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

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.