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Theses

Sur les déformations des systèmes complètement intégrables classiques et semi-classiques

Abstract : In a first part, we consider a regular completely integrable hamiltonian on a symplectic manifold and we try to caracterize the perturbations of this hamiltonian which are deformations, i.e which remain completely integrable when the perturbation is added. After expliciting the class of the non-degenerate hamiltonians we consider, and conjecturing the general form of the regular deformations, we give the conditions, formal in the perturbation parameter, for the hamiltonian to remain completely integrable, in a regular or in a singular way. In a second part, we consider a semi-classical completely integrable system described by means of a pseudodifferantial operator on the torus and we study the spectrum of a perturbation of this operator. For this purpose, we make use of a normal form method which puts the operator in a simple form near each resonance. This normal form is the used in order to construct quasimodes for the perturbed operator.
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https://tel.archives-ouvertes.fr/tel-00003400
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Submitted on : Friday, September 19, 2003 - 4:10:46 PM
Last modification on : Wednesday, November 4, 2020 - 1:59:00 PM
Long-term archiving on: : Friday, April 2, 2010 - 7:30:18 PM

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  • HAL Id : tel-00003400, version 1

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Nicolas Roy. Sur les déformations des systèmes complètement intégrables classiques et semi-classiques. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2003. Français. ⟨tel-00003400⟩

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