Anderson Localization for the Almost Mathieu Operator in Exponential Regime

Abstract

For the almost Mathieu operator $(H_{λ,α,θ}u)_n=u_{n+1}+u_{n-1}+2λ\cos2π(θ+nα)u_n$, Avila and Jitomirskaya guess that for a.e. $θ$, $H_{λ,α,θ}$ satisfies Anderson localization if $ |λ| > e^{ β} $, and they establish this for $ |λ| > e^{\frac{16}{9} β}$. In the present paper, we extend their result to regime $ |λ| > e^{\frac{3}{2} β}$.