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

Article type: Research Article

Authors: Khochman, Abdallah

Affiliations: Université Bordeaux I, Institut de Mathématiques, UMR CNRS 5251, 351, cours de la Libération, 33405 Talence, France. E-mail: Abdallah.Khochman@math.u-bordeaux1.fr

Abstract: We study the Klein paradox for the semi-classical Dirac operator on R with potentials having constant limits, possibly different 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.

Keywords: semi-classical Dirac operator, scattering matrix, Klein paradox, complex WKB method

DOI: 10.3233/ASY-2009-0956

Journal: Asymptotic Analysis, vol. 65, no. 3-4, pp. 223-249, 2009