Volume 55, Issue 10, October 2014
Index of content:

In the previous works of Borzov and Damaskinsky [“ChebyshevKoornwinder oscillator,” Theor. Math. Phys.175(3), 765–772 (2013)] and [“Ladder operators for ChebyshevKoornwinder oscillator,” in Proceedings of the Days on Diffraction, 2013], the authors have defined the oscillatorlike system that is associated with the two variable ChebyshevKoornwinder polynomials. We call this system the generalized ChebyshevKoornwinder oscillator. In this paper, we study the properties of infinitedimensional Lie algebra that is analogous to the Heisenberg algebra for the ChebyshevKoornwinder oscillator. We construct the exact irreducible representation of this algebra in a Hilbert space of functions that are defined on a region which is bounded by the Steiner hypocycloid. The functions are squareintegrable with respect to the orthogonality measure for the ChebyshevKoornwinder polynomials and these polynomials form an orthonormalized basis in the space . The generalized oscillator which is studied in the work can be considered as the simplest nontrivial example of multiboson quantum system that is composed of three interacting oscillators.
 ARTICLES

 Partial Differential Equations

Equivariant spectral asymptotics for hpseudodifferential operators
View Description Hide DescriptionWe prove equivariant spectral asymptotics for hpseudodifferential operators for compact orthogonal group actions generalizing results of El Houakmi and Helffer [“Comportement semiclassique en présence de symétries: Action d'un groupe de Lie compact,” Asymp. Anal.5(2), 91–113 (1991)] and Cassanas [“Reduced Gutzwiller formula with symmetry: Case of a Lie group,” J. Math. Pures Appl.85(6), 719–742 (2006)]. Using recent results for certain oscillatory integrals with singular critical sets [P. Ramacher, “Singular equivariant asymptotics and Weyl's law: On the distribution of eigenvalues of an invariant elliptic operator,” J. Reine Angew. Math. (Crelles J.) (to be published)], we can deduce a weak equivariant Weyl law. Furthermore, we can prove a complete asymptotic expansion for the Gutzwiller trace formula without any additional condition on the group action by a suitable generalization of the dynamical assumptions on the Hamilton flow.

Global solution of the electromagnetic fieldparticle system of equations
View Description Hide DescriptionIn this paper we discuss global existence of the solution of the Maxwell and Newton system of equations, describing the interaction of a rigid charge distribution with the electromagnetic field it generates. A unique solution is proved to exist (for regular charge distributions) on suitable homogeneous and nonhomogeneous Sobolev spaces, for the electromagnetic field, and on coordinate and velocity space for the charge; provided initial data belong to the subspace that satisfies the divergence part of Maxwell's equations.

Solution of the Kortewegde Vries equation on the line with analytic initial potential
View Description Hide DescriptionWe present a theory of SturmLiouville nonsymmetric vessels, realizing an inverse scattering theory for the SturmLiouville operator with analytic potentials on the line. This construction is equivalent to the construction of a matrix spectral measure for the SturmLiouville operator, defined with an analytic potential on the line. Evolving such vessels we generate Kortewegde Vries (KdV) vessels, realizing solutions of the KdV equation. As a consequence, we prove the theorem as follows: Suppose that q(x) is an analytic function on . Then there exists a closed subset and a KdV vessel, defined on Ω. For each one can find T x > 0 such that {x} × [ − T x , T x ]⊆Ω. The potential q(x) is realized by the vessel for t = 0. Since we also show that if q(x, t) is a solution of the KdV equation on , then there exists a vessel, realizing it, the theory of vessels becomes a universal tool to study this problem. Finally, we notice that the idea of the proof applies to a similar existence of a solution for evolutionary nonlinear Schrödinger and Boussinesq equations, since both of these equations possess vessel constructions.

Ground state solutions for nonautonomous dynamical systems
View Description Hide DescriptionWe study the existence of periodic solutions for a second order nonautonomous dynamical system. We allow both sublinear and superlinear problems. We obtain ground state solutions.

