Semiclassical analysis of generic codimension 3 crossings.

We study a family of solutions to a system of two equations which display a codimension 3 generic crossing: the crossing set is a symplectic submanifold of the phase space. We consider initial data which are bounded families in L2. We obtain a quantitative description of the energy transfer above the crossing in terms of two-scale Wigner measures. We prove first that this transfer concerns the part of the energy localized at finite distance, with respect to the scale h, of the sets of all the ingoing classical trajectories. Provided a generalization of two-scale Wigner measures to the case of symplectic submanifolds, we split the Wigner measure of families of solutions into a countable sum of positive Radon measures. The energy transfer at the crossing is described by Landau-Zener formula for the terms of this decomposition. [ABSTRACT FROM AUTHOR]