Absolutely Continuous Spectrum for the Quasi-periodic Schr\"{o}dinger Operator in Exponential Regime

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