Volume 56, Issue 1, January 2015
Index of content:

A 2 fusion rules" title="A new proof of a formula for the type A 2 fusion rules" />
We give a new proof of a formula for the fusion rules for type A 2 due to Bégin, Mathieu, and Walton. Our approach is to symbolically evaluate the KacWalton algorithm.
 ARTICLES

 Partial Differential Equations

A note on a strongly damped wave equation with fast growing nonlinearities
View Description Hide DescriptionA strongly damped wave equation including the displacement depending nonlinear damping term and nonlinear interaction function is considered. The main aim of the note is to show that under the standard dissipativity restrictions on the nonlinearities involved, the initial boundary value problem for the considered equation is globally wellposed in the class of sufficiently regular solutions and the semigroup generated by the problem possesses a global attractor in the corresponding phase space. These results are obtained for the nonlinearities of an arbitrary polynomial growth and without the assumption that the considered problem has a global Lyapunov function.

Eigenvalues variations for AharonovBohm operators
View Description Hide DescriptionWe study how the eigenvalues of a magnetic Schrödinger operator of AharonovBohm type depend on the singularities of its magnetic potential. We consider a magnetic potential defined everywhere in ℝ^{2} except at a finite number of singularities, so that the associated magnetic field is zero. On a fixed planar domain, we define the corresponding magnetic Hamiltonian with Dirichlet boundary conditions and study its eigenvalues as functions of the singularities. We prove that these functions are continuous, and in some cases even analytic. We sketch the connection of this eigenvalue problem to the problem of finding spectral minimal partitions of the domain.

Solutions and reductions for radiative energy transport in laserheated plasma
View Description Hide DescriptionA full symmetry classification is given for models of energy transport in radiant plasma when the mass density is spatially variable and the diffusivity is nonlinear. A systematic search for conservation laws also leads to some potential symmetries and to an integrable nonlinear model. Classical point symmetries, potential symmetries, and nonclassical symmetries are used to effect variable reductions and exact solutions. The simplest timedependent solution is shown to be stable and relevant to a closed system.
 Representation Theory and Algebraic Methods

Oneparameter formal deformations of HomLieYamaguti algebras
View Description Hide DescriptionThis paper studies oneparameter formal deformations of HomLieYamaguti algebras. The first, second, and third cohomology groups on HomLieYamaguti algebras extending ones on LieYamaguti algebras are provided. It is proved that first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations, and rigidity. However, the third cohomology group is not suitable for the obstructions.

Some remarks on representations of YangMills algebras
View Description Hide DescriptionIn this article, we present some new properties of representations of YangMills algebras. We first show that any free Lie algebra with m generators is a quotient of the YangMills algebra 𝔶𝔪(n) on n generators, for n ≥ 2m. We derive from this that any semisimple Lie algebra and even any affine KacMoody algebra is a quotient of 𝔶𝔪(n) for n ≥ 4. Combining this with previous results on representations of YangMills algebras given in [Herscovich and Solotar, Ann. Math. 173(2), 1043–1080 (2011)], one may obtain solutions to the YangMills equations by differential operators acting on sections of twisted vector bundles on the affine space of dimension n ≥ 4 associated to representations of any semisimple Lie algebra. We also show that this quotient property does not hold for n = 3, since any morphism of Lie algebras from 𝔶𝔪(3) to 𝔰𝔩(2, k) has in fact solvable image.

A new proof of a formula for the type A 2 fusion rules
View Description Hide DescriptionWe give a new proof of a formula for the fusion rules for type A 2 due to Bégin, Mathieu, and Walton. Our approach is to symbolically evaluate the KacWalton algorithm.

