| mots-clés : Analyse semi-classique – Analyse microlocale – Système hamiltonien – Système intégrable – Analyse spectrale – Opérateurs pseudo-différentiels – Forme normale de Birkhoff – Monodromie – Asymptotique spectrale – Mécanique quantique – Mécanique classique – Géométrie symplectique. |
| autres localisations : http://ecm.univ-rennes1.fr/nuxeo/site/esupversions/10726a73-0921-4c6f-9927-0dadb82813b3 |
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| Monodromy of non-selfadjoint operators |
| We propose to build in this thesis a combinatorial invariant, called the "spectral monodromy" from the spectrum of a single (non-selfadjoint) h-pseudodifferential operator with two degrees of freedom in the semi-classical limit. Our inspiration comes from the quantum monodromy defined for the joint spectrum of an integrable system of n commuting selfadjoint h-pseudodifferential operators, given by S. Vu Ngoc. The first simple case that we treat in this work is a normal operator. In this case, the discrete spectrum can be identified with the joint spectrum of an integrable quantum system. The second more complex case we propose is a small perturbation of a selfadjoint operator with a classical integrability property. We show that the discrete spectrum (in a small band around the real axis) also has a combinatorial monodromy. The difficulty here is that we do not know the description of the spectrum everywhere, but only in a Cantor type set. In addition, we also show that the monodromy can be identified with the classical monodromy (which is defined by J. Duistermaat). These are the main results of this thesis. |
| mots-clés en anglais : Semi-classical analysis – Microlocal Analysis – Hamiltonian integrable system – Spectral analysis – Pseudodifferential operators – Birkhoff normal form – Monodromy – Asymptotic spectral – Quantum mechanics – Classical mechanics – Symplectic geometry |