The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions
View Description Hide DescriptionThe inverse scattering transform (IST) as a tool to solve the initialvalue problem for the focusing nonlinear Schrödinger (NLS) equation with nonzero boundary values as x → ∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with A l ≠ A r and θ l ≠ θ r . The direct problem is shown to be welldefined for NLS solutions q(x, t) such that with respect to x for all t ⩾ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a RiemannHilbert problem on a single sheet of the scattering variables , where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x → ±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the longtime asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the RiemannHilbert problem, or via matched asymptotic expansions on the Marchenko integral equations.
 Representation Theory and Algebraic Methods

YangMills like instantons in eight and seven dimensions
View Description Hide DescriptionWe consider a gauge theory in which a nonassociative Moufang loop takes the place of a structure group. We construct BelavinPolyakovSchwartzTyupkin (BPST) and t’Hooft like instanton solutions of the gauge theory in seven and eight dimensions.

Coupling coefficients for tensor product representations of quantum SU(2)
View Description Hide DescriptionWe study tensor products of infinite dimensional irreducible ^{*}representations (not corepresentations) of the SU(2) quantum group. We obtain (generalized) eigenvectors of certain selfadjoint elements using spectral analysis of Jacobi operators associated to wellknown qhypergeometric orthogonal polynomials. We also compute coupling coefficients between different eigenvectors corresponding to the same eigenvalue. Since the continuous spectrum has multiplicity two, the corresponding coupling coefficients can be considered as 2 × 2matrixvalued orthogonal functions. We compute explicitly the matrix elements of these functions. The coupling coefficients can be considered as qanalogs of Bessel functions. As a results we obtain several qintegral identities involving qhypergeometric orthogonal polynomials and qBesseltype functions.

Construction of conformally invariant higher spin operators using transvector algebras
View Description Hide DescriptionThis paper deals with a systematic construction of higher spin operators, defined as conformally invariant differential operators acting on functions on flat space with values in an arbitrary halfinteger irreducible representation for the spin group. To be more precise, the higher spin version of the Dirac operator and associated twistor operators will be constructed as generators of a transvector algebra, hereby generalising the wellknown fact that the classical Dirac operator on and its symbol generate the orthosymplectic Lie superalgebra . To do so, we will use the extremal projection operator and its relation to transvector algebras. In the second part of the article, the conformal invariance of the constructed higher spin operators will be proven explicitly.
 Quantum Mechanics

Coherent state transforms and the MackeyStoneVon Neumann theorem
View Description Hide DescriptionMackey showed that for a compact Lie group K, the pair (K, C ^{0}(K)) has a unique nontrivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinitedimensional family of K × K invariant polarizations on T ^{*} K. The Kähler polarizations in the family are generated by (complex) timeτ Hamiltonian flows applied to the (Schrödinger) vertical real polarization. The unitary equivalence of the corresponding quantizations of T ^{*} K is then studied by considering covariant pairs of representations of K defined by geometric prequantization and of representations of C ^{0}(K) defined via Heisenberg time(−τ) evolution followed by time(+τ) geometricquantizationinduced evolution. We show that in the semiclassical and large imaginary time limits, the unitary transform whose existence is guaranteed by Mackey's theorem can be approximated by composition of the time(+τ) geometricquantizationinduced evolution with the time(−τ) evolution associated with the momentum space [W. D. Kirwin and S. Wu, “Momentum space for compact Lie groups and the PeterWeyl theorem” (unpublished)] quantization of the Hamiltonian function generating the flow. In the case of quadratic Hamiltonians, this asymptotic result is exact and unitary equivalence between quantizations is achieved by identifying the Heisenberg imaginary time evolution with heat operator evolution, in accordance with the coherent state transform of Hall.

