Shape Invariant Potentials in Supersymmetric Quantum Cosmology

In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of N = 2 supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schrodinger-Wheeler-DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersymmetric partner Hamiltonians, H-1 and H-2. Moreover, we point out how the shape invariance property can be employed to bring a relation among several factor orderings choices for our Schrodinger-Wheeler-DeWitt equation. The ground state is retrieved, and the excited states easily written. Finally, the Hamiltonians, H-1 and H-2, are explicitly presented within a N = 2 supersymmetric quantum mechanics framework.