Semi-classical measures and eigenvalue crossings

Semiclassical study of multidimensional crystals leads naturally to the following question: how do semiclassical measures propagate through energy level crossings? In this contribution, we discuss a simple 2 x 2 system which displays such a crossing. For that purpose, we introduce two-scaled semi-classical measures, which describe how the usual Wigner transforms are concentrating on trajectories passing through the crossing points. Then we derive explicit formulae for the branching of such measures. These formulae are generalizations of the so-called Landau-Zener formulae.