Twoparameter quantum affine algebra of type , Drinfeld realization and vertex representation
View Description Hide DescriptionIn this paper, we define the twoparameter quantum affine algebra for type and give the (r, s)Drinfeld realization of , as well as establish and prove its Drinfeld isomorphism. We construct and verify explicitly the levelone vertex representation of twoparameter quantum affine algebra , which also supports an evidence in nontwisted type for the uniform defining approach via the twoparameter τinvariant generating functions proposed in Hu and Zhang [Generating functions with τinvariance and vertex representations of twoparameter quantum affine algebras : Simply laced cases eprint arXiv:1401.4925]

A 2categorical state sum model
View Description Hide DescriptionIt has long been argued that higher categories provide the proper algebraic structure underlying state sum invariants of 4manifolds. This idea has been refined recently, by proposing to use 2groups and their representations as specific examples of 2categories. The challenge has been to make these proposals fully explicit. Here, we give a concrete realization of this program. Building upon our earlier work with Baez and Wise on the representation theory of 2groups, we construct a fourdimensional state sum model based on a categorified version of the Euclidean group. We define and explicitly compute the simplex weights, which may be viewed a categorified analogue of RacahWigner 6jsymbols. These weights solve a hexagon equation that encodes the formal invariance of the state sum under the Pachner moves of the triangulation. This result unravels the combinatorial formulation of the Feynman amplitudes of quantum field theory on flat spacetime proposed in A. Baratin and L. Freidel [Classical Quantum Gravity 24, 2027–2060 (2007)] which was shown to lead after gaugefixing to Korepanov’s invariant of 4manifolds.

Structure of classical affine and classical affine fractional algebras
View Description Hide DescriptionWe introduce a classical BRST complex (See Definition 3.2.) and show that one can construct a classical affine algebra via the complex. This definition clarifies that classical affine algebras can be considered as quasiclassical limits of quantum affine algebras. We also give a definition of a classical affine fractional algebra as a Poisson vertex algebra. As in the classical affine case, a classical affine fractional algebra has two compatible λbrackets and is isomorphic to an algebra of differential polynomials as a differential algebra. When a classical affine fractional algebra is associated to a minimal nilpotent, we describe explicit forms of free generators and compute λbrackets between them. Provided some assumptions on a classical affine fractional algebra, we find an infinite sequence of integrable systems related to the algebra, using the generalized Drinfel’d and Sokolov reduction.
 ManyBody and Condensed Matter Physics

Landau superfluids as nonequilibrium stationary states
View Description Hide DescriptionWe define a superfluid state to be a nonequilibrium stationary state (NESS), which, at zero temperature, satisfies certain metastability conditions, which physically express that there should be a sufficiently small energymomentum transfer between the particles of the fluid and the surroundings (e.g., pipe). It is shown that two models, the Girardeau model and the HuangYangLuttinger (HYL) model, describe superfluids in this sense and, moreover, that, in the case of the HYL model, the metastability condition is directly related to Nozières’ conjecture that, due to the repulsive interaction, the condensate does not suffer fragmentation into two (or more) parts, thereby assuring its quantum coherence. The models are rigorous examples of NESS in which the system is not finite, but rather a manybody system.
 Quantum Mechanics

Dirac oscillator and nonrelativistic Snyderde Sitter algebra
View Description Hide DescriptionThree dimensional Dirac oscillator was considered in space with deformed commutation relations known as Snyderde Sitter algebra. Snyderde Sitter commutation relations give rise to appearance of minimal uncertainties in position as well as in momentum. To derive energy spectrum and wavefunctions of the Dirac oscillator, supersymmetric quantum mechanics and shape invariance technique were applied.

On spectral deformations and singular Weyl functions for onedimensional Dirac operators
View Description Hide DescriptionWe investigate the connection between singular Weyl–Titchmarsh–Kodaira theory and the double commutation method for onedimensional Dirac operators. In particular, we compute the singular Weyl function of the commuted operator in terms of the data from the original operator. These results are then applied to radial Dirac operators in order to show that the singular Weyl function of such an operator belongs to a generalized Nevanlinna class N κ 0 with , where κ ∈ ℝ is the corresponding angular momentum.

Removal of ordering ambiguity for a class of position dependent mass quantum systems with an application to the quadratic Liénard type nonlinear oscillators
View Description Hide DescriptionWe consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as nonHermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map onedimensional timeindependent position dependent mass Schrödinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For nonHermitian ordering, we show that the class of systems can also be exactly solvable and is also shown to be isospectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and nonHermitian orderings. We illustrate the technique with the quadratic Liénard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

Solution of coupled integral equations for quantum scattering in the presence of complex potentials
View Description Hide DescriptionIn this paper, we present a method to compute solutions of coupled integral equations for quantum scattering problems in the presence of a complex potential. We show how the elastic and absorption cross sections can be obtained from the numerical solution of these equations in the asymptotic region at large radial distances.

