Almost Mathieu operators with completely resonant phases

TL;DR

This paper proves Anderson localization for almost Mathieu operators with resonant phases under certain conditions, extending previous conjectures by developing new methods to handle frequency and phase resonances.

Contribution

The paper introduces a novel method to address simultaneous frequency and phase resonances in almost Mathieu operators, confirming localization for larger coupling constants than previously conjectured.

Findings

Proves Anderson localization for $|ta| > e^{3eta(ta)}$ with resonant phases.

Develops a new technique to handle combined frequency and phase resonances.

Extends the understanding of spectral properties of almost Mathieu operators with resonant phases.

Abstract

Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$ and $\beta(\alpha) = \limsup _{n \to \infty}(\ln q_{n+1})/ q_n <\infty$, where $p_n/q_n$ is the continued fraction approximations to $\alpha$. Let $(H_{\lambda,\alpha,\theta}u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos2\pi(\theta+n\alpha)u(n)$ be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $\lambda, \theta\in \mathbb{R}$. Avila and Jitomirskaya \cite{avila2009ten} conjectured that for $2\theta \in \alpha \mathbb{Z} + \mathbb{Z}$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $|\lambda|>e^{2\beta(\alpha)}$. In this paper, we developed a method to treat simultaneous frequency and phase resonances and obtain that for $2\theta\in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $|\lambda|>e^{3\beta(\alpha)}$.