Planar Dirac Electron in Coulomb and Magnetic Fields: a Bethe ansatz approach

TL;DR

This paper introduces an algebraic Bethe ansatz method to solve the Dirac equation for an electron in two dimensions under Coulomb and magnetic fields, revealing unique features not present in similar Schrödinger or Klein-Gordon problems.

Contribution

It presents a novel algebraic Bethe ansatz approach for the Dirac equation in this context, including the treatment of additional parameters and the non-$sl_2$ nature of the differential equation.

Findings

Derived algebraic Bethe ansatz equations for the problem

Identified the unique role of additional parameters in the solution

Showed the differential equation does not belong to $sl_2$ class

Abstract

The Dirac equation for an electron in two spatial dimensions in the Coulomb and homogeneous magnetic fields is an example of the so-called quasi-exactly solvable models. The solvable parts of its spectrum was previously solved from the recursion relations. In this work we present a purely algebraic solution based on the Bethe ansatz equations. It is realised that, unlike the corresponding problems in the Schr\"odinger and the Klein-Gordon case, here the unknown parameters to be solved for in the Bethe ansatz equations include not only the roots of wave function assumed, but also a parameter from the relevant operator. We also show that the quasi-exactly solvable differential equation does not belong to the classes based on the algebra $sl_2$.