Factorization of nonlinear supersymmetry in one-dimensional Quantum Mechanics. I: general classification of reducibility and analysis of the third-order algebra

TL;DR

This paper classifies and analyzes the reducibility of supersymmetric transformations in one-dimensional quantum mechanics, focusing on third-order cases and their algebraic structures, to better understand their factorization into elementary Darboux transformations.

Contribution

It provides a comprehensive classification of irreducible SUSY transformations and analyzes the algebraic structure of third-order SUSY operators in quantum mechanics.

Findings

Classification of irreducible (almost) isospectral transformations

Detailed analysis of third-order SUSY algebras

Insights into factorization into elementary Darboux transformations

Abstract

We study possible factorizations of supersymmetric (SUSY) transformations in the one-dimensional quantum mechanics into chains of elementary Darboux transformations with nonsingular coefficients. A classification of irreducible (almost) isospectral transformations and of related SUSY algebras is presented. The detailed analysis of SUSY algebras and isospectral operators is performed for the third-order case.