Small denominators and large numerators of quasiperiodic Schrdinger operators

We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schr"odinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let $ (H_{lambda,alpha, heta}u) (n)=u(n+1)+u(n-1)+ 2lambda cos2pi( heta+nalpha)u(n)$ be the almost Mathieu operator on $ell^2(mathbb{Z})$, where $lambda, alpha, hetain mathbb{R}$. Let $$ \beta(alpha)=limsup_{k
ightarrow infty}-frac{ln ||kalpha||_{mathbb{R}/mathbb{Z}}}{|k|}.$$ We prove that for any $ heta$ with $2 hetain alpha mathbb{Z}+mathbb{Z}$, $H_{lambda,alpha, heta}$ satisfies Anderson localization if $|lambda|>e^{2\beta(alpha)}$. This confirms a conjecture of Avila and Jitomirskaya [The Ten Martini Problem. Ann. of Math. (2) 170 (2009), no. 1, 303--342] and a particular case of a conjecture of Jitomirskaya [Almost everything about the almost Mathieu operator. II. XIth International Congress of Mathematical Physics (Paris, 1994), 373--382, Int. Press, Cambridge, MA, 1995].

Small denominators and large numerators of quasiperiodic Schrdinger operators Institutional Repository Document