Homogeneity of the spectrum for quasi-periodic Schr\"odinger operators

Abstract

We consider the one-dimensional discrete Schr\"odinger operator $$ \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n)\ , $$ $n \in \mathbb{Z}$, $x,\omega \in [0, 1]$ with real-analytic potential $V(x)$. Assume $L(E,\omega)>0$ for all $E$. Let $\mathcal{S}_\omega$ be the spectrum of $H(x,\omega)$. For all $\omega$ obeying the Diophantine condition $\omega \in \mathbb{T}_{c,a}$, we show the following: if $\mathcal{S}_\omega \cap (E',E")\neq \emptyset$, then $\mathcal{S}_\omega \cap (E',E")$ is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if $G_i$, $i=1,2$ are two gaps with $1 > |G_1| \ge |G_2|$, then $|G_2|\lesssim \exp\left(-(\log \mathrm{dist} (G_1,G_2))^A\right)$, $A\gg 1$. Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for $\mathcal S_{N,\omega}:=\cup_{x\in\mathbb T}\mathrm{spec} H_{[-N,N]}(x,\omega) $, $N \ge 1 $, where $H_{[-N, N]}(x, \omega)$ is the Schr\"odinger operator restricted to the interval $[-N,N]$ with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with $|\lambda| \neq 1$. For the supercritical almost Mathieu operator, we combine the methods of [GolSch08] with Jitomirskaya's approach from [Jit99] to establish most of the results from [GolSch08] with $\omega$ obeying a strong Diophantine condition.