Unified Supersymmetric Description of Shape-Invariant Potentials within and beyond the Natanzon Class

Abstract

The transformations of supersymmetric quantum mechanics are discussed within a formalism that employs a six-parameter function, from which the superpotential and the supersymmetric partner potentials V−(x) and V+(x) are constructed in a general form. By specific choice of the parameters, V−(x) and V+(x) are matched with the general form of PI class potentials and their rationally extended versions. The choice of the parameters also determines which of the four possible SUSY transformations Ti, i=1,…4 is in effect. After this general discussion, the formulae are specified to the three members of this potential class, the Scarf I, Scarf II and generalized Pöschl–Teller potentials. Due to the different domains of definition and their consequences on the boundary conditions, the results turn out to be rather diverse for the three potentials, while the mathematical formalism and the network of the potentials interconnected by the SUSYQM transformations still remains common to a large extent. The general framework allows a unified and consistent interpretation of earlier isolated findings. It also helps to connect the results to further potential classes and to place them into a more general context within the zoo of exactly solvable potentials.