Small denominators and large numerators of quasiperiodic Schr\"odinger operators

TL;DR

This paper develops a new approach to analyze the numerators and denominators of Green's functions in quasi-periodic Schrödinger operators, leading to results on Anderson localization for the almost Mathieu operator under certain conditions.

Contribution

It introduces a method to handle both numerators and denominators simultaneously, confirming a conjecture about localization in the almost Mathieu operator for resonant phases.

Findings

Proves Anderson localization for the almost Mathieu operator when ||>e^{2}() for specific phases.

Confirms a conjecture of Avila and Jitomirskaya on localization.

Addresses resonant phases of the almost Mathieu operator.

Abstract

We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schr\"odinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. 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 $\lambda, \alpha, \theta\in \mathbb{R}$. Let $$ \beta(\alpha)=\limsup_{k\rightarrow \infty}-\frac{\ln ||k\alpha||_{\mathbb{R}/\mathbb{Z}}}{|k|}.$$ We prove that for any $\theta$ with $2\theta\in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $|\lambda|>e^{2\beta(\alpha)}$. This confirms a conjecture of Avila and Jitomirskaya [The Ten Martini Problem. Ann. of Math. (2) 170 (2009), no. 1, 303--342] and a particular case of a conjecture of Jitomirskaya [Almost everything about the almost Mathieu operator. II. XIth International Congress of Mathematical Physics (Paris, 1994), 373--382, Int. Press, Cambridge, MA, 1995].