Unified treatment of a class of spherically symmetric potentials: quasi-exact solution

TL;DR

This paper presents a unified Lie algebraic method to obtain quasi-exact solutions for a class of spherically symmetric potentials in the Schrödinger equation, revealing a common underlying differential equation.

Contribution

It introduces a unified approach using sl(2) Lie algebra to solve a broad class of potentials, simplifying and generalizing previous methods.

Findings

All models reduce to the same differential equation.

Exact solutions are derived using sl(2) representation theory.

The approach unifies the treatment of various spherically symmetric potentials.

Abstract

In this paper, we investigate the Schr\"odinger equation for a class of spherically symmetric potentials in a simple and unified manner using the Lie algebraic approach within the framework of quasi-exact solvability. We illustrate that all models give rise to the same basic differential equation, which is expressible as an element of the universal enveloping algebra of sl(2). Then, we obtain the general exact solutions of the problem by employing the representation theory of sl(2) Lie algebra.