Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials

TL;DR

This paper classifies all standard potentials in 2D Euclidean space that allow separation in polar coordinates and admit a fourth-order integral, revealing new quantum superintegrable systems beyond known models.

Contribution

It provides a complete characterization of such potentials, including new quantum superintegrable systems with classical limits of free motion.

Findings

Classical potentials match TTW and PW models.

New quantum superintegrable systems identified.

Classical limit corresponds to free motion.

Abstract

Superintegrable Hamiltonian systems in a two-dimensional Euclidean space are considered. We present all real standard potentials that allow separation of variables in polar coordinates and admit an independent fourth-order integral of motion. The general form of the potentials satisfies a linear ODE. In the classical case, the standard potentials coincide with the Tremblay-Turbiner-Winternitz (TTW) or Post-Winternitz (PW) models. In the quantum case new superintegrable systems are obtained, in addition to the TTW and PW ones. Their classical limit is free motion.