The semiclassical study of multidimensional crystals leads naturally to the following question: how do Wigner measures propagate through energy level crossings?

In this talk, we discuss a simple $2\times 2$ system which displays such a crossing. For that purpose, we introduce two-scaled Wigner measures which describe how the usual Wigner measures are concentrating on trajectories passing through the crossing points. Then we derive explicit formulae for the branching of these measures. These formulae are very similar to the so-called Landau-Zener formulae.

These results are issued from a recent jointwork with Clotilde Fermanian.

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