Supersymmetry,Shape Invariance and Exactly Solvable Noncentral Potentials

TL;DR

This paper demonstrates how supersymmetry and shape invariance techniques enable the exact algebraic solution of a broad class of noncentral potentials, extending the set of potentials with closed-form eigenvalues and eigenfunctions.

Contribution

It introduces a generalized operator method leveraging supersymmetry and shape invariance to solve noncentral potentials exactly, broadening the scope of algebraically solvable quantum systems.

Findings

Eigenvalues and eigenfunctions obtained in closed form

Extended list of exactly solvable potentials

Illustrated with a complex seven-parameter potential

Abstract

Using the ideas of supersymmetry and shape invariance we show that the eigenvalues and eigenfunctions of a wide class of noncentral potentials can be obtained in a closed form by the operator method. This generalization considerably extends the list of exactly solvable potentials for which the solution can be obtained algebraically in a simple and elegant manner. As an illustration, we discuss in detail the example of the potential $$V(r,\theta,\phi)={\omega^2\over 4}r^2 + {\delta\over r^2}+{C\over r^2 sin^2\theta}+{D\over r^2 cos^2\theta} + {F\over r^2 sin^2\theta sin^2 \alpha\phi} +{G\over r^2 sin^2\theta cos^2\alpha\phi}$$ with 7 parameters.Other algebraically solvable examples are also given.