Superintegrable Systems in Darboux spaces

TL;DR

This paper classifies all superintegrable potentials with quadratic integrals of motion in four Darboux spaces, revealing their equivalence to known potentials in complex Euclidean and spherical spaces.

Contribution

It exhaustively finds and classifies superintegrable potentials in Darboux spaces, extending the understanding beyond constant curvature spaces.

Findings

All such potentials are equivalent to known superintegrable potentials in Euclidean or spherical spaces.

The classification is exhaustive for potentials with at least two quadratic integrals of motion.

Results are summarized in comprehensive tables.

Abstract

Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via "coupling constant metamorphosis" (or equivalently, via Staeckel multiplier transformations). We present tables of the results.