On the Kunz-Souillard approach to localization for the discrete one dimensional generalized Anderson model

We prove dynamical and spectral localization at all energies for the discrete generalized Anderson model via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schr\"odinger operators, to the case where a single random v...

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Bibliographic Details
Published in:arXiv.org 2016-10
Main Author: Bucaj, Valmir
Format: Article
Language:English
Subjects:
Chaos theory
Liapunov exponents
Localization
Random variables
Online Access:Get full text
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Summary:We prove dynamical and spectral localization at all energies for the discrete generalized Anderson model via the Kunz-Souillard approach to localization. This is an extension of the original Kunz-Souillard approach to localization for Schr\"odinger operators, to the case where a single random variable determines the potential on a block of an arbitrary, but fixed, size \(\alpha\). For this model, we also give a description of the almost sure spectrum as a set and prove uniform positivity of the Lyapunov exponents. In fact, regarding positivity of the Lyapunov exponents, we prove a stronger statement where we also allow finitely supported distributions. We also show that for any size \(\alpha\) {\it generalized Anderson model}, there exists some finitely supported distribution \(\nu\) for which the Lyapunov exponent will vanish for at least one energy. Moreover, restricting to the special case \(\alpha=1\), we describe a pleasant consequence of this modified technique to the original Kunz-Souillard approach to localization. In particular, we demonstrate that actually the single operator \(T_1\) is a strict contraction in \(L^2(\mathbb{R})\), whereas before it was only shown that the second iterate of \(T_1\) is a strict contraction.
ISSN:2331-8422