Deformed su(1,1) Algebra as a Model for Quantum Oscillators

Jafarov, E. I.; Stoilova, N. I.; Jeugt, J. Van der

doi:10.3842/sigma.2012.025

Cited by 9 publications

(9 citation statements)

References 32 publications

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“…Substantial work has been done on these systems which are now well understood and classified (see [4,37,16,17,18,19,20,21,30] and references therein). Further developments in the study of integrable systems include progress in the classification of superintegrable systems with higher order symmetry [22,38,39], the examination of discrete/finite superintegrable models [25] and the exploration of systems involving reflection operators [7,14,15,27,28,29,31,32,34,35].…”

Section: Introductionmentioning

confidence: 99%

The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients

Genest

Ismail

Vinet

et al. 2013

J. Phys. A: Math. Theor.

102

143

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Abstract. The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra generated by the constants of motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie algebra u(2) with involutions. The system admits separation of variables in both Cartesian and polar coordinates. The separated wavefunctions are respectively expressed in terms of generalized Hermite polynomials and products of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi-Dunkl polynomials appear as eigenfunctions of the symmetry operator responsible for the separation of variables in polar coordinates. The expansion coefficients between the Cartesian and polar bases (overlap coefficients) are given as linear combinations of dual −1 Hahn polynomials. The connection with the Clebsch-Gordan problem of the sl −1 (2) algebra is explained.The Dunkl oscillator in the plane I 2

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Section: Introductionmentioning

confidence: 99%

The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients

Genest

Ismail

Vinet

et al. 2013

J. Phys. A: Math. Theor.

102

143

View full text Add to dashboard Cite

show abstract

“…In view of the special properties and applications of superintegrable models, there is considerable interest in enlarging the set of documented systems with this property. Recent advances in this perspective include the study of superintegrable systems with higher order symmetries [25,26,34,50,51], the construction of new superintegrable models from exceptional polynomials [35,42], the search for discretized superintegrable systems [36] and the examination of models described by Hamiltonians involving reflection operators [10,11,21,22,40,41,43,44].…”

Section: Introductionmentioning

confidence: 99%

The singular and the 2:1 anisotropic Dunkl oscillators in the plane

Genest¹,

Vinet²,

Zhedanov³

2013

J. Phys. A: Math. Theor.

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Abstract. Two Dunkl oscillator models are considered: one singular and the other with a 2 : 1 frequency ratio. These models are defined by Hamiltonians which include the reflection operators in the two variables x and y. The singular or caged Dunkl oscillator is second-order superintegrable and admits separation of variables in both Cartesian and polar coordinates. The spectrum of the Hamiltonian is obtained algebraically and the separated wavefunctions are given in the terms of Jacobi, Laguerre and generalized Hermite polynomials. The symmetry generators are constructed from the su(1, 1) dynamical operators of the one-dimensional model and generate a cubic symmetry algebra. In terms of the symmetries responsible for the separation of variables, the symmetry algebra of the singular Dunkl oscillator is quadratic and can be identified with a special case of the Askey-Wilson algebra AW (3) with central involutions. The 2 : 1 anisotropic Dunkl oscillator model is also second-order superintegrable. The energies of the system are obtained algebraically, the symmetry generators are constructed using the dynamical operators and the resulting symmetry algebra is quadratic. The general system appears to admit separation of variables only in Cartesian coordinates. Special cases where separation occurs in both Cartesian and parabolic coordinates are considered. In the latter case the wavefunctions satisfy the biconfluent Heun equation and depend on a transcendental separation constant.

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“…The Casimir operator of H then takes the form 5) which is clearly a simple one-parameter deformation of the standard sl(2) Casimir operator. Interestingly, one-parameter versions of this algebra (with either ν or χ equal to zero) have been introduced in studies of finite analogues of the parabosonic oscillator 6,7 . The wave functions that were found in this context turn out to be symmetrized dual −1 Hahn polynomials.…”

Section: Resultsmentioning

confidence: 99%

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

Genest

Vinet

Zhedanov

2013

Journal of Mathematical Physics

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The algebra H of the dual −1 Hahn polynomials is derived and shown to arise in the ClebschGordan 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 sl q (2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(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 ClebschGordan 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.MSC: 81R50, 17B37, 33C47, 33C80, 15A30

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Deformed su(1,1) Algebra as a Model for Quantum Oscillators

Cited by 9 publications

References 32 publications

The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients

The Dunkl oscillator in the plane: I. Superintegrability, separated wavefunctions and overlap coefficients

The singular and the 2:1 anisotropic Dunkl oscillators in the plane

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

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