# Homogeneity of the spectrum for quasi-periodic Schrödinger operators

### David Damanik

Rice University, Houston, USA### Michael Goldstein

University of Toronto, Canada### Wilhelm Schlag

University of Chicago, USA### Mircea Voda

University of Chicago, USA

## Abstract

We consider the one-dimensional discrete Schrödinger operator

$n∈Z$, $x,ω∈[0,1]$ with real-analytic potential $V(x)$. Assume $L(E,ω)>0$ for all $E$. Let $S_{ω}$ be the spectrum of $H(x,ω)$. For all $ω$ obeying the Diophantine condition $ω∈T_{c,a}$, we show the following: if $S_{ω}∩(E_{′},E_{′′})=∅$, then $S_{ω}∩(E_{′},E_{′′})$ is homogeneous in the sense of Carleson (see [Car83]). Furthermore, we prove, that if $G_{i}$, $i=1,2$ are two gaps with $1>∣G_{1}∣≥∣G_{2}∣$, then $∣G_{2}∣≲exp(−(gdist(G_{1},G_{2}))_{A})$, $A≫1$. Moreover, the same estimates hold for the gaps in the spectrum on a finite interval, that is, for $S_{N,ω}:=∪_{x∈T}specH_{[−N,N]}(x,ω)$ , $N≥1$ , where $H_{[−N,N]}(x,ω)$ 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 $∣λ∣=1$. 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 $ω$ obeying a strong Diophantine condition.

## Cite this article

David Damanik, Michael Goldstein, Wilhelm Schlag, Mircea Voda, Homogeneity of the spectrum for quasi-periodic Schrödinger operators. J. Eur. Math. Soc. 20 (2018), no. 12, pp. 3073–3111

DOI 10.4171/JEMS/829