Four-dimensional double singular oscillator

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.