Monday, July 29, 2013

1307.7047 (Victor Chulaevsky)

Uniform Anderson localization, unimodal eigenstates and simple spectra
in a class of "haarsh" deterministic potentials

Victor Chulaevsky
We study a particular class of families of deterministic random (including quasi-periodic), multi-dimensional lattice Schroedinger operators with potentials depending upon an infinite number of parameters in an auxiliary measurable space. In the strong disorder regime, we prove Anderson localization for generic operator families, using a variant of the Multi-Scale Analysis, and show that all localized eigenfunctions are unimodal and feature uniform exponential decay away from their respective localization centers. Until now, such properties have been established only in the exactly solvable Maryland model and some of its generalizations. Using the Klein--Molchanov argument and a variant of the Minami estimate for deterministic potentials, we also prove pointwise simplicity of spectra in our model.
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