Two-step Shape Invariance in the Framework of N-fold Supersymmetry

TL;DR

This paper explores two-step shape invariance within N-fold supersymmetry, revealing new potentials, their properties, and their conditional solvability, thus advancing the understanding of supersymmetric quantum systems.

Contribution

It demonstrates that two-step shape invariance implies type A 2-fold supersymmetry and introduces new, conditionally two-step shape-invariant potentials beyond known cases.

Findings

Established the link between two-step shape invariance and type A 2-fold supersymmetry.

Constructed several new two-step shape-invariant potentials.

Identified some potentials as conditionally solvable.

Abstract

We extensively investigate two-step shape invariance in the framework of N-fold supersymmetry. We first show that any two-step shape-invariant system possesses type A 2-fold supersymmetry with an intermediate Hamiltonian and thus has second-order parasupersymmetry as well. Employing the general form of type A 2-fold supersymmetry, we systematically construct two-step shape-invariant potentials. In addition to the well-known ordinary shape-invariant potentials, we obtain several new and novel two-step shape-invariant ones which are not ordinary shape invariant. Furthermore, some of the latter potentials are conditionally two-step shape invariant and thus are conditionally solvable.