Semi-Classical Analysis of a Dirac Equation Without Adiabatic Decoupling

Abstract.

We study adiabatic decoupling for Dirac equation with some scaling which yields that the mass appears with a coefficient ɛ^α where ɛ is the semi-classical parameter and α > 0. Therefore, the system presents an avoided crossing. The scale α = 1/2 is critical: adiabatic decoupling holds for α∈ (0,1/2) while for α ≥ 1/2, there is energy transfer at leading order between the two modes. We describe this transfer in terms of two-scale Wigner measures by means of Landau-Zener formula which takes into account the change of polarization of the measures after the crossing.