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      Eigensystem multiscale analysis for Anderson localization in energy intervals

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          Abstract

          We present an eigensystem multiscale analysis for proving localization (pure point spectrum with exponentially decaying eigenfunctions, dynamical localization) for the Anderson model in an energy interval. In particular, it yields localization for the Anderson model in a nonempty interval at the bottom of the spectrum. This eigensystem multiscale analysis in an energy interval treats all energies of the finite volume operator at the same time, establishing level spacing and localization of eigenfunctions with eigenvalues in the energy interval in a fixed box with high probability. In contrast to the usual strategy, we do not study finite volume Green's functions. Instead, we perform a multiscale analysis based on finite volume eigensystems (eigenvalues and eigenfunctions). In any given scale we only have decay for eigenfunctions with eigenvalues in the energy interval, and no information about the other eigenfunctions. For this reason, going to a larger scale requires new arguments that were not necessary in our previous eigensystem multiscale analysis for the Anderson model at high disorder, where in a given scale we have decay for all eigenfunctions.

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          LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS

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            Localization for Some Continuous, Random Hamiltonians in d-Dimensions

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              Moment Analysis for Localization in Random Schroedinger Operators

              We study localization effects of disorder on the spectral and dynamical properties of Schroedinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
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                Author and article information

                Journal
                2016-11-08
                Article
                1611.02650
                2724ff80-49b8-4ba8-b21c-6915e7143933

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                arXiv admin note: text overlap with arXiv:1509.08521
                math-ph math.MP

                Mathematical physics,Mathematical & Computational physics
                Mathematical physics, Mathematical & Computational physics

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