Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials

A new approach is proposed to general systems possessing SU(1,1)(+)SU(1,1) dynamical symmetry. On the base of this approach, the quadratic Hahn algebra QH(3) is shown to serve as a hidden symmetry (in both quantum and classical pictures) for two potentials generalizing the Hartmann and the oscillator ring-shaped potentials. The overlap coefficients between wavefunctions in spherical and parabolic (cylindrical) coordinates are shown to coincide with Clebsch-Gordan coefficients for SU(1,1) algebra.