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.
 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.
 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.
 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.
 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.

Duality between the trigonometric BC _{ n } Sutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system
View Description Hide DescriptionWe present a new case of duality between integrable manybody systems, where two systems live on the actionangle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one. Our investigation utilizes an idea that was exploited previously to provide grouptheoretic interpretation for several dualities discovered originally by Ruijsenaars. In the grouptheoretic framework, one applies Hamiltonian reduction to two Abelian Poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space. Taking the cotangent bundle of U(2n) as the upstairs space, we demonstrate how this mechanism leads to a new dual pair involving the BC n trigonometric Sutherland system. Thereby, we generalize earlier results pertaining to the A n trigonometric Sutherland system as well as a recent work by Pusztai on the hyperbolic BC n Sutherland system.

The negative of regular cosmological time function is a viscosity solution
View Description Hide DescriptionIn this paper we show that for a spacetime (M, g) with regular cosmological time function τ, −τ is a viscosity solution to the HamiltonJacobi equation g(∇u, ∇u) = −1 and shares some analogous properties with the socalled weak KAM solutions.
 Classical Mechanics and Classical Fields

Dynamical realizations of lconformal Galilei superalgebra
View Description Hide DescriptionDynamical systems which are invariant under supersymmetric extension of the lconformal Galilei algebra are constructed. These include a free superparticle which is governed by higher derivative equations of motion and an supersymmetric PaisUhlenbeck oscillator for a particular choice of its frequencies. A Niedererlike transformation which links the models is proposed.
 Methods of Mathematical Physics

Bilinear covariants and spinor fields duality in quantum Clifford algebras
View Description Hide DescriptionClassification of quantum spinor fields according to quantum bilinear covariants is introduced in a context of quantum Clifford algebras on Minkowski spacetime. Once the bilinear covariants are expressed in terms of algebraic spinor fields, the duality between spinor and quantum spinor fields can be discussed. Thus, by endowing the underlying spacetime with an arbitrary bilinear form with an antisymmetric part in addition to a symmetric spacetime metric, quantum algebraic spinor fields and deformed bilinear covariants can be constructed. They are thus compared to the classical (non quantum) ones. Classes of quantum spinor fields classes are introduced and compared with Lounesto's spinor field classification. A physical interpretation of the deformed parts and the underlying grading is proposed. The existence of an arbitrary bilinear form endowing the spacetime already has been explored in the literature in the context of quantum gravity[S. W. Hawking, “The unpredictability of quantum gravity,” Commun. Math. Phys.87, 395 (1982)]. Here, it is shown further to play a prominent role in the structure of Dirac, Weyl, and Majorana spinor fields, besides the most general flagpoles and flagdipoles. We introduce a new duality between the standard and the quantum spinor fields, by showing that when Clifford algebras over vector spaces endowed with an arbitrary bilinear form are taken into account, a mixture among the classes does occur. Consequently, novel features regarding the spinor fields can be derived.

Riemann–Hilbert problem approach for twodimensional flow inverse scattering^{a)}
View Description Hide DescriptionWe consider inverse scattering for the timeharmonic wave equation with firstorder perturbation in two dimensions. This problem arises in particular in the acoustic tomography of moving fluid. We consider linearized and nonlinearized reconstruction algorithms for this problem of inverse scattering. Our nonlinearized reconstruction algorithm is based on the nonlocal Riemann–Hilbert problem approach. Comparisons with preceding results are given.

Lie group analysis of a generalized KricheverNovikov differentialdifference equation
View Description Hide DescriptionThe symmetry algebra of the differentialdifference equation with D = u n+1 − u n−1 and N = P(u n )u n+1 u n−1 + Q(u n )(u n+1 + u n−1) + R(u n ), where P, Q, and R are arbitrary analytic functions is shown to have the dimension 1 ⩽ dimL ⩽ 5. When P, Q, and R are specific second order polynomials in u n (depending on 6 constants) this is the integrable discretization of the Krichever–Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies dimL = 2. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity u n+N = u n and thus to reduce the differentialdifference equation to a system of N coupled ordinary three points difference equations.

Analytical and numerical studying of the perturbed Kortewegde Vries equation
View Description Hide DescriptionThe perturbed Kortewegde Vries equation is considered. This equation is used for the description of onedimensional viscous gas dynamics, nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves. The integrability of this equation is investigated using the Painlevé approach. The condition on parameters for the integrability of the perturbed Kortewegde Vries equation equation is established. New classical and nonclassical symmetries admitted by this equation are found. All corresponding symmetry reductions are obtained. New exact solutions of these reductions are constructed. They are expressed via trigonometric and Airy functions. Stability of the exact solutions of the perturbed Kortewegde Vries equation is investigated numerically.