Generation of a Complete Set of Supersymmetric Shape Invariant Potentials from an Euler Equation

TL;DR

This paper links shape invariance in supersymmetric quantum mechanics to an Euler equation, deriving all such superpotentials and developing an algorithm to generate both $bar$-independent and dependent cases.

Contribution

It establishes a novel connection between shape invariance and fluid dynamics, solving related PDEs to generate all conventional superpotentials and creating an algorithm for more general cases.

Findings

All conventional additive shape invariant superpotentials satisfy specific PDEs.

The PDEs are equivalent to an Euler equation from fluid mechanics.

An algorithm is developed to generate all additive shape invariant superpotentials, including those with explicit $bar$ dependence.

Abstract

In supersymmetric quantum mechanics, shape invariance is a sufficient condition for solvability. We show that all conventional additive shape invariant superpotentials that are independent of $\hbar$ obey two partial differential equations. One of these is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow, and it is closed by the other. We solve these equations, generate the set of all conventional shape invariant superpotentials, and show that there are no others in this category. We then develop an algorithm for generating all additive shape invariant superpotentials including those that depend on $\hbar$ explicitly.