Third order superintegrable systems separating in polar coordinates

TL;DR

This paper classifies quantum and classical superintegrable systems in two-dimensional space that separate in polar coordinates and possess a third-order integral of motion, discovering new systems involving Painlevé transcendents and elliptic functions.

Contribution

It provides a complete classification of such systems and introduces new quantum superintegrable potentials related to special functions.

Findings

New quantum superintegrable systems with Painlevé VI potentials

Identification of systems involving Weierstrass elliptic functions

Complete classification of third-order superintegrable systems in polar coordinates

Abstract

A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlev\'e transcendent or in terms of the Weierstrass elliptic function.