Formes normales semi‐classiques des systèmes complètement intégrables au voisinage d'un point critique de l'application moment

Abstract

The semi‐classical study of a 1‐dimensional Schrödinger operator near a non‐degenerate maximum of the potential has lead Colin de Verdière and Parisse to prove a microlocal normal form theorem for any 1‐dimensional pseudo‐differential operator with the same kind of singularity. We present here a generalization of this result to pseudo‐differential integrable systems of any finite degree of freedom with a Morse singularity. Our results are based upon Eliasson's classical mechanical study of critical integrable systems.