PT Symmetry in Natanzon-class Potentials

The conditions under which P T symmetry can be applied to the general six-parameter Natanzon potential class are investigated. For this the transformation of the differential equation of the Jacobi polynomials to the Schrödinger equation is considered in its most general form. The parity and P T -parity properties of the y ( x ) function that is responsible for the transformation are studied in order to implement the P T symmetry of the potential V ( x ). Situations in which the bound-state energy eigenvalues can or cannot become complex are identified. A number of known Natanzon-class potentials are analyzed. As a by-product, the relation of two variable transformation methods is clarified.