The Hahn superalgebra and supersymmetric Dunkl oscillator models

Genest, Vincent X.; Lemay, Jean-Michel; Vinet, Luc; Zhedanov, Alexei

doi:10.1088/1751-8113/46/50/505204

Cited by 18 publications

(13 citation statements)

References 41 publications

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“…where T read as the kinetic energy of a free particle in the absence of Dunkl space, and it is similar to what is given by the classical state which has been used. However, since the three-dimensional Dunkl oscillator Hamiltonain (1) contains first order terms in the derivative, let us consider the following gauge transformation introduced in [28] for each dimension separately…”

Section: Superintegrabilitymentioning

confidence: 99%

Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

Dong,

Najafizade,

Panahi

et al. 2021

Preprint

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This paper has studied the three-dimensional Dunkl oscillator models in a generalization of superintegrable Euclidean Hamiltonian systems to curved ones. These models are defined based on curved Hamiltonians, which depend on a deformation parameter of underlying space and involve reflection operators. Their symmetries are obtained by the Jordan-Schwinger representations in the family of the Cayley-Klein orthogonal algebras using the creation and annihilation operators of the dynamical sl −1 (2) algebra of the one-dimensional Dunkl oscillator. The resulting algebra is a deformation of so κ 1 κ 2 (4) with reflections, which is known as the Jordan-Schwinger-Dunkl algebra jsd κ 1 κ 2 (4). Hence, this model is shown to be maximal superintegrable. On the other hand, the superintegrability of the three-dimensional Dunkl oscillator model is studied from the factorization approach viewpoint. The spectrum of this system is derived through the separation of variables in geodesic polar coordinates, and the resulting eigenfunctions are algebraically given in terms of Jacobi polynomials.

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Section: Superintegrabilitymentioning

confidence: 99%

Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

Dong,

Najafizade,

Panahi

et al. 2021

Preprint

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“…(iv) Rescaling s i → rs i and taking the limit as r → ∞ gives the Hamiltonian of the Dunkl oscillator [27,33] H = −[D 2 x 1 + D 2 x 2 ] + µ 2 3 (x 2 1 + x 2 2 ), after appropriate renormalization; see also [34,35,36].…”

Section: A Superintegrable Model On S 2 With Bannai-ito Symmetrymentioning

confidence: 99%

“…The first step is to obtain the spectra of the intermediate Casimir operators. Simple considerations based on the nature of the sl −1 (2) representation show that the eigenvalues q 12 and q 23 of Q 12 and Q 23 take the form [29,30,28,20]:…”

Section: The Racah Problem For Sl −1 (2) and The Bannai-ito Algebramentioning

confidence: 99%

The Bannai-Ito algebra and some applications

Bie

Genest

Tsujimoto

et al. 2015

J. Phys.: Conf. Ser.

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The Bannai-Ito algebra is presented together with some of its applications. Its relations with the Bannai-Ito polynomials, the Racah problem for the sl −1 (2) algebra, a superintegrable model with reflections and a Dirac-Dunkl equation on the 2-sphere are surveyed. the systems in which they arise. This can be represented by the following virtuous circle: Exact solvabilityThe classical path is the following: start with a model, find its symmetries, determine how these symmetries are mathematically described, work out the representations of that mathematical structure and obtain its relation to special functions to arrive at the solution of the model. However, one can profitably start from any node on this circle. For instance, one can identify and characterize new special functions, determine the algebraic structure they encode, look for models that have this structure as symmetry algebra and proceed to the solution. In this paper, the following path will be taken:The outline of the paper is as follows. In section 2, the Bannai-Ito algebra is introduced and some of its special cases are presented. In section 3, a realization of the Bannai-Ito algebra in terms of discrete shift and reflection operators is exhibited. The Bannai-Ito polynomials and their properties are discussed in section 4. In section 5, the Bannai-Ito algebra is used to derive the recurrence relation satisfied by the Bannai-Ito polynomials. In section 6, the paraboson algebra and the sl −1 (2) algebra are introduced. In section 7, the realization of sl −1 (2) in terms of Dunkl operators is discussed. In section 8, the Racah problem for sl −1 (2) and its relation with the Bannai-Ito algebra is examined. A superintegrable model on the 2-sphere with Bannai-Ito symmetry is studied in section 9. In section 10, a Dunkl-Dirac equation on the 2-sphere with Bannai-Ito symmetry is discussed. A list of open questions is provided in lieu of a conclusion.

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“…One-dimensional quantum systems involving the Dunkl operator or reflection operators were studied in the context of Calogero models and supersymmetric quantum mechanics [13,14,15,16,17,18,19]. In recent years, a 2D superintegrable system involving reflection operators has been discovered and studied [20] and a series of papers introduced isotropic 2D and 3D Dunkl and singular Dunkl oscillators and a particular case of anisotropic with frequency 2 : 1 [20,21,22,23,24,26]. These works made interesting connections between the obtained quadratic algebras and orthogonal polynomials of various types such as Jacobi-Dunkl polynomials and dual-1 Hahn polynomials.…”

Section: Introductionmentioning

confidence: 99%

Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum

Isaac¹,

Marquette²

2016

J. Phys. A: Math. Theor.

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We generalise the construction of integrals of motion for quantum superintegrable models and the deformed oscillator algebra approach. This is presented in the context of 1D systems admitting ladder operators satisfying a parabosonic algebra involving reflection operators and more generally c λ extended oscillator algebras with grading. We apply the construction on two-dimensional c λ oscillators. We also introduce two new superintegrable Hamiltonians that are the anisotropic Dunkl and the singular Dunkl oscillators. We construct the integrals and using this extended approach of the Daskaloyannis method with grading and we present an algebraic derivation of the energy spectrum of the two models from the finite dimensional unitary representations and show how their spectrum divides into different sectors and relates to the physical spectrum.

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The Hahn superalgebra and supersymmetric Dunkl oscillator models

Cited by 18 publications

References 41 publications

Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

Superintegrability on the Dunkl oscillator model in three-Dimensional spaces of constant curvature

The Bannai-Ito algebra and some applications

Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum

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