Exact solutions of the Schroedinger equation

Textbooks on quantum mechanics often give the impression that the Schroedinger equation can be solved exactly only for a few simple potential models. However, exact solutions are available in terms of hypergeometric functions and their confluent variants for the so-called Natanzon potentials. These potentials include the Poeschl-Teller, Manning-Rosen and Rosen-Morse potentials that are also special cases of the Eckart potential. The Natanzon potentials are reviewed and connections are made to problems in classical physics like propagation of electromagnetic waves in inhomogeneous media and of acoustic waves in a variable speed profile. The availability of exact solutions is of particular interest for the explicit construction of time evolution operators and in the solution of inverse scattering problems