Bohr-Sommerfeld phases for avoided crossings

We consider the adiabatic limit in quantum mechanics with several avoided crossings. We compute the interferences effects uniformly w.r. to the gaps and the adiabatic parameter. This way we get the asymoptotic expansion of the global scattering matrix. We use the technics of normal forms introduced in the paper written with B. Parisse (Commun. Math. Phys. 205 pp. 459--500 (1999)).