The Coulomb-Oscillator Relation on n-Dimensional Spheres and Hyperboloids

TL;DR

This paper establishes a mathematical relation between Coulomb and oscillator systems on n-dimensional spheres and hyperboloids, enabling the derivation of Coulomb energy spectra and wave functions from oscillator solutions.

Contribution

It introduces a novel relation connecting Coulomb and oscillator problems on curved spaces, extending known Euclidean results to n-dimensional spheres and hyperboloids.

Findings

Quasiradial equations for Coulomb and oscillator systems coincide on curved spaces.

Energy spectra and wave functions for Coulomb systems are derived from oscillator solutions.

The relation holds for dimensions n ≥ 2.

Abstract

In this paper we establish a relation between Coulomb and oscillator systems on $n$-dimensional spheres and hyperboloids for $n\geq 2$. We show that, as in Euclidean space, the quasiradial equation for the $n+1$ dimensional Coulomb problem coincides with the $2n$-dimensional quasiradial oscillator equation on spheres and hyperboloids. Using the solution of the Schr\"odinger equation for the oscillator system, we construct the energy spectrum and wave functions for the Coulomb problem.