A New Approach to Analysis of 2D Higher Order Quantum Superintegrable Systems

Abstract

We revise a method by Kalnins et al. (J Phys A Math Theor 43:265205, 2010) for constructing a canonical form for symmetry operators of arbitrary order for the Schrödinger eigenvalue equation HΨ ≡ (Δ ₂ + V )Ψ = EΨ on any 2D Riemannian manifold, real or complex, that admits a separation of variables in some orthogonal coordinate system. Most of this paper is devoted to describing the method. Details will be provided elsewhere. As examples we revisit the Tremblay and Winternitz derivation of the Painlevé VI potential for a third order superintegrable flat space system that separates in polar coordinates and, as new results, we show that the Painlevé VI potential also appears for a third order superintegrable system on the 2-sphere that separates in spherical coordinates, as well as a third order superintegrable system on the 2-hyperboloid that separates in spherical coordinates and one that separates in horocyclic coordinates. The purpose of this project is to develop tools for analysis and classification of higher order superintegrable systems on any 2D Riemannian space, not just Euclidean space.