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Étude de la stabilité de systèmes dynamiques quantiques

Abstract : The dynamics of a periodic time-dependent quantum system may be described by means of a Floquet operator on a suitable Hilbert space. The spectral properties of this operator give some information on the long time behaviour of the system. Two models are considered here. First, we investigate the spectral properties of the Floquet operators of some stationary quantum systems with a discrete and simple spectrum, periodically perturbed by a rank one kick. Our first result is the following: if the perturbation is suitably chosen and if the eigenvalues of the stationary system are given by a polynomial with some arithmetic condition on its coefficients, the spectrum of the Floquet operator is purely singular continuous. Then, we prove this is still true for some rank one perturbation, if the eigenvalues grow suitably fast and if the period belongs to a set of full Lebesgue measure. These results complete a previous study of Combescure. The spectral properties of a family of Floquet operators with a matrix representation displaying a band structure are also analyzed. Such operators appear in the study of some electronic conduction models. Although they depend on a larger number of parameters, we prove their spectral properties depend in some limit, on the choice of two infinite sequences of phases. We prove therefore that the spectrum remains purely singular if the phases are determined by some ergodic processes. However, the spectrum is absolutely continuous and may possess a finite number of isolated simple eigenvalues if the phases are built according to a periodic procedure.
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Submitted on : Wednesday, December 18, 2002 - 8:48:35 AM
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Olivier Bourget. Étude de la stabilité de systèmes dynamiques quantiques. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2002. Français. ⟨tel-00002171⟩

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