How to relate the oscillator and Coulomb systems on spheres and pseudospheres?

TL;DR

This paper explores the mathematical relationship between oscillator and Coulomb systems on spheres and pseudospheres, revealing transformations that connect these systems and extend to higher dimensions, including a generalized MIC-Kepler problem.

Contribution

It demonstrates how Bohlin and Kustaanheimo-Stiefel transformations relate oscillator and Coulomb systems on curved spaces, introducing a pseudospheres generalization and a charge-dyon system.

Findings

Oscillator states correspond to Coulomb systems on pseudospheres.

Odd oscillator states relate to Coulomb systems with magnetic flux.

Higher-dimensional generalizations include a pseudospheres MIC-Kepler problem.

Abstract

We show that the oscillators on a sphere and pseudosphere are related, by the so-called Bohlin transformation, with the Coulomb systems on the pseudosphere: the even states of an oscillator yields the conventional Coulomb system on pseudosphere, while the odd states yield the Coulomb system on pseudosphere in the presence of magnetic flux tube generating half spin. In the higher dimensions the oscillator and Coulomb(-like) systems are connected in the similar way. In particular, applying the Kustaanheimo-Stiefel transformation to the oscillators on sphere and pseudosphere, we obtained the preudospherical generalization of MIC-Kepler problem describing three-dimensional charge-dyon system.