Volume 54, Issue 7, July 2013
Index of content:

We consider a polyharmonic operator in dimension two with l ⩾ 2, l being an integer, and a quasiperiodic potential . We prove that the absolutely continuous spectrum of H contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves at the high energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.
 ARTICLES


Partial Differential Equations

Chemically reacting mixtures in terms of degenerated parabolic setting
View Description Hide DescriptionThe paper analyzes basic mathematical questions for a model of chemically reacting mixtures. We derive a model of several (finite) component compressible gas taking rigorously into account the thermodynamical regime. Mathematical description of the model leads to a degenerate parabolic equation with hyperbolic deviation. The thermodynamics implies that the diffusion terms are nonsymmetric, not positively defined, and crossdiffusion effects must be strongly marked. The mathematical goal is to establish the existence of weak solutions globally in time for arbitrary number of reacting species. A key point is an entropylike estimate showing possible renormalization of the system.

Soliton solutions for quasilinear Schrödinger equations
View Description Hide DescriptionIt is established the existence of nontrivial solutions for quasilinear Schrödinger equations with subcritical or critical exponents, which appear from plasma physics as well as highpower ultrashort laser in matter.

Riemann problems and exact solutions to a traffic flow model
View Description Hide DescriptionWithin the theoretical framework of differential constraints method a nonhomogeneous model describing traffic flows is considered. Classes of exact solutions to the governing equations under interest are determined. Furthermore, Riemann problems and generalized Riemann problems which model situations of interest for traffic flows are solved.

On onedimensional compressible NavierStokes equations with degenerate viscosity and constant state at far fields
View Description Hide DescriptionIn this paper, we are concerned with the Cauchy problem for onedimensional compressible isentropic NavierStokes equations with densitydependent viscosity μ(ρ) = ρα(α > 0) and pressure P(ρ) = ργ (γ > 1). We will establish the global existence and asymptotic behavior of weak solutions for any α > 0 and γ > 1 under the assumption that the density function keeps a constant state at far fields. In particular, in the case that , we obtain the large time behavior of the strong solution obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).

Representation Theory and Algebraic Methods

Subsingular vectors in Verma modules, and tensor product of weight modules over the twisted HeisenbergVirasoro algebra and W(2, 2) algebra
View Description Hide DescriptionWe show that subsingular vectors may exist in Verma modules over W(2, 2), and present the subquotient structure of these modules. We prove conditions for irreducibility of the tensor product of intermediate series module with a highest weight module. Relation to intertwining operators over vertex operator algebra associated with W(2, 2) is discussed. Also, we study the tensor product of intermediate series and a highest weight module over the twisted HeisenbergVirasoro algebra, and present series of irreducible modules with infinitedimensional weight spaces.

Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras
View Description Hide DescriptionWe introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finitedimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently.

Quantum Mechanics

On a solution of the Schrödinger equation with a hyperbolic doublewell potential
View Description Hide DescriptionWe report a solution of the onedimensional Schrödinger equation with a hyperbolic doublewell confining potential via a transformation to the socalled confluent Heun equation. We discuss the requirements on the parameters of the system in which a reduction to confluent Heun polynomials is possible, representing the wavefunctions of bound states.

Topologies on quantum topoi induced by quantization
View Description Hide DescriptionIn the present paper, we consider effects of quantization in a topos approach of quantum theory. A quantum system is assumed to be coded in a quantum topos, by which we mean the topos of presheaves on the context category of commutative subalgebras of a von Neumann algebra of bounded operators on a Hilbert space. A classical system is modeled by a Lie algebra of classical observables. It is shown that a quantization map from the classical observables to selfadjoint operators on the Hilbert space naturally induces geometric morphisms from presheaf topoi related to the classical system to the quantum topos. By means of the geometric morphisms, we give LawvereTierney topologies on the quantum topos (and their equivalent Grothendieck topologies on the context category). We show that, among them, there exists a canonical one which we call a quantization topology. We furthermore give an explicit expression of a sheafification functor associated with the quantization topology.

