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

TL;DR

This paper demonstrates the existence of extended states in a high-dimensional discrete Schrödinger operator with a decaying Bernoulli potential, extending previous results to a broader decay rate and employing advanced renormalization techniques.

Contribution

It extends Bourgain's earlier work by establishing extended states for $eta > 1/4$ in $d \\geq 5$, using a novel sixth-order renormalization scheme and new analytical tools.

Findings

Constructed extended states for most Bernoulli potentials under specified conditions.

Developed a generalized Khintchine inequality via Bonami's lemma.

Applied fractional Gagliardo-Nirenberg inequality to control non-random operators.

Abstract

In this paper, we investigate the delocalization property of the discrete Schr\"odinger operator $H_\omega=-\Delta+v_n\omega_n\delta_{n,n'}$, 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 $d\geq 5$, $\alpha>\frac14$ and $0<\kappa\ll1$, 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 $\alpha>\frac13$ 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.