Tools for Verifying Classical and Quantum Superintegrability

TL;DR

This paper organizes and proves the superintegrability of various classical and quantum systems, highlighting mechanisms behind their symmetries and extending known results to higher dimensions and new systems.

Contribution

It introduces a large class of superintegrable systems, many new, and provides proofs of their classical and quantum superintegrability, especially for systems with rational parameter conditions.

Findings

Classical superintegrability of many systems proven

Quantum superintegrability established for specific 2D systems

Extension of superintegrability results to higher dimensions

Abstract

Recently many new classes of integrable systems in n dimensions occurring in classical and quantum mechanics have been shown to admit a functionally independent set of 2n-1 symmetries polynomial in the canonical momenta, so that they are in fact superintegrable. These newly discovered systems are all separable in some coordinate system and, typically, they depend on one or more parameters in such a way that the system is superintegrable exactly when some of the parameters are rational numbers. Most of the constructions to date are for n=2 but cases where n>2 are multiplying rapidly. In this article we organize a large class of such systems, many new, and emphasize the underlying mechanisms which enable this phenomena to occur and to prove superintegrability. In addition to proofs of classical superintegrability we show that the 2D caged anisotropic oscillator and a Stackel transformed version on the 2-sheet hyperboloid are quantum superintegrable for all rational relative frequencies, and that a deformed 2D Kepler-Coulomb system is quantum superintegrable for all rational values of a parameter k in the potential.