The partnership of potentials in Quantum Mechanics and Shape Invariance

TL;DR

This paper explores the partnership of potentials in quantum mechanics, focusing on non-uniqueness issues and generating new factorizations through invariance under parameter transformations, supported by illustrative examples.

Contribution

It introduces a method to generate new factorizations of potentials by exploiting invariance properties, addressing ambiguities in the factorization process.

Findings

Identified non-uniqueness in potential partnerships due to factorization ambiguities

Developed a technique to produce new factorizations via parameter transformation invariance

Provided examples demonstrating the application of the theory

Abstract

The concept of partnership of potentials is studied in detail and in particular the non-uniqueness due to the ambiguity in the election of the factorization energy and in the choice of the solution of certain Riccati equation. We generate new factorizations from old ones using invariance under parameter transformations. The theory is illustrated with some examples.