The dynamical algebra of the generic superintegrable model on the two-sphere

TL;DR

This paper identifies the Jacobi algebra as the dynamical algebra of a superintegrable model on the two-sphere, providing an algebraic solution and expressing wavefunctions via two-variable Jacobi polynomials.

Contribution

It introduces the Jacobi algebra as the dynamical algebra for the superintegrable model and derives its exact solution algebraically from an embedding in ^{\u00b3}.

Findings

Wavefunctions expressed in terms of two-variable Jacobi polynomials.

Algebraic derivation of the exact solution.

Identification of the Jacobi algebra as the dynamical algebra.

Abstract

The rank two Jacobi algebra $\mathfrak{J}_2$ is identified as the dynamical algebra of the generic quadratic superintegrable model on the two-sphere. The physical representation of this algebra is obtained from its embedding in $\mathfrak{su}(1,1)^{\otimes 3}$. The exact solution of the model is derived algebraically from this representation. The wavefunctions are found to be expressed in terms of two-variable Jacobi polynomials whose characterization is a by-product of the algebraic treatment of the model.