Lévy flights and nonlocal quantum dynamics
View Description Hide DescriptionWe develop a fully fledged theory of quantum dynamical patterns of behavior that are nonlocally induced. To this end we generalize the standard Laplacianbased framework of the Schrödinger picture quantum evolution to that employing nonlocal (pseudodifferential) operators. Special attention is paid to the Salpeter (here, m ⩾ 0) quasirelativistic equation and the evolution of various wave packets, in particular to their radial expansion in 3D. Foldy's synthesis of “covariant particle equations” is extended to encompass free Maxwell theory, which however is devoid of any “particle” content. Links with the photon wave mechanics are explored.

Quantum Information and Computation

Complete positivity of the map from a basis to its dual basis
View Description Hide DescriptionThe dual of a matrix ordered space has a natural matrix ordering that makes the dual space matrix ordered as well. The purpose of these notes is to give a condition that describes when the linear map taking a basis of M n to its dual basis is a complete order isomorphism. We exhibit “natural” orthonormal bases for M n such that this map is an order isomorphism, but not a complete order isomorphism. Included among such bases is the Pauli basis. Our results generalize the Choi matrix by giving conditions under which the role of the standard basis {E ij } can be replaced by other bases.

Exclusivity structures and graph representatives of local complementation orbits
View Description Hide DescriptionWe describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lovász number which is equal to its fractional packing number. The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated with G, and exclusive events are adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under local complementation. Physically, the construction translates into graphtheoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics. In the context of zeroerror information theory, the construction suggests a protocol achieving the maximum rate of entanglementassisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G). The violation of the Bell inequality is expressed by the oneshot version of this capacity being strictly larger than the independence number. Finally, given the correspondence between graphs and exclusivity structures, we are able to compute the independence number for certain infinite families of graphs with the use of quantum nonlocality, therefore highlighting an application of quantum theory in the proof of a purely combinatorial statement.

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

GroenewoldMoyal product, α*cohomology, and classification of translationinvariant noncommutative structures
View Description Hide DescriptionThe theory of α*cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2cocycle, the harmonic form, which generates a particular GroenewoldMoyal star product. This leads to an algebraic classification of translationinvariant noncommutative structures and shows that any general translationinvariant noncommutative quantum field theory is physically equivalent to a GroenewoldMoyal noncommutative quantum field theory.

Operator product expansion algebra
View Description Hide DescriptionWe establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ4quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hepth/1105.3375, that the 3point OPE, , usually interpreted only as an asymptotic short distance expansion, actually converges at finite, and even large, distances. We further show that the factorization identity is satisfied for suitable configurations of the spacetime arguments. Again, the infinite sum is shown to be convergent. Our proofs rely on explicit bounds on the remainders of these expansions, obtained using refined versions, mostly due to Kopper et al., of the renormalization group flow equation method. These bounds also establish that each OPE coefficient is a real analytic function in the spacetime arguments for noncoinciding points. Our results hold for arbitrary but finite loop orders. They lend support to proposals for a general axiomatic framework of quantum field theory, based on such “consistency conditions” and akin to vertex operator algebras, wherein the OPE is promoted to the defining structure of the theory.

Pointform dynamics of quasistable states
View Description Hide DescriptionWe present a field theoretical model of pointform dynamics which exhibits resonance scattering. In particular, we construct pointform Poincaré generators explicitly from field operators and show that in the vector spaces for the instates and outstates (endowed with certain analyticity and topological properties suggested by the structure of the Smatrix) these operators integrate to furnish differentiable representations of the causal Poincaré semigroup, the semidirect product of the semigroup of spacetime translations into the forward lightcone and the group of Lorentz transformations. We also show that there exists a class of irreducible representations of the Poincaré semigroup defined by a complex mass and a halfinteger spin. The complex mass characterizing the representation naturally appears in the construction as the square root of the pole position of the propagator. These representations provide a description of resonances in the same vein as Wigner's unitary irreducible representations of the Poincaré group provide a description of stable particles.

(Pre)Hilbert spaces in twistor quantization
View Description Hide DescriptionIn twistor theory, the canonical quantization procedure, called twistor quantization, is performed with the twistor operators represented as and . However, it has not been clarified what kind of function spaces this representation is valid in. In the present paper, we intend to find appropriate (pre)Hilbert spaces in which the above representation is realized as an adjoint pair of operators. To this end, we define an inner product for the helicity eigenfunctions by an integral over the product space of the circular space S 1 and the upper half of projective twistor space. Using this inner product, we define a Hilbert space in some particular case and indefinitemetric preHilbert spaces in other particular cases, showing that the abovementioned representation is valid in these spaces. It is also shown that only the Penrose transform in the first particular case yields positivefrequency massless fields without singularities, while the Penrose transforms in the other particular cases yield positivefrequency massless fields with singularities.

