Algebraic methods for the Natanzon potentials

It is shown that the Schrödinger equation can be solved by means of spectrum-generating algebra techniques for the most general class of Natanzon potentials based on the SO(2, 1) algebra. This paper describes in detail thelinear spectrum generating algebra method which is then applied to solve the Natanzon confluent potentials, and it is extended to one example with spin-orbit coupling. Further, the method is used to explain in detail how to find the energy spectrum for the Dirac equation with a Coulomb potential. Afterwards thequadratic spectrum generation algebra method is presented, and it is used to solve the most general hypergeometric Natanzon potential: The bound state problem and the corresponding wave functions are given. A simple example further illustrates the use of the quadratic method.Partially supported by grant FONDECYT 90-1240.