On integrable rational potentials of the Dirac equation

TL;DR

This paper investigates the integrability of the one-dimensional Dirac equation with rational potentials using differential Galois theory, identifying classes of potentials that lead to solvable equations like the Whittaker equation.

Contribution

It introduces a method to analyze the integrability of rational potentials in the Dirac equation and finds a class of potentials related to the Whittaker equation.

Findings

Reduction of Dirac equation to Riccati-like ODEs

Use of differential Galois theory for integrability analysis

Identification of potentials leading to Whittaker equation

Abstract

The one dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.