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Theses

Commutators, spectral analysis, and applications to discrete Schrödinger operators

Abstract : This thesis deals with the analysis of spectral and dynamical properties of quantum mechanical systems using techniques of operator commutators. Two of the three research papers that are presented deal exclusively with the discrete Schrödinger operators on the lattice. The first article proves a limiting absorption principle for the multi-dimensional discrete Laplacian perturbed by the sum of a Wigner-von Neumann potential and long-range potential. This result notably implies the absolute continuity of the spectrum of this Hamiltonian at certain energies. The second article proves that eigenfunctions corresponding to non-threshold eigenvalues of multidimensional discrete Schrödinger operators decay sub-exponentially. In one dimension, it is further proven that these eigenfunctions decay exponentially. A consequence of this is the absence of eigenvalues when the middle portion of the spectrum does not contain any thresholds. The third article investigates dynamical properties of Hamiltonians under very minimal assumptions in the theory of commutators. Based on minimal escape velocities and an improved version of the RAGE Theorem, we derive propagation estimates for these types of Hamiltonians. These estimates indicate that the states of the system behave dynamically very much like scattering states. Nonetheless, the existence of singularly continuous states cannot be disproved.
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Submitted on : Wednesday, January 24, 2018 - 5:45:31 PM
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Marc Adrien Mandich. Commutators, spectral analysis, and applications to discrete Schrödinger operators. General Mathematics [math.GM]. Université de Bordeaux, 2017. English. ⟨NNT : 2017BORD0725⟩. ⟨tel-01681183v2⟩

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