On quantized Liénard oscillator and momentum dependent mass
View Description Hide DescriptionWe examine the analytical structure of the nonlinear Liénard oscillator and show that it is a biHamiltonian system depending upon the choice of the coupling parameters. While one has been recently studied in the context of a quantized momentumdependent mass system, the other Hamiltonian also reflects a similar feature in the mass function and also depicts an isotonic character. We solve for such a Hamiltonian and give the complete solution in terms of a confluent hypergeometric function.

Minimum time optimal synthesis for two level quantum systems
View Description Hide DescriptionFor the time optimal problem of an invariant system on SU (2), with two independent controls and a bound on the norm of the control, the extremals of the Pontryagin maximum principle are explicit functions of time. We use this fact here to perform the optimal synthesis for these systems, i.e., to find all time optimal trajectories. Although the Lie group SU (2) is three dimensional, time optimal trajectories can be described in the unit disk of the complex plane. We find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others. Inside this separatrix circle, another trajectory (the critical trajectory) plays an important role in that all optimal trajectories end at an intersection with this curve. The results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system.

Quantum systems with positiondependent mass and spinorbit interaction via Rashba and Dresselhaus terms
View Description Hide DescriptionWe consider a particle with spin 1/2 with positiondependent mass moving in a plane. Considering separately Rashba and Dresselhaus spinorbit interactions, we write down the Hamiltonian for this problem and solve it for Dirichlet boundary conditions. Our radial wavefunctions have two contributions: homogeneous ones which are written as Bessel functions of noninteger orders—that depend on angular momentum m—and particular solutions which are obtained after decoupling the nonhomogeneous system. In this process, we find nonhomogeneous Bessel equation, Laguerre, as well as biconfluent Heun equation. We also present the probability densities for m = 0, 1, 2 in an annular quantum well. Our results indicate that the background as well as the spinorbit interaction naturally splits the spinor components.
 Quantum Information and Computation

Exponential of a matrix, a nonlinear problem, and quantum gates
View Description Hide DescriptionWe describe solutions of the matrix equation exp(z(A − In )) = A, where z ∈ ℂ. Applications in quantum computing are given. Both normal and nonnormal matrices are studied. For normal matrices, the Lambert Wfunction plays a central role.

Quantum chisquared and goodness of fit testing
View Description Hide DescriptionA quantum mechanical hypothesis test is presented for the hypothesis that a certain setup produces a given quantum state. Although the classical and the quantum problems are very much related to each other, the quantum problem is much richer due to the additional optimization over the measurement basis. A goodness of fit test for i.i.d quantum states is developed and a maxmin characterization for the optimal measurement is introduced. We find the quantum measurement which leads both to the maximal Pitman and Bahadur efficiencies, and determine the associated divergence rates. We discuss the relationship of the quantum goodness of fit test to the problem of estimating multiple parameters from a density matrix. These problems are found to be closely related and we show that the largest error of an optimal strategy, determined by the smallest eigenvalue of the Fisher information matrix, is given by the divergence rate of the goodness of fit test.
 Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

The EinsteinHilbert action with cosmological constant as a functional of generic form
View Description Hide DescriptionThe geometrical underpinnings of a specific class of Dirac operators are discussed. It is demonstrated how this class of Dirac operators allows to relate various geometrical functionals like the YangMills action and the functional of nonlinear σ − models (i.e., of (Dirac) harmonic maps). These functionals are shown to be similar to the EinsteinHilbert action with cosmological constant (EHC). The EHC may thus be regarded as a “generic functional.” As a byproduct, the geometrical setup presented also allows to avoid the issue of “fermion doubling” as usually encountered, for instance, in the geometrical discussion of the Standard Model in terms of Dirac operators. Furthermore, it is demonstrated how the geometrical setup presented allows to derive the cosmological constant term of the EHC from the EinsteinHilbert functional and the action of a purely gauge coupling Higgs field.