Small Denominators and Large Numerators of Quasiperiodic Schrödinger Operators

Abstract

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_{\lambda ,\alpha ,\theta }u) (n)=u(n+1)+u(n-1)+ 2\lambda \cos 2\pi (\theta +n\alpha )u(n)$ be the almost Mathieu operator on $\ell ^2({\mathbb {Z}})$, where $\lambda , \alpha , \theta \in {\mathbb {R}}$. Let

$\begin{aligned} \beta (\alpha )=\limsup _{k\rightarrow \infty }-\frac{\ln \Vert k\alpha \Vert _{{\mathbb {R}}/{\mathbb {Z}}}}{|k|}. \end{aligned}$

We prove that for any $\theta $ with $2\theta \in \alpha {\mathbb {Z}}+{\mathbb {Z}}$, $H_{\lambda ,\alpha ,\theta }$ satisfies Anderson localization if $|\lambda |>e^{2\beta (\alpha )}$. This confirms a conjecture of Avila and Jitomirskaya (Ann Math (2) 170(1):303–342, 2009) and a particular case of the second spectral transition line conjecture of Jitomirskaya (XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, pp. 373–382, 1995).

Keywords

Almost Mathieu operators / Small denominators / Large numerators / Completely resonant phase / Second spectral transition line / Primary 47A10 / Secondary 81Q10 / 47B39

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Funding

Division of Mathematical Sciences(2000345)

RIGHTS & PERMISSIONS

Peking University