Four-dimensional double-singular oscillator

The Schrödinger equation for the four-dimensional double-singular oscillator is separable in Eulerian, double-polar, and spheroidal coordinates in ℝ4. It is shown that the coefficients for the expansion of the double-polar basis in terms of the Eulerian basis can be expressed through the Klebsch-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.