A Superintegrable Model With Reflections on S^3 and the Rank Two Bannai-Ito Algebra

Abstract

A quantum superintegrable model with reflections on the three-sphere is presented. Its symmetry algebra is identified with the rank-two Bannai-Ito algebra. It is shown that the Hamiltonian of the system can be constructed from the tensor product of four representations of the superalgebra osp(1|2) and that the superintegrability is naturally understood in that setting. The exact separated solutions are obtained through the Fischer decomposition and a Cauchy-Kovalevskaia extension theorem. This paper is dedicated with admiration and gratitude to Jiří Patera and Pavel Winternitz on the occasion of their 80th birthdays.

Bivariate Bannai-Ito polynomials

“…We have initiated this exploration of the Bannai-Ito polynomials in many variables within the Tratnik framework because of the expected occurence of extensions of that type in the representation theory of the higher rank Bannai-Ito algebra [26] as well as in certain superintegrable models that have been constructed [27,28]. Let us mention the following to be concrete.…”

“…Let us mention the following to be concrete. A Hamiltonian system on the 3-sphere whose symmetries realize the Bannai-Ito algebra of rank 2 has been constructed in [27] and various bases of wavefunctions have been explicitly obtained using the Cauchy-Kovalevskaia extension theorem. It is expected that bivariate Bannai-Ito polynomials arise in the interbasis connection coefficients.…”

A superintegrable model with reflections onSn−1and the higher rank Bannai–Ito algebra

“…We have stressed that the Hamiltonian H with reflections on S n−1 actually commutes with all reflection operators and that these can hence be diagonalized simultaneously with H. In each of the sectors with definite parity, H reduces to a scalar Hamiltonian that extends to S n−1 the generic model on S 2 known to have the Racah algebra as symmetry algebra. It was shown in [5] that these scalar models on S n−1 admit the (more involved) higher rank Racah algebra identified in the same article as the algebra generated by intermediate Casimir operators in the n-fold tensor product of realizations of sl (2). This indicates that, as in the rank 1 case [14], there is an embedding of the Racah algebra in the Bannai-Ito one for higher ranks also.…”

Superintegrability and the dual −1 Hahn algebra in superconformal quantum mechanics

“…While the bivariate Bannai-Ito polynomials have recently been introduced and studied [38], this is not the case for the bivariate complementary Bannai-Ito polynomials from which the bivariate dual −1 Hahn polynomials should descend. With respect to contractions, these three-dimensional singular oscillators should relate to systems on the three sphere obtained by considering the addition of four realizations of osp(1|2) (see [39] in this connection).…”