Inverse-root and inverse-root-exponential potentials: Darboux transformations and elementary Darboux partners

Abstract

Abstract The Schrödinger equations for an inverse-root potential and an inverse-root-exponential potential are studied. We derive conditions under which the hypergeometric solutions of the Schrödinger equation degenerate to elementary functions. By means of Darboux transformations we construct new exactly-solvable Schrödinger equations, focusing on bound state solutions and on cases where the transformed potentials are elementary functions.