The generic quantum superintegrable system on the sphere and Racah operators

TL;DR

This paper explores the symmetry algebra of a quantum superintegrable system on the sphere, revealing connections to Racah operators, multivariable Jacobi polynomials, and Lie algebra representations, with implications for bispectral properties.

Contribution

It introduces a new Lie-theoretic framework for understanding multivariable Racah operators within quantum superintegrable systems on the sphere.

Findings

Symmetry operators form a representation of the Kohno-Drinfeld Lie algebra.

Jacobi polynomials diagonalize Gaudin subalgebras in the system.

Multivariable Racah operators describe the action of symmetry generators.

Abstract

We consider the generic quantum superintegrable system on the $d$-sphere with potential $V(y)=\sum_{k=1}^{d+1}\frac{b_k}{y_k^2}$, where $b_k$ are parameters. Appropriately normalized, the symmetry operators for the Hamiltonian define a representation of the Kohno-Drinfeld Lie algebra on the space of polynomials orthogonal with respect to the Dirichlet distribution. The Gaudin subalgebras generated by Jucys-Murphy elements are diagonalized by families of Jacobi polynomials in $d$ variables on the simplex. We define a set of generators for the symmetry algebra and we prove that their action on the Jacobi polynomials is represented by the multivariable Racah operators introduced in arXiv:0705.1469. The constructions also yield a new Lie-theoretic interpretation of the bispectral property for Tratnik's multivariable Racah polynomials.