lagrangian relative equilibria of the curved 3body problem have equal masses
View Description Hide DescriptionWe consider the 3body problem in 3dimensional spaces of nonzero constant Gaussian curvature and study the relationship between the masses of the Lagrangian relative equilibria, which are orbits that form a rigidly rotating equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3dimensional spaces of constant nonzero curvature: positive elliptic and positive ellipticelliptic, on 3spheres, and negative elliptic, on hyperbolic 3spheres. We prove that all these Lagrangian relative equilibria exist only for equal values of the masses.

The 3dimensional cored and logarithm potentials: Periodic orbits
View Description Hide DescriptionWe study analytically families of periodic orbits for the cored and logarithmic Hamiltonians with 3 degrees of freedom, which are relevant in the analysis of the galactic dynamics. First, after introducing a scale transformation in the coordinates and momenta with a parameter ɛ, we show that both systems give essentially the same set of equations of motion up to first order in ɛ. Then the conditions for finding families of periodic orbits, using the averaging theory up to first order in ɛ, apply equally to both systems in every energy level H = h > 0 showing the existence of at least 3 periodic orbits, for ɛ small enough, and also provides an analytic approximation for the initial conditions of these periodic orbits. We prove that at every positive energy level the cored and logarithmic Hamiltonians with 3 degrees of freedom have at least three periodic solutions. The technique used for proving such a result can be applied to other Hamiltonian systems.

Singularity analysis in planar vector fields
View Description Hide DescriptionThe aim of singularity analysis is to build local solutions around their singularities in the complex time. Integrability properties or the existence of particular solutions follow from the detailed analysis of the series expansions of the local solutions once they are constructed. In this work we made a survey of the most important results of the singularity analysis which we apply to planar vector fields.

The geometry and integrability of the Suslov problem
View Description Hide DescriptionIn this paper, we discuss the integrability of a nonholonomic mechanical system—a generalized Klebsh–Tisserand case of the Suslov problem. Using the theory of Hamiltonization and the Poincaré–Hopf theorem we analyze the topology of the invariant manifolds and in particular describe their genus. We contrast the results with those for Hamiltonian systems.
 Classical Mechanics and Classical Fields

Involutive constrained systems and HamiltonJacobi formalism
View Description Hide DescriptionIn this paper, we study singular systems with complete sets of involutive constraints. The aim is to establish, within the HamiltonJacobi theory, the relationship between the Frobenius’ theorem, the infinitesimal canonical transformations generated by constraints in involution with the Poisson brackets, and the lagrangian point (gauge) transformations of physical systems.