Klein paradox and Scattering theory for the semi-classical Dirac equation

TL;DR

This paper investigates the Klein paradox and scattering phenomena for the semi-classical Dirac equation with variable potentials, employing complex WKB methods to analyze the scattering matrix and its asymptotics.

Contribution

It introduces a comprehensive scattering theory framework for the semi-classical Dirac operator with potentials having different limits at infinity, including the zero mass case.

Findings

Derived the unitarity of the scattering matrix.

Provided asymptotic expansions of the scattering matrix for various scattering regimes.

Analyzed the zero mass case within the scattering framework.

Abstract

We study the Klein paradox for the semi-classical Dirac operator on $\R$ with potentials having constant limits, not necessarily the same at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter $h$, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.