Generalized MICZ-Kepler systems on three-dimensional sphere and hyperboloid

TL;DR

The authors extend the MICZ-Kepler framework to curved three-dimensional spaces by introducing generalized MICZ-Kepler potentials on the sphere and the hyperboloid with a Dirac monopole, described via a conformal metric g(r). They solve the quantum problem by separation of variables in hyperspherical coordinates, obtaining explicit energy spectra that depend on two quantum numbers and closed-form normalized radial and angular wavefunctions. They show these systems are minimally superintegrable and discuss limiting connections to the standard curved-space Kepler problem and the flat MICZ-Kepler system, as well as avenues for higher-dimensional and symmetry-reduced generalizations. The work thus broadens the class of exactly solvable, superintegrable systems in curved spaces and sets the stage for perturbations and extensions that preserve integrability.

Abstract

We propose analogs of the generalized MICZ-Kepler system on the three-dimensional sphere and (two-sheet) hyperboloid. We then construct their energy spectra and normalized wave functions, concluding that the suggested systems are minimally superintegrable.