Charged particles in external fields as physical examples of quasi-exactly solvable models: a unified treatment

TL;DR

This paper unifies the treatment of three quasi-exactly solvable charged particle problems in external fields, revealing a common algebraic structure and providing systematic solutions for energies and frequencies.

Contribution

It introduces a unified algebraic approach to three different charged particle problems, highlighting a shared $sl_2$ structure and deriving explicit solutions.

Findings

All three cases reduce to a common quasi-exactly solvable equation.

Explicit energy and frequency expressions are obtained via Bethe ansatz equations.

The method applies to Schrödinger and Klein-Gordon equations for charged particles.

Abstract

We present a unified treatment of three cases of quasi-exactly solvable problems, namely, charged particle moving in Coulomb and magnetic fields, for both the Schr\"odinger and the Klein-Gordon case, and the relative motion of two charged particles in an external oscillator potential. We show that all these cases are reducible to the same basic equation, which is quasi-exactly solvable owing to the existence of a hidden $sl_2$ algebraic structure. A systematic and unified algebraic solution to the basic equation using the method of factorization is given. Analytic expressions of the energies and the allowed frequencies for the three cases are given in terms of the roots of one and the same set of Bethe ansatz equations.