On solvable Dirac equation with polynomial potentials

TL;DR

This paper investigates the solvability of the one-dimensional Dirac equation with polynomial potentials, establishing that only linear potentials admit exact solutions based on Liouvillian functions.

Contribution

It introduces a criterion for solvability of the Dirac equation with polynomial potentials using Liouvillian functions, showing non-solvability for all but linear cases.

Findings

Only linear polynomial potentials yield exact solutions.

Non-linear polynomial potentials are not solvable in closed form.

Liouvillian functions are used to determine solvability.

Abstract

One dimensional Dirac equation is analysed with regard to the existence of exact (or closed-form) solutions for polynomial potentials. The notion of Liouvillian functions is used to define solvability, and it is shown that except for the linear potentials the equation in question is not solvable.