Supersymmetry, shape invariance and the Legendre equations

TL;DR

This paper explores the connection between supersymmetry, shape invariance, and Legendre equations in physics, revealing underlying symmetries that explain their solvability in spherical systems.

Contribution

It demonstrates the shape invariance symmetry in Legendre equations, providing a new operator-based perspective on their solutions in physical systems.

Findings

Shape invariance underpins the solvability of Legendre equations.

Operator approach reveals supersymmetry in spherical problems.

Insights applicable to multipole expansions and related physics problems.

Abstract

In three space dimensions, when a physical system possesses spherical symmetry, the dynamical equations automatically lead to the Legendre and the associated Legendre equations, with the respective orthogonal polynomials as their standard solutions. This is a very general and important result and appears in many problems in physics (for example, the multipole expansion etc). We study these equations from an operator point of view, much like the harmonic oscillator, and show that there is an underlying shape invariance symmetry in these systems responsible for their solubility. We bring out various interesting features resulting from this analysis from the shape invariance point of view.