Small denominators and large numerators of quasiperiodic Schrödinger operators

We initiate an approach to simultaneously treat numerators and denominators of Green's functions arising from quasi-periodic Schrödinger operators, which in particular allows us to study completely resonant phases of the almost Mathieu operator. Let $(H_{λ,α,θ}u) (n)=u(n+1)+u(n-1)+ 2λ\cos2π(θ+nα)u(n)$ be the almost Mathieu operator on $\ell^2(\mathbb{Z})$, where $λ, α, θ\in \mathbb{R}$. Let $β(α)=\limsup_{k\rightarrow \infty}-\frac{\ln ||kα||_{\mathbb{R}/\mathbb{Z}}}{|k|}.$ We prove that for any $θ$ with $2θ\in α\mathbb{Z}+\mathbb{Z}$, $H_{λ,α,θ}$ satisfies Anderson localization if $|λ|>e^{2β(α)}$. 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].

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