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Feb 2013

Volume 54, Issue 2, Articles (02xxxx)

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J. Math. Phys. 54, 021506 (2013); http://dx.doi.org/10.1063/1.4790887 (24 pages)

Gui-Qiang Chen, Vaibhav Kukreja, and Hairong Yuan
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back to top Methods of Mathematical Physics

Supertransvectants, cohomology, and deformations

Nizar Ben Fraj, Ismail Laraiedh, and Salem Omri

J. Math. Phys. 54, 023501 (2013); http://dx.doi.org/10.1063/1.4789539 (19 pages)

Online Publication Date: 1 February 2013

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Over the (1, N)-dimensional real superspace, N = 2, 3, we classify math(N|2)-invariant binary differential operators acting on the superspaces of weighted densities, where math(N|2) is the orthosymplectic Lie superalgebra. This result allows us to compute the first differential math(N|2)-relative cohomology of the Lie superalgebra K(N) of contact vector fields with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We classify generic formal math(3|2)-trivial deformations of the K(3)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal math(3|2)-trivial deformation of this K(3)-module is equivalent to its infinitesimal part. This work is the simplest generalization of a result by the first author et al. [Basdouri, I., Ben Ammar, M., Ben Fraj, N., Boujelbene, M., and Kammoun, K., “Cohomology of the Lie superalgebra of contact vector fields on 1|1K1|1 and deformations of the superspace of symbols,” J. Nonlinear Math. Phys. 16, 373 (2009)10.1142/S1402925109000431].
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02.10.Ud Linear algebra
02.60.Lj Ordinary and partial differential equations; boundary value problems

Symmetry classification of variable coefficient cubic-quintic nonlinear Schrödinger equations

C. Özemir and F. Güngör

J. Math. Phys. 54, 023502 (2013); http://dx.doi.org/10.1063/1.4789543 (13 pages)

Online Publication Date: 4 February 2013

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A Lie-algebraic classification of the variable coefficient cubic-quintic nonlinear Schrödinger equations involving 5 arbitrary functions of space and time is performed under the action of equivalence transformations. It is shown that the symmetry group can be at most four-dimensional in the case of genuine cubic-quintic nonlinearity. It may be five-dimensional (isomorphic to the Galilei similitude algebra math(1)) when the equation is of cubic type, and six-dimensional (isomorphic to the Schrödinger algebra math(1)) when it is of quintic type.
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03.65.Ge Solutions of wave equations: bound states
02.20.Sv Lie algebras of Lie groups
02.30.Hq Ordinary differential equations
03.65.Fd Algebraic methods

Characterization of compact and self-adjoint operators on free Banach spaces of countable type over the complex Levi-Civita field

José Aguayo, Miguel Nova, and Khodr Shamseddine

J. Math. Phys. 54, 023503 (2013); http://dx.doi.org/10.1063/1.4789541 (19 pages)

Online Publication Date: 5 February 2013

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Let C be the complex Levi-Civita field and let E be a free Banach space over C of countable type. Then E is isometrically isomorphic to c0math: = math, where s:math→(0,∞). If the range of s is contained in |C∖{0}|, it is enough to study c0math, which will be denoted by c0(C) or, simply, c0. In this paper, we define a natural inner product on c0, which induces the sup-norm of c0. Of course, c0 is not orthomodular, so we characterize those closed subspaces of c0 with an orthonormal complement with respect to this inner product; that is, those closed subspaces M of c0 such that c0 = M ⊕ M⊥. Such a subspace, together with its orthonormal complement, defines a special kind of projection, the so-called normal projection. We present a characterization of such normal projections as well as a characterization of another kind of operators, the compact operators on c0.
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02.30.Tb Operator theory

Linearisable ultradiscrete systems with sign variables and the confinement of singularities

N. Mimura, J. Satsuma, A. Ramani, and B. Grammaticos

J. Math. Phys. 54, 023504 (2013); http://dx.doi.org/10.1063/1.4776188 (30 pages)

Online Publication Date: 8 February 2013

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We present the singularity analysis of the ultradiscrete analogue of linearisable mappings of Quispel-Roberts-Thompson (QRT) type. The ultradiscretisation method used here is one which keeps track of signs and thus can be applied without the positivity restrictions of the classical ultradiscretisation approach. We show that in all cases the mappings possess confined singularities. The same is true for two non-autonomous equations, which are equally linearisable in the discrete case. We construct explicitly the solutions of the ultradiscrete mappings analysed here.
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02.30.Hq Ordinary differential equations

Maxwell-Chern-Simons vortices on compact surfaces: Nonequivalence of the first and the second order equations

