On a Generalized Kepler-Coulomb System: Interbasis Expansions

TL;DR

This paper explores a generalized Kepler-Coulomb system, demonstrating variable separation in prolate spheroidal coordinates and analyzing interbasis expansion coefficients, revealing connections to SU(2) Clebsch-Gordan coefficients and recursion relations.

Contribution

It introduces a generalized Kepler-Coulomb system, studies interbasis expansions, and links expansion coefficients to analytically continued SU(2) Clebsch-Gordan coefficients.

Findings

Coefficients for parabolic-spherical basis expansion relate to SU(2) Clebsch-Gordan coefficients.

Spheroidal basis expansion coefficients satisfy three-term recursion relations.

System allows separation of Schrödinger equation in prolate spheroidal coordinates.

Abstract

This paper deals with a dynamical system that generalizes the Kepler-Coulomb system and the Hartmann system. It is shown that the Schr\"odinger equation for this generalized Kepler-Coulomb system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic and spheroidal) are studied in detail. It is found that the coefficients for the expansion of the parabolic basis in terms of the spherical basis, and vice-versa, can be expressed through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the spheroidal basis in terms of the spherical and parabolic bases are proved to satisfy three-term recursion relations.