Extended states for the Random Schrödinger operator on ${\mathbb{Z}}^d$ ($d\ge 5$) with decaying Bernoulli potential

Abstract

In this paper, we investigate the delocalization property of the discrete Schrödinger operator $H_{\omega }=-\Delta +v_n{\omega }_n{\delta }_{n,n^{\prime }}$, where $v_n=\kappa |n{|}^{-\alpha }$ and $\omega =\{{\omega }_n{\}}_{n\in {\mathbb{Z}}^d}\in \{\pm 1{\}}^{{\mathbb{Z}}^d}$ is a sequence of i.i.d. Bernoulli random variables. Under the assumptions of $d\ge 5$, $\alpha >\frac{1}{4}$ and $0<\kappa \ll 1$, we construct the extended states for a deterministic renormalization of $H_{\omega }$ for most $\omega$. This extends the work of Bourgain [{\it Geometric Aspects of Functional Analysis}, LNM 1807: 70--98, 2003], where the case $\alpha >\frac{1}{3}$ was handled. Our proof is based on Green's function estimates via a $6$th-order renormalization scheme. Among the main new ingredients are the proof of a generalized Khintchine inequality via Bonami's lemma, and the application of the fractional Gagliardo-Nirenberg inequality to control a new type of non-random operators arising from the $6$th-order renormalization.

Comments welcome; 68 pages