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

TL;DR

This paper extends the Kunz-Souillard approach to prove dynamical and spectral localization for the discrete generalized Anderson model with blocks of fixed size, analyzing spectrum, Lyapunov exponents, and operator contractions.

Contribution

It generalizes the Kunz-Souillard method to block potentials of arbitrary fixed size and provides new insights into spectrum description and Lyapunov exponent positivity.

Findings

Proves localization at all energies for the generalized Anderson model.

Describes the almost sure spectrum as a set.

Shows existence of finitely supported distributions with zero Lyapunov exponent at some energies.

Abstract

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.