Solving simultaneously Dirac and Ricatti equations

TL;DR

This paper explores solving the Dirac equation in 1+1 dimensions with scalar potentials using the factorization method, linking it to Ricatti equations, and demonstrates the approach with a model relevant to fermion number fractionalization.

Contribution

It introduces a method to generate solvable Dirac potentials by integrating Ricatti equations, expanding the class of solvable models in supersymmetric quantum mechanics.

Findings

Constructed one-parameter families of solvable potentials.

Linked Dirac equation solutions to Ricatti equations.

Applied method to a model demonstrating fermion number fractionalization.

Abstract

We analyse the behaviour of the Dirac equation in $d=1+1$ with Lorentz scalar potential. As the system is known to provide a physical realization of supersymmetric quantum mechanics, we take advantage of the factorization method in order to enlarge the restricted class of solvable problems. To be precise, it suffices to integrate a Ricatti equation to construct one-parameter families of solvable potentials. To illustrate the procedure in a simple but relevant context, we resort to a model which has proved useful in showing the phenomenon of fermion number fractionalization.