Almost Mathieu operators with completely resonant phases

Let $\alpha \in \mathbb{R}\mathrm{\backslash }\mathbb{Q}$ and $\beta(\alpha) = \limsup {n \to \infty}(\ln q{n+1})/ q_n <\infty$,where$p_n/q_n$ is the continued fraction approximations to $\alpha$. 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 ${\mathrm{\ell }}^2(\mathbb{Z})$, where $\lambda ,\theta \in \mathbb{R}$. Avila and Jitomirskaya \cite{avila2009ten} conjectured that for $2\theta \in \alpha $\mathbb{Z}$ + $\mathbb{Z}$$,$H_{\lambda,\alpha,\theta}$ satisfies Anderson localization if $\mathrm{\mid }\lambda \mathrm{\mid }>e^{2\beta (\alpha )}$. In this paper, we developed a method to treat simultaneous frequency and phase resonances and obtain that for $2\theta \in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda ,\alpha ,\theta }$ satisfies Anderson localization if $\mathrm{\mid }\lambda \mathrm{\mid }>e^{3\beta (\alpha )}$.