Singular Lagrangian manifolds and semiclassical analysis

Abstract

Lagrangian submanifolds of symplectic manifolds are very central objects in classical mechanics and microlocal analysis. These manifolds are frequently singular (integrable systems, bifurcations, reduction). There have been many works on singular Lagrangian manifolds initiated by V. Arnold, A. Givental, and others. The goal of our paper is to extend the classical and semiclassical normal forms of completely integrable systems near nondegenerate (Morse-Bott) singularities to more singular systems. It turns out that there is a nicely working way to do that, leading to normal forms and universal unfoldings. We obtain in this way natural ansatzes extending the Wentzel-Kramers-Brillouin(WKB)-Maslov ansatz. We give more details on the simplest non-Morse example, the cusp, which corresponds to a saddle-node bifurcation.