Anderson localization for the completely resonant phases
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abstract
For the almost Mathieu operator $ (H_{lambda,alpha, heta}u) (n)=u(n+1)+u(n-1)+ lambda v( heta+nalpha)u(n)$, Avila and Jitomirskaya guess that for every phase $ heta in mathscr{R} riangleq{ hetain mathbb{R};| ; 2 heta + alpha mathbb{Z} in mathbb{Z}}$, $H_{lambda,alpha, heta}$ satisfies Anderson localization if $ |lambda| > e^{ 2 \beta}$. In the present paper, we show that for every phase $ heta in mathscr{R} $, $H_{lambda,alpha, heta}$ satisfies Anderson localization if $ |lambda| > e^{ 7 \beta}$.