General Relativity and Gravitation

Collineations of a symmetric 2covariant tensor: Ricci collineations
View Description Hide DescriptionThe infinitesimal transformations that leave invariant a twocovariant symmetric tensor are studied. The interest of these symmetry transformations lays in the fact that this class of tensors includes the energymomentum and Ricci tensors. We find that in most cases the class of infinitesimal generators of these transformations is a finite dimensional Lie algebra, but in some cases exhibiting a higher degree of degeneracy, this class is infinite dimensional and may fail to be a Lie algebra. As an application, we study the Ricci collineations of a type B warped spacetime.

Dynamical Systems

On the relaxation dynamics of the Kuramoto oscillators with small inertia
View Description Hide DescriptionFor the Kuramoto oscillators with small inertia, we present several quantitative estimates on the relaxation dynamics and formational structure of a phaselocked state (PLS) for some classes of initial configurations. In a supercritical regime where the coupling strength is strictly larger than the diameter of natural frequencies, we present quantitative relaxation dynamics on the collision numbers and the structure of PLS. In a critical coupling regime where the coupling strength is exactly the diameter of natural frequencies, we provide a sufficient condition for an asymptotically PLS solution. In particular, we show the existence of slow relaxation to a PLS, when there are exactly two natural frequencies. This generalizes the earlier results of Choi et al. [“Asymptotic formation and orbital stability of phase locked states for the Kuramoto model,” Physica D241, 735–754 (Year: 2012)10.1016/j.physd.2011.11.011; Choi et al. “Complete synchronization of Kuramoto oscillators with finite inertia,” Physica D240, 32–44 (Year: 2011)]10.1016/j.physd.2010.08.004

On the formulation of new explicit conditions for steepness from a former result of N.N. Nekhoroshev
View Description Hide DescriptionThe application of the Nekhoroshev theorem to many problems arising in different fields of Physics and Astronomy depends on a nondegeneracy property, called steepness, that a suitable Hamiltonian approximation must satisfy. Since steepness is implicitly defined, we have the problem of recognizing whether a given function is steep or not. For this purpose, we here consider some sufficient conditions for steepness provided by Nekhoroshev in 1979, based on the solvability of a collection of systems depending on the number n of degrees of freedom, the derivatives of the function up to a certain order r, and some auxiliary parameters. These conditions are really explicit only for r = 2, corresponding to quasiconvexity , and for r = 3. Instead, for r ⩾ 4, the conditions are implicit, since they require an elaborate computation of the closure of a certain set. In this paper, we first revisit Nekhoroshev's result and we show that the number of parameters in the collections of systems can be suitably reduced. Then, we show that for r = 4 Nekhoroshev's result is interesting only for n = 2, 3, and 4, and in these cases we find explicit conditions for steepness which are formulated in a purely algebraic form.

The 4body problem in a (1+1)dimensional selfgravitating system
View Description Hide DescriptionWe report on the results of a study of the motion of a four particle nonrelativistic onedimensional selfgravitating system. We show that the system can be visualized in terms of a single particle moving within a potential whose equipotential surfaces are shaped like a box of pyramidshaped sides. As such this is the largest Nbody system that can be visualized in this way. We describe how to classify possible states of motion in terms of Braid Group operators, generalizing this to N bodies. We find that the structure of the phase space of each of these systems yields a large variety of interesting dynamics, containing regions of quasiperiodicity and chaos. Lyapunov exponents are calculated for many trajectories to measure stochasticity and previously unseen phenomena in the Lyapunov graphs are observed.

The free rigid body dynamics: Generalized versus classic
View Description Hide DescriptionIn this paper we analyze some normal forms of a general quadratic Hamiltonian system defined on the dual of the Lie algebra of real Kskewsymmetric matrices, where K is an arbitrary 3×3 real symmetric matrix. A consequence of the main results is that any firstorder autonomous threedimensional differential equation possessing two independent quadratic constants of motion, which admit a positive/negative definite linear combination, is affinely equivalent to the classical “relaxed” free rigid body dynamics with linear control parameters.
