Superpotential for novel symmetry beyond shape invariance

TL;DR

This paper introduces a new superpotential concept that breaks traditional supersymmetric energy conditions and shape invariance, revealing a novel energy relation between partner Hamiltonians, especially under discontinuities or distortions.

Contribution

The paper proposes a new superpotential that invalidates existing SUSY energy conditions and shape invariance, and establishes a new energy relation between partner Hamiltonians.

Findings

Traditional SUSY energy conditions do not hold for the new superpotential.

A new energy relation $E_{n}^{+}-E_{n}^{(-)}=2$ is established.

Discontinuities or distortions invalidate standard SUSY conditions.

Abstract

We propose a new "superpotential" and find that neither the supersymmetric energy conditions nor the associated shape invariance condition remain valid. On the other hand a new energy condition $E_{n}^{+}-E_{n}^{(-)}=2$ between the two partner Hamiltonian $H^{(\pm)}$ emerges. Mathematical proof supported the present findings with examples are presented. It is observed that, when the superpotential is associated with discontinuity or distortion, SUSY energy conditions and the shape invariance condition will no longer hold good.