Superintegrability with third order invariants in quantum and classical mechanics

TL;DR

This paper classifies potentials in classical and quantum 2D mechanics that admit both first- and third-order integrals of motion, revealing quantum superintegrable systems with no classical counterparts.

Contribution

It explicitly finds all potentials with first- and third-order integrals and identifies quantum superintegrable systems lacking classical analogs.

Findings

All potentials with such integrals are explicitly determined.

Quantum superintegrable systems without classical analogs are identified.

Potentials proportional to ^2 are found to have no classical limit.

Abstract

We consider here the coexistence of first- and third-order integrals of motion in two dimensional classical and quantum mechanics. We find explicitly all potentials that admit such integrals, and all their integrals. Quantum superintegrable systems are found that have no classical analog, i.e. the potentials are proportional to \hbar^2, so their classical limit is free motion.