Superintegrable Systems with a Third Order Integrals of Motion

TL;DR

This paper reviews 2D superintegrable systems with third order integrals, highlighting differences between classical and quantum cases, and presents new results on their use in both frameworks.

Contribution

It introduces new findings on classical and quantum third order integrals in superintegrable systems, emphasizing their distinct roles in classical and quantum mechanics.

Findings

Classical and quantum third order integrals differ significantly.

New methods for using third order integrals are developed.

Insights into superintegrability in 2D systems are provided.

Abstract

Two-dimensional superintegrable systems with one third order and one lower order integral of motion are reviewed. The fact that Hamiltonian systems with higher order integrals of motion are not the same in classical and quantum mechanics is stressed. New results on the use of classical and quantum third order integrals are presented in Section 5 and 6.