Jongmin Han and Seongtag Kim

J. Math. Phys. 54, 023505 (2013); http://dx.doi.org/10.1063/1.4790416 (15 pages)

Online Publication Date: 11 February 2013

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In this paper we study the Maxwell-Chern-Simons-Higgs and the Chern-Simons-Higgs vortices on a compact Riemann surface. We establish the existence of a solution of the static Maxwell-Chern-Simons-Higgs vortex equations, which is a minimizer of the static energy functional. This shows the nonequivalence of the first and the second order Maxwell-Chern-Simons-Higgs vortex equations. The nonequivalence is also proved for the Chern-Simons-Higgs vortices by verifying the Chern-Simons limit.
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11.15.Yc Chern-Simons gauge theory
03.50.De Classical electromagnetism, Maxwell equations

The algebra of dual −1 Hahn polynomials and the Clebsch-Gordan problem of sl−1(2)

Vincent X. Genest, Luc Vinet, and Alexei Zhedanov

J. Math. Phys. 54, 023506 (2013); http://dx.doi.org/10.1063/1.4790417 (13 pages)

Online Publication Date: 11 February 2013

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The algebra H of the dual −1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl−1(2). The dual −1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from the q → −1 limit of the dual q-Hahn polynomials. The Hopf algebra sl−1(2) has four generators including an involution, it is also a q → −1 limit of the quantum algebra slq(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of math(2) with an involution as additional generator, is first derived from the recurrence relation of the −1 Hahn polynomials. It is then shown that H can be realized in terms of the generators of two added sl−1(2) algebras, so that the Clebsch-Gordan coefficients of sl−1(2) are dual −1 Hahn polynomials. An irreducible representation of H involving five-diagonal matrices and connected to the difference equation of the dual −1 Hahn polynomials is constructed.
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03.65.Fd Algebraic methods
02.10.De Algebraic structures and number theory
02.30.Hq Ordinary differential equations

Localization in abelian Chern-Simons theory

B. D. K. McLellan

J. Math. Phys. 54, 023507 (2013); http://dx.doi.org/10.1063/1.4790565 (24 pages)

Online Publication Date: 11 February 2013

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Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected, and abelian. The abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. The partition function is then formally computed using the technique of non-abelian localization. This study leads to a natural identification of the abelian Reidemeister-Ray-Singer torsion as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections for the class of Sasakian three-manifolds. The torsion part of the abelian Chern-Simons partition function is computed explicitly in terms of Seifert data for a given Sasakian three-manifold.
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11.15.Yc Chern-Simons gauge theory
02.20.Sv Lie algebras of Lie groups
02.40.Pc General topology

The Poincaré algebra in rank 3 simple Lie algebras

Andrew Douglas, Hubert de Guise, and Joe Repka

J. Math. Phys. 54, 023508 (2013); http://dx.doi.org/10.1063/1.4790415 (18 pages)

Online Publication Date: 15 February 2013

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We classify embeddings of the Poincaré algebra math(3,1) into the rank 3 simple Lie algebras. Up to inner automorphism, we show that there are exactly two embeddings of math(3,1) into math(4,math), which are, however, related by an outer automorphism of math(4,math). Next, we show that there is a unique embedding of math(3,1) into math(7,math), up to inner automorphism, but no embeddings of math(3,1) into math(6,math). All embeddings are explicitly described. As an application, we show that each irreducible highest weight module of math(4,math) (not necessarily finite-dimensional) remains indecomposable when restricted to math(3,1), with respect to any embedding of math(3,1) into math(4,math).
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02.20.Sv Lie algebras of Lie groups
11.30.Cp Lorentz and Poincaré invariance

Damping and pseudo-fermions

F. Bagarello

J. Math. Phys. 54, 023509 (2013); http://dx.doi.org/10.1063/1.4790514 (12 pages)

Online Publication Date: 15 February 2013

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After a short abstract introduction on the time evolution driven by non-self-adjoint Hamiltonians, we show how the recently introduced concept of pseudo-fermion can be used in the description of damping in finite dimensional quantum systems, and we compare the results deduced adopting the Schrödinger and the Heisenberg representations.
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05.30.Fk Fermion systems and electron gas

Spectrum generating algebra for the continuous spectrum of a free particle in Lobachevski space

M. Gadella, J. Negro, G. P. Pronko, and M. Santander

J. Math. Phys. 54, 023510 (2013); http://dx.doi.org/10.1063/1.4791683 (15 pages)

Online Publication Date: 19 February 2013

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In this paper, we construct a Spectrum Generating Algebra (SGA) for a quantum system with purely continuous spectrum: the quantum free particle in a Lobachevski space with constant negative curvature. The SGA contains the geometrical symmetry algebra of the system plus a subalgebra of operators that give the spectrum of the system and connects the eigenfunctions of the Hamiltonian among themselves. In our case, the geometrical symmetry algebra is math(3,1) and the SGA is math(4,2). We start with a representation of math(4,2) by functions on a realization of the Lobachevski space given by a two-sheeted hyperboloid, where the Lie algebra commutators are the usual Poisson-Dirac brackets. Then, we introduce a quantized version of the representation in which functions are replaced by operators on a Hilbert space and Poisson-Dirac brackets by commutators. Eigenfunctions of the Hamiltonian are given and “naive” ladder operators are identified. The previously defined “naive” ladder operators shift the eigenvalues by a complex number so that an alternative approach is necessary. This is obtained by a non-self-adjoint function of a linear combination of the ladder operators, which gives the correct relation among the eigenfunctions of the Hamiltonian. We give an eigenfunction expansion of functions over the upper sheet of a two-sheeted hyperboloid in terms of the eigenfunctions of the Hamiltonian.
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03.65.Fd Algebraic methods
03.65.Ta Foundations of quantum mechanics; measurement theory
03.65.Aa Quantum systems with finite Hilbert space
02.10.Ud Linear algebra
02.20.Sv Lie algebras of Lie groups
02.40.-k Geometry, differential geometry, and topology

