Absolutely Continuous Spectrum for the Quasi-periodic Schrödinger Operator in Exponential Regime

Abstract

Avila and Jitomirskaya prove that the quasi-periodic Schrödinger operator $H_{λv,α,θ}$ has purely absolutely continuous spectrum for $α$ in sub-exponential regime (i.e., $β(α)=0$) with small $λ$, if $v$ is real analytic in a strip of real axis. In the present paper, we show that for all $α$ with $0<β(α)<\infty$, $H_{λv,α,θ}$ has purely absolutely continuous spectrum with small $λ$, if $v$ is real analytic in strip $ |\Im x|< Cβ$, where $C$ is a large absolute constant.