Infinite families of position-dependent mass Schrödinger equations with known ground and first excited states

A construction method of infinite families of quasi-exactly solvable position-dependent mass Schrödinger equations with known ground and first excited states is proposed in a deformed supersymmetric background. Such families correspond to extensions of known potentials endowed with a deformed shape invariance property. Two different approaches are combined. The first one is a generating function method, which enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential, from some generating function $W_{+}\left(x\right)$ [and its accompanying function $W_{-}\left(x\right)$]. The second approach is the conditionally deformed shape invariance method, wherein the deformed shape invariance property of the starting potentials is generalized to their extensions by adding some constraints on the parameters and by imposing compatibility conditions between sets of constraints. Detailed results are given for some extensions of the linear and radial harmonic oscillators, as well as the Kepler–Coulomb and Morse potentials.