Quasi-exact Solvability of Dirac Equations

TL;DR

This paper introduces a general method to identify quasi-exact solvability in Dirac and Pauli equations by leveraging $sl(2)$ symmetry and supersymmetry, demonstrated through a spherical electric field example.

Contribution

It provides a systematic procedure for analyzing quasi-exact solvability in relativistic quantum equations using algebraic and supersymmetric techniques.

Findings

Established a connection between quasi-exact solvability and $sl(2)$ symmetry.

Applied the method to the Dirac-Pauli equation with spherical electric field.

Illustrated the procedure with concrete examples.

Abstract

We present a general procedure for determining quasi-exact solvability of the Dirac and the Pauli equation with an underlying $sl(2)$ symmetry. This procedure makes full use of the close connection between quasi-exactly solvable systems and supersymmetry. The Dirac-Pauli equation with spherical electric field is taken as an example to illustrate the procedure.