Some Generalizations in Supersymmetric Quantum Mechanics and the Supersymmetric $\varepsilon$-System Revisited

TL;DR

This paper explores new operator classes in supersymmetric quantum mechanics and revisits the supersymmetric epsilon-system, demonstrating broader supersymmetry connections and refining energy level calculations.

Contribution

It introduces a generalized class of operators involving superpotentials and reanalyzes the supersymmetric epsilon-system using Hamilton hierarchy and delta-expansion methods.

Findings

New operators reduce to conventional supersymmetric operators.

Supersymmetry connects a broader class of differential operators.

Accurate energy level for the first excited state of the epsilon-system.

Abstract

We discuss two distinct aspects in supersymmetric quantum mechanics. First, we introduce a new class of operators A and $\bar{A}$ in terms of anticommutators between the momentum operator and N+1 arbitrary superpotentials. We show that these operators reduce to the conventional ones which are the starting point in standard supersymmetric quantum mechanics. In this context, we argue furthermore that supersymmetry does not only connect Schr\"odinger-like operators, but also a more general class of differential operators. Second, we revisit the supersymmetric $\varepsilon$-system recently introduced in the literature by exploiting its intrinsic supersymmetry. Specifically, combining the Hamilton hierarchy method and the $\delta$-expansion method, we determine an energy for the first excited state of the bosonic Hamiltonian close to that calculated in earlier works.