On the squared eigenfunction symmetry of the Toda lattice hierarchy

Jipeng Cheng and Jingsong He

J. Math. Phys. 54, 023511 (2013); http://dx.doi.org/10.1063/1.4791702 (14 pages)

Online Publication Date: 19 February 2013

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The squared eigenfunction symmetry for the Toda lattice hierarchy is explicitly constructed in the form of the Kronecker product of the vector eigenfunction and the vector adjoint eigenfunction, which can be viewed as the generating function for the additional symmetries when the eigenfunction and the adjoint eigenfunction are the wave function and the adjoint wave function, respectively. Then after the Fay-like identities and some important relations about the wave functions are investigated, the action of the squared eigenfunction related to the additional symmetry on the tau function is derived, which is equivalent to the Adler-Shiota-van Moerbeke formulas.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.10.Ud Linear algebra

Translationally invariant semi-classical electrodynamics of magnetic media to electric octopole-magnetic quadrupole order

A. Welter, R. E. Raab, and O. L. de Lange

J. Math. Phys. 54, 023512 (2013); http://dx.doi.org/10.1063/1.4791999 (10 pages)

Online Publication Date: 21 February 2013

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We consider semi-classical macroscopic electrodynamics that is translationally invariant (independent of the choice of an arbitrary, implicit set of coordinate origins for molecule-fixed axes) for linear, homogeneous, anisotropic media interacting with harmonic, plane electromagnetic waves. We extend a previous formulation at electric octopole-magnetic quadrupole order to include media comprising magnetic molecules (those possessing both time-even and time-odd properties). This requires two additional invariant, time-odd molecular polarizabilities. Overall, the electrodynamics depends on 10 invariant polarizabilities—5 time even (one each of electric dipole and electric quadrupole–magnetic dipole order, and three of electric octopole-magnetic quadrupole order) and 5 time odd (one, two, and two, respectively)—that are required for the description of linear transmission and reflection phenomena, and material constants. The two additional time-odd polarizabilities account for certain predicted effects, and one of them contributes to the inverse ac permeability of magnetic media. The results are presented in a form that is suitable for numerical computation.
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03.50.De Classical electromagnetism, Maxwell equations
33.15.Kr Electric and magnetic moments (and derivatives), polarizability, and magnetic susceptibility
02.60.-x Numerical approximation and analysis

The generalized additional symmetries of the two-Toda lattice hierarchy

Jipeng Cheng, Ye Tian, Zhaowen Yan, and Jingsong He

J. Math. Phys. 54, 023513 (2013); http://dx.doi.org/10.1063/1.4792479 (18 pages)

Online Publication Date: 21 February 2013

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The generalized additional symmetries of the two-Toda lattice hierarchy are investigated in this paper. The algebraic structure of this generalized additional symmetry is showed as w∞⊗w∞. And the actions of the generalized additional symmetries on τ-function are also discussed, by restricting the two-Toda lattice hierarchy to the semi-infinite case.
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05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.10.-v Logic, set theory, and algebra

Representing elements of the Weyl algebra by labeled trees

Walaa Asakly, Toufik Mansour, and Matthias Schork

J. Math. Phys. 54, 023514 (2013); http://dx.doi.org/10.1063/1.4792655 (17 pages)

Online Publication Date: 25 February 2013

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It is shown how arbitrary elements of the Weyl algebra can be represented by labeled plane trees using normal ordering. Several examples are treated and combinatorial aspects are discussed. Also, possible avenues for future research are described.
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02.10.Ox Combinatorics; graph theory

Poisson bialgebras

Xiang Ni and Chengming Bai

J. Math. Phys. 54, 023515 (2013); http://dx.doi.org/10.1063/1.4792668 (14 pages)

Online Publication Date: 28 February 2013

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We introduce a notion of Poisson bialgebra as an analogue of a Lie bialgebra of Drinfeld. Poisson bialgebras exhibit many familiar properties of Lie bialgebras. In particular, they can be constructed from a combination of the classical Yang-Baxter equation and the associative Yang-Baxter equation and there exists a natural Drinfeld classical double. Moreover, Poisson bialgebras are related to certain algebraic structures and they fit naturally into a framework to construct compatible Poisson brackets in integrable systems.
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02.20.Sv Lie algebras of Lie groups
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