A Recurrence Relation Approach to Higher Order Quantum Superintegrability

TL;DR

This paper introduces a recurrence relation method to rigorously prove quantum superintegrability of 2D systems, explicitly construct higher order symmetry generators, and analyze their algebraic structures across multiple families of systems.

Contribution

The paper presents a novel recurrence relation approach for proving superintegrability and constructing symmetry algebras, including new proofs for systems with rational parameter values.

Findings

Proved quantum superintegrability for five families of systems.

Explicitly constructed higher order symmetry generators.

Determined the structure equations of the symmetry algebras.

Abstract

We develop our method to prove quantum superintegrability of an integrable 2D system, based on recurrence relations obeyed by the eigenfunctions of the system with respect to separable coordinates. We show that the method provides rigorous proofs of superintegrability and explicit constructions of higher order generators for the symmetry algebra. We apply the method to 5 families of systems, each depending on a parameter $k$, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system, and we give proofs of quantum superintegrability for all rational values of $k$, new for 4 of these systems. In addition, we show that the explicit information supplied by the special function recurrence relations allows us to prove, for the first time in 4 cases, that the symmetry algebra generated by our lowest order symmetries closes and to determine the associated structure equations of the algebras for each $k$. We have no proof that our generating symmetries are of lowest possible order, but we have no counterexamples, and we are confident we can can always find any missing generators from our raising and lowering operator recurrences. We also get for free, one variable models of the action of the symmetry algebra in terms of difference operators. We describe how the St\"ackel transform acts and show that it preserves the structure equations.