Exact analytical calculations for Natanzon-class potentials

Local potentials have been used to describe the interactions of the subatomic world ever since the beginnings of quantum mechanics. Practical application of these potentials is addressed. The reason for this is that a large number of analytical results obtained for the simplest (shape-invariant) potentials (like the Coulomb, harmonic oscillator, Poeschl-Teller potentials etc.) can be generalized to the much wider class of Natanzon potentials. The example considered is the generalized Coulomb potential, which is a simultaneous generalization of the Coulomb and harmonic oscillator potentials. The following areas are discussed: S-matrix, Coulomb-Sturmian basis, matrix elements of the Green's operator and an SU(1,1) algebra associated with the problem. (K.A.)