Systemes Integrables Semi-Classiques

This book presents a panorama of finite dimensional completely integrable Hamiltonian systems, in which classical aspects and quantum aspects will be living side by side, with similar appearances. Classical mechanics is considered from the viewpoint of the geometric study of the singular Lagrangian foliation, whose regular leaves are the famous Liouville tori. Singularities are tackled using local and semi-global normal forms, which involve topological and symplectic invariants. Some relationships with toric varieties are explored. Quantum integrable systems are treated in the framework of semiclassical microlocal analysis. Pseudo-differential calculus and Fourier integral operators offer efficient tools for discovering how the geometric features of these systems influence their spectral properties.