A new way to classify 2D higher order quantum superintegrable systems

TL;DR

This paper revises a method for classifying higher order superintegrable systems on 2D Riemannian manifolds, providing new potential classifications and extending previous results to curved spaces like the 2-sphere and hyperboloid.

Contribution

It introduces a revised method for constructing symmetry operators, applies it to classify potentials on curved spaces, and extends the understanding of superintegrable systems beyond Euclidean space.

Findings

Classified potentials on the 2-sphere and hyperboloid that separate in specific coordinates.

Identified the Painlevé VI potential in new superintegrable systems on curved spaces.

Extended the classification of superintegrable systems to include non-Euclidean geometries.

Abstract

We revise a method by Kalnins, Kress and Miller (2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schr\"odinger eigenvalue equation $H\Psi \equiv (\Delta_2 +V)\Psi=E\Psi$ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. We apply the method, as an example, to revisit the Tremblay and Winternitz (2010) derivation of the Painlev\'e VI potential for a 3rd order superintegrable flat space system that separates in polar coordinates and, as new results, we give a listing of the possible potentials on the 2-sphere that separate in spherical coordinates and 2-hyperbolic (two-sheet) potentials separating in horocyclic coordinates. In particular, we show that the Painlev\'e VI potential also appears for a 3rd order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a 3rd order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. Our aim is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.