Quantization of a particle on a twodimensional manifold of constant curvature
View Description Hide DescriptionThe formulation of quantum mechanics on spaces of constant curvature is studied. It is shown how a transition from a classical system to the quantum case can be accomplished by the quantization of the Noether momenta. These can be determined by means of Lie differentiation of the metric which defines the manifold. For the metric examined here, it is found that the resulting Schrödinger equation is separable and the spectrum and eigenfunctions can be investigated in detail.

Topos quantum theory on quantizationinduced sheaves
View Description Hide DescriptionIn this paper, we construct a sheafbased topos quantum theory. It is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von Neumann algebras of bounded operators on a Hilbert space. Also, it is already known that quantization naturally induces a LawvereTierney topology on the presheaf topos. We show that a topos quantum theory akin to the presheafbased one can be constructed on sheaves defined by the quantizationinduced LawvereTierney topology. That is, starting from the spectral sheaf as a state space of a given quantum system, we construct sheafbased expressions of physical propositions and truth objects, and thereby give a method of truthvalue assignment to the propositions. Furthermore, we clarify the relationship to the presheafbased quantum theory. We give translation rules between the sheafbased ingredients and the corresponding presheafbased ones. The translation rules have “coarsegraining” effects on the spaces of the presheafbased ingredients; a lot of different proposition presheaves, truth presheaves, and presheafbased truthvalues are translated to a proposition sheaf, a truth sheaf, and a sheafbased truthvalue, respectively. We examine the extent of the coarsegraining made by translation.

Full spectrum of the twophoton and the twomode quantum Rabi models
View Description Hide DescriptionThis paper is concerned with the rigorous analytical determination of the spectrum of the twophoton and the twomode quantum Rabi models. To reach this goal, we exploit the hidden symmetries in these models by means of the unitary and similarity transformations in addition to the BargmannFock space description. In each case, the purely quantum mechanical problem of the Rabi model studied is reduced to solutions for differential equations. This eventually gives a thirdorder differential equation for each of these models, which is reduced to a secondorder differential equation by additional transformations. The analytical expressions of the wave functions describing the energy levels are obtained in terms of the confluent hypergeometric functions.

Analytical solutions of a generalized noncentral potential in Ndimensions
View Description Hide DescriptionWe present that Ndimensional nonrelativistic wave equation for the generalized noncentral potential with arbitrary angular momentum is analytically solvable in the hyperspherical coordinates. Asymptotic iteration method as a different approach is applied to obtain Ndimensional energy eigenvalues and the corresponding eigenfunctions. In hyperspherical coordinates, the wave function solutions are obtained in terms of hypergeometric functions and Jacobi polynomials. The bound states of quantum systems under consideration for some special cases, such as Hartmann and Makarov potentials, have been discussed in Ndimensions.
 Quantum Information and Computation

Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination
View Description Hide DescriptionWe consider the multiple hypothesis testing problem for symmetric quantum state discrimination between r given states σ1, …, σ r . By splitting up the overall test into multiple binary tests in various ways we obtain a number of upper bounds on the optimal error probability in terms of the binary error probabilities. These upper bounds allow us to deduce various bounds on the asymptotic error rate, for which it has been hypothesized that it is given by the multihypothesis quantum Chernoff bound (or Chernoff divergence) C(σ1, …, σ r ), as recently introduced by Nussbaum and Szkoła in analogy with Salikhov's classical multihypothesis Chernoff bound. This quantity is defined as the minimum of the pairwise binary Chernoff divergences . It was known already that the optimal asymptotic rate must lie between C/3 and C, and that for certain classes of sets of states the bound is actually achieved. It was known to be achieved, in particular, when the state pair that is closest together in Chernoff divergence is more than 6 times closer than the next closest pair. Our results improve on this in two ways. First, we show that the optimal asymptotic rate must lie between C/2 and C. Second, we show that the Chernoff bound is already achieved when the closest state pair is more than 2 times closer than the next closest pair. We also show that the Chernoff bound is achieved when at least r − 2 of the states are pure, improving on a previous result by Nussbaum and Szkoła. Finally, we indicate a number of potential pathways along which a proof (or disproof) may eventually be found that the multihypothesis quantum Chernoff bound is always achieved.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

