New superintegrable models on spaces of constant curvature

Abstract

A general class of superintegrable systems on 2D spaces of constant curvature is known for which the potential is not spherically symmetric but allows separation of variables in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalized Kepler potential. Unlike the radial parts, the angular ones are given implicitly. In the present paper new two-parameter families of angular potentials are constructed in terms of elementary functions. It is shown that for appropriate choice of parameters a family corresponding to the oscillator or Kepler type radial potential reduces to the Poschl–Teller potential. This allows considering Hamiltonian systems defined by this family as generalizations of Tremblay–Turbiner–Winternitz (TTW) or Post–Winternitz (PW) models, both on the plane and on curved spaces of constant curvature.