On General Form of N-fold Supersymmetry

TL;DR

This paper explores the general structure of N-fold supersymmetry in quantum mechanics, introducing a systematic framework and constructing explicit forms for N=2, 3, and 4 cases.

Contribution

It provides a comprehensive analysis of N-fold supersymmetry's structure and introduces a new systematic approach for constructing weakly quasi-solvable quantum systems.

Findings

Derived the most general form of N-fold supersymmetric systems for N=2, 3, and 4.

Established a framework using dimensional analysis and differential operators.

Facilitated the construction of weakly quasi-solvable quantum models.

Abstract

We analyze general structure of N-fold supersymmetry which provides a systematic framework to construct weakly quasi-solvable quantum mechanical systems. Main ingredients of our analysis are dimensional analysis and introduction of an equivalent class of linear differential operators associated with N-fold supersymmetry for each N. To illustrate how they work, we construct the most general form of N-fold supersymmetric systems for N=2, 3, and 4.