Small denominators and large numerators of quasiperiodic Schrödinger operators

Abstract

We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schrödinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let $(H_{\lambda ,\alpha ,\theta }u)(n)=u(n+1)+u(n-1)+2\lambda \cos 2\pi (\theta +n\alpha )u(n)$ be the almost Mathieu operator on ${\ell }^2(\mathbb{Z})$, where $\lambda ,\alpha ,\theta \in \mathbb{R}$. Let $$\beta (\alpha )=\underset{k\to \infty }{\mathrm{limsup}}-\frac{\ln ||k\alpha |{|}_{\mathbb{R}/\mathbb{Z}}}{|k|}.$$ 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].