Four-dimensional double singular oscillator

TL;DR

This paper analyzes the four-dimensional double singular oscillator, demonstrating separability in multiple coordinate systems and deriving analytical expressions for basis expansion coefficients using group theory.

Contribution

It introduces new analytical expressions for basis expansion coefficients and establishes recursion relations for spheroidal basis expansions in the context of this oscillator.

Findings

Coefficients for double polar to Eulerian basis expansion expressed via SU(2) Clebsch-Gordan coefficients.

Spheroidal basis expansion coefficients satisfy three-term recursion relations.

Separable solutions in Eulerian, double polar, and spheroidal coordinates.

Abstract

The Schr\"odinger equation for the four-dimensional double singular oscillator is separable in Eulerian, doble polar and spheroidal coordinates in ${\rm I R}^4$. It is shown that the coefficients for the expansion of double polar basis in terms of the Eulerian basis can be expressed through the Clebsch-Gordan coefficients of 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 Eulerian and double polar bases are proved to satisfy three-term recursion relations.