More on superintegrable models on spaces of constant curvature

TL;DR

This paper explores superintegrable models on 2D spaces of constant curvature, introducing a new family of angular potentials for Kepler-type radial potentials, including a special case of the spherical Higgs oscillator.

Contribution

It constructs a new two-parameter family of angular potentials in superintegrable systems, expanding the understanding of models with Kepler-type radial potentials on curved spaces.

Findings

Introduced a new family of angular potentials in superintegrable models.

Reduced the family to the asymmetric spherical Higgs oscillator for specific parameters.

Provided explicit elementary function expressions for these potentials.

Abstract

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalised Kepler potential. The angular components, on the contrary, are given implicitly by a transcendental, in general, equation. In the present note, devoted to the previously less studied models with the radial potential of the generalised Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.