Homogeneity of the spectrum for quasi-periodic Schrödinger operators

Abstract

We consider the one-dimensional discrete Schrödinger operator

$n\in \mathbb{Z}$, $x,\omega \in [0,1]$ 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 ${\mathcal{S}}_{\omega }\cap (E^{\prime },E^{\prime \prime })\ne \varnothing$, then ${\mathcal{S}}_{\omega }\cap (E^{\prime },E^{\prime \prime })$ 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 Schrödinger operator restricted to the interval $[-N,N]$ with Dirichlet boundary conditions. In particular, all these results hold for the almost Mathieu operator with $|\lambda |\ne 1$. 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.