# The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous

### David Damanik

Rice University, Houston, United States### Michael Goldstein

University of Toronto, Canada### Milivoje Lukic

Rice University, Houston, USA

## Abstract

We consider the quasi-periodic Schrödinger operator

$[Hψ](x)=−ψ_{′′}(x)+V(x)ψ(x)$

in $L_{2}(R)$, where the potential is given by

$V(x)=m∈Z_{ν}∖{0}∑ c(m)exp(2πimωx)$

with a Diophantine frequency vector $ω=(ω_{1},…,ω_{ν})∈R_{ν}$ and exponentially decaying Fourier coefficients $∣c(m)∣≤εexp(−κ_{0}∣m∣)$. In the regime of small $ε>0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson.

## Cite this article

David Damanik, Michael Goldstein, Milivoje Lukic, The spectrum of a Schrödinger operator with small quasi-periodic potential is homogeneous. J. Spectr. Theory 6 (2016), no. 2, pp. 415–427

DOI 10.4171/JST/128