01-164 Colin de Verdi re, Yves

SINGULAR LAGRANGIAN MANIFOLDS AND SEMI-CLASSICAL ANALYSIS (398K, Postscript) May 2, 01

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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 has been a lot of works on singular Lagrangian manifolds initiated by Arnold, Givental and others. The goal of our paper is to extend the classical and semi-classical normal forms of completely integrable systems near non degenerate (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 this way natural Ansatz's extending the WKB-Maslov Ansatz. We give more details on the simplest non Morse example, the cusp, which corresponds to a saddle-node bifurcation.

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