Deformed Shape Invariant Superpotentials in Quantum Mechanics and Expansions in Powers of $\hbar$

TL;DR

This paper generalizes a method for solving shape invariant potentials in supersymmetric quantum mechanics to include deformed potentials, using an infinite set of PDEs, with examples illustrating both $$-independent and $$-dependent superpotentials.

Contribution

The paper extends the existing method for shape invariance to deformed potentials in deformed supersymmetric quantum mechanics, broadening the class of solvable models.

Findings

Method successfully applied to several examples.

Extension includes superpotentials depending explicitly on .

Demonstrates the approach's versatility with both -independent and -dependent cases.

Abstract

We show that the method developed by Gangopadhyaya, Mallow, and their coworkers to deal with (translationally) shape invariant potentials in supersymmetric quantum mechanics and consisting in replacing the shape invariance condition, which is a difference-differential equation, by an infinite set of partial differential equations can be generalized to deformed shape invariant potentials in deformed supersymmetric quantum mechanics. The extended method is illustrated by several examples, corresponding both to $\hbar$-independent superpotentials and to a superpotential explicitly depending on $\hbar$.