The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere

Abstract

A quantum superintegrable model with reflections on the 2-sphere is introduced. Its two algebraically independent constants of motion generate a central extension of the Bannai--Ito algebra. The Schrodinger equation separates in spherical coordinates and its exact solutions are presented. It is further observed that the Hamiltonian of the system arises in the addition of three representations of the sl_{-1}(2) algebra (the dynamical algebra of the one-dimensional parabosonic oscillator). The contraction from the two-sphere to the Euclidean plane yields the Dunkl oscillator in two dimensions and its Schwinger-Dunkl symmetry algebra sd(2).

16 pp; Added references and comments

References in corpus (5)

• Classical and Quantum Superintegrability with Applications
• Two-Variable Wilson Polynomials and the Generic Superintegrable System on the 3-Sphere
• Bispectrality of the Complementary Bannai-Ito Polynomials
• A Bochner Theorem for Dunkl Polynomials
• A "Continuous" Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials

Cited by in corpus (11)

• The Bannai-Ito algebra and some applications
• A Dirac-Dunkl equation on $S^2$ and the Bannai-Ito algebra
• An embedding of the universal Askey-Wilson algebra into $U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)\otimes U_q({\mathfrak{sl}}_2)$
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• The Racah algebra as a commutant and Howe duality
• The quantum superalgebra ${\mathfrak{osp}}_q(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• An embedding of the Bannai-Ito algebra in $\mathcal{U}(\mathfrak{osp}(1,2))$ and $-1$ polynomials
• The equitable presentation of ${\mathfrak{osp}}_q(1|2)$ and a $q$-analog of the Bannai-Ito algebra
• Bivariate Bannai-Ito polynomials
• Superintegrability and the dual $-1$ Hahn algebra in superconformal quantum mechanics