Superintegrability and higher order constants for quantum systems

TL;DR

This paper refines a method to identify higher order symmetry operators in quantum systems on 2D manifolds, demonstrating quantum superintegrability for specific potentials previously known to be classically superintegrable.

Contribution

It introduces a canonical form approach for symmetry operators, proving quantum superintegrability for certain potentials across all rational parameters, extending classical results.

Findings

Established quantum superintegrability for specific 2D potentials for all rational k

Developed a constructive proof using canonical operator method

Created a classical analog of the quantum canonical form

Abstract

We refine a method for finding a canonical form for symmetry operators of arbitrary order for the Schroedinger eigenvalue equation on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. As examples we treat two potentials with parameter k (one of which is the Tremblay, Turbiner, and Winternitz system) that have been shown to be classically superintegrable for all rational numbers k. We apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k. We also develop the classical analog of the quantum canonical form for a symmetry. It is clear that our methods will generalize to other Hamiltonian systems.