Volume 56, Issue 4, April 2015
Index of content:

In usual realistic classical dynamical systems, the Hamiltonian depends explicitly on time. In this work, a class of classical systems with time dependent nonlinear Hamiltonians is analyzed. This type of problems allows to find invariants by a family of Veronese maps. The motivation to develop this method results from the observation that the PoissonLie algebra of monomials in the coordinates and momenta is clearly defined in terms of its brackets and leads naturally to an infinite linear set of differential equations, under certain circumstances. To perform explicit analytic and numerical calculations, two examples are presented to estimate the trajectories, the first given by a nonlinear problem and the second by a quadratic Hamiltonian with three time dependent parameters. In the nonlinear problem, the Veronese approach using jets is shown to be equivalent to a direct procedure using elliptic functions identities, and linear invariants are constructed. For the second example, linear and quadratic invariants as well as stability conditions are given. Explicit solutions are also obtained for stepwise constant forces. For the quadratic Hamiltonian, an appropriated set of coordinates relates the geometric setting to that of the three dimensional manifold of central conic sections. It is shown further that the quantum mechanical problem of scattering in a superlattice leads to mathematically equivalent equations for the wave function, if the classical time is replaced by the space coordinate along a superlattice. The mathematical method used to compute the trajectories for stepwise constant parameters can be applied to both problems. It is the standard method in quantum scattering calculations, as known for locally periodic systems including a space dependent effective mass.
 ARTICLES

 Partial Differential Equations

Uniqueness of topological multivortex solutions for a skewsymmetric ChernSimons system
View Description Hide DescriptionIn this paper, we consider a skewsymmetric ChernSimons system problem with a coupling parameter. Our main goal is that, when the coupling parameter is small, the topological type solutions to this system problem are uniquely determined by the location of their vortex points. This result follows by the bubbling analysis and the nondegeneracy of linearized equations.
 Representation Theory and Algebraic Methods

Wigner–Eckart theorem for the noncompact algebra 𝔰𝔩(2, ℝ)
View Description Hide DescriptionThe Wigner–Eckart theorem is a well known result for tensor operators of 𝔰𝔲(2) and, more generally, any compact Lie algebra. In this paper, the theorem will be generalized to the particular noncompact case of 𝔰𝔩(2, ℝ). In order to do so, recoupling theory between representations that are not necessarily unitary will be studied, namely, between finitedimensional and infinitedimensional representations. As an application, the Wigner–Eckart theorem will be used to construct an analogue of the Jordan–Schwinger representation, previously known only for representations in the discrete class, which also covers the continuous class.

On the generating function of weight multiplicities for the representations of the Lie algebra C 2
View Description Hide DescriptionWe use the generating function of the characters of C 2 to obtain a generating function for the multiplicities of the weights entering in the irreducible representations of that simple Lie algebra. From this generating function, we derive some recurrence relations among the multiplicities and a simple graphical recipe to compute them.

On the cohomology of Leibniz conformal algebras
View Description Hide DescriptionWe construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The lowdimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.

Generation of excited coherent states for a charged particle in a uniform magnetic field
View Description Hide DescriptionWe introduce excited coherent states, , where n is an integer and states denote the coherent states of a charged particle in a uniform magnetic field. States minimize the SchrödingerRobertson uncertainty relation while having the nonclassical properties. It has been shown that the resolution of identity condition is realized with respect to an appropriate measure on the complex plane. Some of the nonclassical features such as subPoissonian statistics and quadrature squeezing of these states are investigated. Our results are compared with similar Agarwal’s type photon added coherent states (PACSs) and it is shown that, while photoncounting statistics of are the same as PACSs, their squeezing properties are different. It is also shown that for large values of , while they are squeezed, they minimize the uncertainty condition. Additionally, it has been demonstrated that by changing the magnitude of the external magnetic field, Bext , the squeezing effect is transferred from one component to another. Finally, a new scheme is proposed to generate states in cavities.

φimaginary Verma modules and their generalizations for the toroidal Lie algebras
View Description Hide DescriptionIn this paper, we first construct a class of weight modules, called φimaginary Verma modules, for the toroidal Lie algebras. Then a criterion for the irreducibility of the φimaginary Verma modules is obtained and the irreducible quotients for the reducible ones are studied. Furthermore, we construct a more general class of ℤ^{ n }graded modules for the toroidal Lie algebras and we discuss their irreducibility. This class of modules includes the φimaginary Verma modules as special examples.
 Quantum Mechanics

Superintegrable systems with position dependent mass
View Description Hide DescriptionFirst order integrals of motion for Schrödinger equations with position dependent masses are classified. Eighteen classes of such equations with nonequivalent symmetries are specified. They include integrable, superintegrable, and maximally superintegrable systems. Among them is a system invariant with respect to the Lie algebra of Lorentz group and a system whose integrals of motion form algebra so(4). Three of the obtained systems are solved exactly.