The twoloop sunrise graph in two spacetime dimensions with arbitrary masses in terms of elliptic dilogarithms
View Description Hide DescriptionWe present the twoloop sunrise integral with arbitrary nonzero masses in two spacetime dimensions in terms of elliptic dilogarithms. We find that the structure of the result is as simple and elegant as in the equal mass case, only the arguments of the elliptic dilogarithms are modified. These arguments have a nice geometric interpretation.
 General Relativity and Gravitation

Liouvillian integrability of gravitating static isothermal fluid spheres
View Description Hide DescriptionWe examine the integrability properties of the Einstein field equations for static, spherically symmetric fluid spheres, complemented with an isothermal equation of state, ρ = np. In this case, Einstein's equations can be reduced to a nonlinear, autonomous second order ordinary differential equation (ODE) for m/R (m is the mass inside the radius R) that has been solved analytically only for n = −1 and n = −3, yielding the cosmological solutions by De Sitter and Einstein, respectively, and for n = −5, case for which the solution can be derived from the De Sitter's one using a symmetry of Einstein's equations. The solutions for these three cases are of Liouvillian type, since they can be expressed in terms of elementary functions. Here, we address the question of whether Liouvillian solutions can be obtained for other values of n. To do so, we transform the second order equation into an equivalent autonomous Lotka–Volterra quadratic polynomial differential system in , and characterize the Liouvillian integrability of this system using Darboux theory. We find that the Lotka–Volterra system possesses Liouvillian first integrals for n = −1, −3, −5, which descend from the existence of invariant algebraic curves of degree one, and for n = −6, a new solvable case, associated to an invariant algebraic curve of higher degree (second). For any other value of n, eventual first integrals of the Lotka–Volterra system, and consequently of the second order ODE for the mass function must be nonLiouvillian. This makes the existence of other solutions of the isothermal fluid sphere problem with a Liouvillian metric quite unlikely.

Relativistic Bessel cylinders
View Description Hide DescriptionA set of cylindrical solutions to Einstein's field equations for power law densities is described. The solutions have a Bessel function contribution to the metric. For matter cylinders regular on axis, the first two solutions are the constant density GottHiscock string and a cylinder with a metric Airy function. All members of this family have the Vilenkin limit to their mass per length. Some examples of Bessel shells and Bessel motion are given.
 Dynamical Systems

AblowitzLadik hierarchy of integrable equations on a timespace scale
View Description Hide DescriptionWe derive the Toda's lattice, the Hirota's network, and the nonlinear Schrodinger dynamic equations on a timespace scale by extension on a timespace scale the AblowitzLadik hierarchy of integrable dynamic systems.

Point vortices on the hyperbolic plane
View Description Hide DescriptionWe investigate the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map. The relative equilibrium conditions are found and the trajectories of relative equilibria with nonzero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on 2 and 3 vortices. Unlike the system on the sphere, this system has relative equilibria with noncompact momentum isotropy subgroup, and these are used to illustrate the different stability types of relative equilibria.

Selftrapping transition for a nonlinear impurity within a linear chain
View Description Hide DescriptionIn the present work, we revisit the issue of the selftrapping dynamical transition at a nonlinear impurity embedded in an otherwise linear lattice. For our Schrödinger chain example, we present rigorous arguments that establish necessary conditions and corresponding parametric bounds for the transition between linear decay and nonlinear persistence of a defect mode. The proofs combine a contraction mapping approach applied to the fully dynamical problem in the case of a 3Dlattice, together with variational arguments for the derivation of parametric bounds for the creation of stationary states associated with the expected fate of the selftrapping dynamical transition. The results are relevant for both power law nonlinearities and saturable ones. The analytical results are corroborated by numerical computations. The latter are performed for cases of different dimension.