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

Abstract

In this paper, we investigate the delocalization property of the discrete Schrödinger operator $H_ω=-Δ+v_nω_nδ_{n,n'}$, where $v_n=κ|n|^{-α}$ and $ω=\{ω_n\}_{n\in\mathbb{Z}^d}\in \{\pm 1\}^{\mathbb{Z}^d}$ is a sequence of i.i.d. Bernoulli random variables. Under the assumptions of $d\geq 5$, $α>\frac14$ and $0<κ\ll1$, we construct the extended states for a deterministic renormalization of $H_ω$ for most $ω$. This extends the work of Bourgain [{\it Geometric Aspects of Functional Analysis}, LNM 1807: 70--98, 2003], where the case $α>\frac13$ 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.