Classical and quantum integrability in 3D systems

TL;DR

This paper explores conditions under which three-dimensional classical and quantum systems are integrable, focusing on axial symmetry, ellipsoidal coordinates, and the Kepler problem on a sphere, revealing specific integrability criteria.

Contribution

It analyzes three distinct scenarios for 3D integrability, providing new insights into conditions for classical and quantum integrability in these systems.

Findings

Integrability conditions depend on symmetries and coordinate choices.

Separation of variables is key for non-symmetric systems.

Specific criteria are identified for Kepler problem on a sphere.

Abstract

In this contribution, we discuss three situations in which complete integrability of a three dimensional classical system and its quantum version can be achieved under some conditions. The former is a system with axial symmetry. In the second, we discuss a three dimensional system without spatial symmetry which admits separation of variables if we use ellipsoidal coordinates. In both cases, and as a condition for integrability, certain conditions arise in the integrals of motion. Finally, we study integrability in the three dimensional sphere and a particular case associated with the Kepler problem in $S^3$.