Fivebody Moshinsky brackets
View Description Hide DescriptionIn variational calculations with harmonic oscillator wavefunctions as trial bases, the transformation coefficients that relate harmonic oscillator wavefunctions in two different sets of internal coordinates are convenient to the evaluation of some matrix elements. Here, we present the explicit expression of these transformation coefficients for fivebody systems. These transformation coefficients can be collected in a matrix according to the quantum number N of harmonic oscillator shell and can be programmed for arbitrary N.
 Quantum Information and Computation

Ancillaapproximable quantum state transformations
View Description Hide DescriptionWe consider the transformations of quantum states obtainable by a process of the following sort. Combine the given input state with a specially prepared initial state of an auxiliary system. Apply a unitary transformation to the combined system. Measure the state of the auxiliary subsystem. If (and only if) it is in a specified final state, consider the process successful, and take the resulting state of the original (principal) system as the result of the process. We review known information about exact realization of transformations by such a process. Then we present results about approximate realization of finite partial transformations. We not only consider primarily the issue of approximation to within a specified positive ε, but also address the question of arbitrarily close approximation.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

A relativistically interacting exactly solvable multitime model for two massless Dirac particles in 1 + 1 dimensions
View Description Hide DescriptionThe question how to Lorentz transform an Nparticle wave function naturally leads to the concept of a socalled multitime wave function, i.e., a map from (spacetime)^{ N } to a spin space. This concept was originally proposed by Dirac as the basis of relativistic quantum mechanics. In such a view, interaction potentials are mathematically inconsistent. This fact motivates the search for new mechanisms for relativistic interactions. In this paper, we explore the idea that relativistic interaction can be described by boundary conditions on the set of coincidence points of two particles in spacetime. This extends ideas from zerorange physics to a relativistic setting. We illustrate the idea at the simplest model which still possesses essential physical properties like Lorentz invariance and a positive definite density: twotime equations for massless Dirac particles in 1 + 1 dimensions. In order to deal with a spatiotemporally nontrivial domain, a necessity in the multitime picture, we develop a new method to prove existence and uniqueness of classical solutions: a generalized version of the method of characteristics. Both mathematical and physical considerations are combined to precisely formulate and answer the questions of probability conservation, Lorentz invariance, interaction, and antisymmetry.

Generalizing the Connes Moscovici Hopf algebra to contain all rooted trees
View Description Hide DescriptionThis paper defines a generalization of the ConnesMoscovici Hopf algebra, , that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in noncommutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.
 General Relativity and Gravitation

On a zerogravity limit of the Kerr–Newman spacetimes and their electromagnetic fields
View Description Hide DescriptionWe discuss the limit of vanishing G (Newton’s constant of universal gravitation) of the maximal analytically extended Kerr–Newman electrovacuum spacetimes represented in Boyer–Lindquist coordinates. We investigate the topologically nontrivial spacetime emerging in this limit and show that it consists of two copies of flat Minkowski spacetime crosslinked at a timelike solid cylinder (spacelike 2disk × timelike ℝ). As G → 0, the electromagnetic fields of the Kerr–Newman spacetimes converge to nontrivial solutions of Maxwell’s equations on this background spacetime . We show how to obtain these fields by solving Maxwell’s equations with singular sources supported only on a circle in a spacelike slice of . These sources do not suffer from any of the pathologies that plague the alternate sources found in previous attempts to interpret the Kerr–Newman fields on the topologically simple Minkowski spacetime. We characterize the singular behavior of these sources and prove that the Kerr–Newman electrostatic potential and magnetic scalar potential are the unique solutions of the Maxwell equations among all functions that have the same blowup behavior at the ring singularity.

Existence of static dyonic black holes in 4d N = 1 supergravity with finite energy
View Description Hide DescriptionWe prove the existence and the uniqueness of the static dyonic black holes in four dimensional N = 1 supergravity theory coupled vector and scalar multiplets. We set the nearhorizon geometry to be a product of two Einstein surfaces, whereas the asymptotic geometry has to be a space of constant scalar curvature. Using these data, we show that there exists a unique solution for scalar fields which interpolates these regions.
 Dynamical Systems

Blowup phenomena for polytropic equation with inhomogeneous density and source
View Description Hide DescriptionThe subject of this investigation is the blowup phenomena of the positive solutions of the mixed problem for the onedimensional polytropic filtration equation with inhomogeneous density and source. It is shown that under certain conditions on the nonlinearities and data, blowup will occur at some finite time. Note that the technique applied for the proof does not use the Zel’dovichKompaneetsBarenblatt solutions, since the construction of such type of function is more complicated in our case. Therefore, we obtain a result by multiplying on a special factor which has convenient properties. In particular, by choosing the parameters of the factor and using the properties of the solution, we obtain the inequality which allows us to show the blow up phenomena.

