The Coulomb problem on a 3-sphere and Heun polynomials

TL;DR

This paper investigates the quantum Coulomb problem on a 3-sphere, providing explicit spectra, eigenvalue equations, and wave functions expressed through Heun polynomials, advancing understanding of quantum systems on curved spaces.

Contribution

It introduces a specialized coordinate system for separation of variables and derives explicit solutions and eigenvalues for the Coulomb problem on a 3-sphere.

Findings

Explicit spectrum of the Coulomb problem on a 3-sphere

Eigenvalue equation for the Runge-Lentz vector

Wave functions expressed via Heun polynomials

Abstract

The paper studies the quantum mechanical Coulomb problem on a 3-sphere. We present a special parametrization of the ellipto-spheroidal coordinate system suitable for the separation of variables. After quantization we get the explicit form of the spectrum and present an algebraic equation for the eigenvalues of the Runge-Lentz vector. We also present the wave functions expressed via Heun polynomials.