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Autour de la dynamique semi-classique de certains systèmes complètement intégrables

Abstract : The semi-classical dynamics of a pseudo-differential operator on a manifold is the quantum analogous of the classical flow of his main symbol on the manifold . This semi-classical dynamics is described by the Schrödinger equation of the operator whereas the classical Hamiltonian flow is given by the Hamilton's equations associated with the function . Thus the spectrum of the pseudo-differential operator enable to describe the general solutions of the associated Schrödinger equation. The long time behavior of these solutions remains in many ways mysterious. The semi-classical dynamics depends directly on the spectrum of the operator and consequently also on the underlying geometry into induced by the classical symbol . In this thesis, we first describe the long time semi-classical dynamics of an Hamiltonian in the one-dimensional case with a symbol function with no singularity or with non-degenerate elliptic singularity type : the associated fibers are closed elliptic orbits. The regular Bohr-Sommerfeld rules supply the spectrum of the operator. We are also interested in the elliptic case of the dimension 2 which leads to some discussion of numbers theory. Finally we consider the case of a one-dimensionnal pseudo-differential operator with a non-degenerate hyperbolic singularity : the singular fiber of in is a “ hyperbolic eight ” (this model is diffeomorphic to the Schrödinger operator with a double wells).
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Contributor : Olivier Lablée <>
Submitted on : Tuesday, December 8, 2009 - 9:56:48 AM
Last modification on : Wednesday, November 4, 2020 - 2:01:48 PM
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  • HAL Id : tel-00439641, version 1



Olivier Lablée. Autour de la dynamique semi-classique de certains systèmes complètement intégrables. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2009. Français. ⟨tel-00439641⟩



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