Factorization of non-linear supersymmetry in one-dimensional Quantum Mechanics. II: proofs of theorems on reducibility

TL;DR

This paper rigorously proves theorems on the reducibility of supersymmetric transformations in one-dimensional quantum mechanics, focusing on factorization into elementary Darboux transformations and spectral equivalence.

Contribution

It provides formal proofs and a detailed analysis of the reducibility of SUSY transformations, extending previous conjectures in the field.

Findings

Defined the class of potentials invariant under Darboux-Crum transformations

Proved lemmas and theorems on the reducibility of differential operators

Analyzed the general case with comprehensive proofs

Abstract

In this paper, we continue to study factorization of supersymmetric (SUSY) transformations in one-dimensional Quantum Mechanics into chains of elementary Darboux transformations with nonsingular coefficients. We define the class of potentials that are invariant under the Darboux - Crum transformations and prove a number of lemmas and theorems substantiating the formulated formerly conjectures on reducibility of differential operators for spectral equivalence transformations. Analysis of the general case is performed with all the necessary proofs.