Quasi-exact solvability of Dirac equation with Lorentz scalar potential

TL;DR

This paper investigates the conditions under which the Dirac equation with a Lorentz scalar potential can be solved exactly or quasi-exactly, using algebraic methods in different coordinate systems.

Contribution

It introduces a framework for analyzing exact and quasi-exact solvability of the Dirac equation with Lorentz scalar potentials in various coordinate systems.

Findings

Identification of exactly solvable potentials in Cartesian coordinates.

Development of $sl(2)$-based quasi-exact solvability criteria.

Extension of solvability analysis to spherical coordinates with a Dirac monopole.

Abstract

We consider exact/quasi-exact solvability of Dirac equation with a Lorentz scalar potential based on factorizability of the equation. Exactly solvable and $sl(2)$-based quasi-exactly solvable potentials are discussed separately in Cartesian coordinates for a pure Lorentz potential depending only on one spatial dimension, and in spherical coordinates in the presence of a Dirac monopole.