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Théorie mathématique du transport topologique dans des modèles unitaires sur réseaux

Abstract : We study certain discrete quantum dynamical systems which are described by unitary operators U acting on the space of square integrable functions defined on the vertices of a countably infinite graph. For initial conditions x the orbits of the system are defined by the iterations Unx for integer n. We consider classes of operators U which depend on parameters. We are interested in topological spectral properties meaning that they are characterized by integers which depend continuously on the parameters of the system. We answer questions which are based on recent observations and applications in physical and information sciences. We state and prove proper mathematical results which apply to some of these observations. In this the sis we present four results: in one spatial dimension and for a class of quantum walks we proved the existence of eigenvalues which are constant with respect to continuous and compact perturbations. In two dimensions we have obtained results on occurrence of stable absolutely continuous spectrum covering the whole unit circle for three different lattice models. We employed several mathematical tools: We used the theory of fibered operators to exhibit systematically the spectral properties of translation invariant operators U. For our operator topological considerations, we put to use the class of Fredholm operators and in particular the complete characterization of its connected components by the index. For the case of the constant eigenvalues we proved a non-trivial and explicit lower bound on their number using the index theorem for Toeplitz operators. We employed the theory of the relative index of a pair of orthogonal projections for our study of the full absolutely continuous spectrum. For each of the three cases we established its non-triviality for a pair involvingU, and then made use of recently proved results concerning the implications on the spectrum of U.
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Mohamed Mouneime M'Madi Issimail. Théorie mathématique du transport topologique dans des modèles unitaires sur réseaux. Physique mathématique [math-ph]. Université de Toulon; Ecole nationale d'enseignement supérieur des Comores, 2019. Français. ⟨NNT : 2019TOUL0015⟩. ⟨tel-02503119⟩

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