Heisenberg-type higher order symmetries of superintegrable systems separable in cartesian coordinates

TL;DR

This paper characterizes a broad class of classical and quantum superintegrable systems with higher order Heisenberg-type symmetries, focusing on those separable in Cartesian coordinates, and explores their integrability and classical-quantum correspondence.

Contribution

It provides a comprehensive characterization of higher order Heisenberg-type symmetries in superintegrable systems separable in Cartesian coordinates, extending previous specific cases to a general framework.

Findings

Quantum symmetries reduce to classical analogs as 0 a0a0

Some systems are only defined in regions of 0a0a0

Identifies integrability properties of these superintegrable systems

Abstract

Heisenberg-type higher order symmetries are studied for both classical and quantum mechanical systems separable in cartesian coordinates. A few particular cases of this type of superintegrable systems were already considered in the literature, but here they are characterized in full generality together with their integrability properties. Some of these systems are defined only in a region of $\mathbb R^n$, and in general they do not include bounded solutions. The quantum symmetries and potentials are shown to reduce to their superintegrable classical analogs in the $\hbar \to0$ limit.