On nonautonomous dynamical systems
View Description Hide DescriptionIn usual realistic classical dynamical systems, the Hamiltonian depends explicitly on time. In this work, a class of classical systems with time dependent nonlinear Hamiltonians is analyzed. This type of problems allows to find invariants by a family of Veronese maps. The motivation to develop this method results from the observation that the PoissonLie algebra of monomials in the coordinates and momenta is clearly defined in terms of its brackets and leads naturally to an infinite linear set of differential equations, under certain circumstances. To perform explicit analytic and numerical calculations, two examples are presented to estimate the trajectories, the first given by a nonlinear problem and the second by a quadratic Hamiltonian with three time dependent parameters. In the nonlinear problem, the Veronese approach using jets is shown to be equivalent to a direct procedure using elliptic functions identities, and linear invariants are constructed. For the second example, linear and quadratic invariants as well as stability conditions are given. Explicit solutions are also obtained for stepwise constant forces. For the quadratic Hamiltonian, an appropriated set of coordinates relates the geometric setting to that of the three dimensional manifold of central conic sections. It is shown further that the quantum mechanical problem of scattering in a superlattice leads to mathematically equivalent equations for the wave function, if the classical time is replaced by the space coordinate along a superlattice. The mathematical method used to compute the trajectories for stepwise constant parameters can be applied to both problems. It is the standard method in quantum scattering calculations, as known for locally periodic systems including a space dependent effective mass.
 Classical Mechanics and Classical Fields

Nonperturbative aspects of particle acceleration in nonlinear electrodynamics
View Description Hide DescriptionWe undertake an investigation of particle acceleration in the context of nonlinear electrodynamics. We deduce the maximum energy that an electron can gain in a nonlinear density wave in a magnetised plasma, and we show that an electron can “surf” a sufficiently intense BornInfeld electromagnetic plane wave and be strongly accelerated by the wave. The first result is valid for a large class of physically reasonable modifications of the linear Maxwell equations, whilst the second result exploits the special mathematical structure of BornInfeld theory.
 Statistical Physics

Uniform asymptotics of areaweighted Dyck paths
View Description Hide DescriptionUsing the generalized method of steepest descents for the case of two coalescing saddle points, we derive an asymptotic expression for the bivariate generating function of Dyck paths, weighted according to their length and their area in the limit of the area generating variable tending towards 1. The result is valid uniformly for a range of the length generating variable, including the tricritical point of the model.
 Methods of Mathematical Physics

A unified construction for the algebrogeometric quasiperiodic solutions of the LotkaVolterra and relativistic LotkaVolterra hierarchy
View Description Hide DescriptionIn this paper, a new type of integrable differentialdifference hierarchy, namely, the generalized relativistic LotkaVolterra (GRLV) hierarchy, is introduced. This hierarchy is closely related to LotkaVolterra lattice and relativistic LotkaVolterra lattice, which allows us to provide a unified and effective way to obtain some exact solutions for both the LotkaVolterra hierarchy and the relativistic LotkaVolterra hierarchy. In particular, we shall construct algebrogeometric quasiperiodic solutions for the LV hierarchy and the RLV hierarchy in a unified manner on the basis of the finite gap integration theory.

Integrable semidiscretization of a multicomponent short pulse equation
View Description Hide DescriptionIn the present paper, we mainly study the integrable semidiscretization of a multicomponent short pulse equation. First, we briefly review the bilinear equations for a multicomponent short pulse equation proposed by Matsuno [J. Math. Phys. 52, 123702 (2011)] and reaffirm its Nsoliton solution in terms of pfaffians. Then by using a Bäcklund transformation of the bilinear equations and defining a discrete hodograph (reciprocal) transformation, an integrable semidiscrete multicomponent short pulse equation is constructed. Meanwhile, its Nsoliton solution in terms of pfaffians is also proved.

Spectral functions for regular SturmLiouville problems
View Description Hide DescriptionIn this paper, we provide a detailed analysis of the analytic continuation of the spectral zeta function associated with onedimensional regular SturmLiouville problems endowed with selfadjoint separated and coupled boundary conditions. The spectral zeta function is represented in terms of a complex integral and the analytic continuation in the entire complex plane is achieved by using the wellknown LiouvilleGreen (or WKB) asymptotic expansion of the eigenfunctions associated with the problem. The analytically continued expression of the spectral zeta function is then used to compute the functional determinant of the SturmLiouville operator and the coefficients of the asymptotic expansion of the associated heat kernel.