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

Abstract

We consider the quasi-periodic Schrödinger operator

in $L^2(\mathbb{R})$, where the potential is given by

with a Diophantine frequency vector $\omega =({\omega }_1,\dots ,{\omega }_{\nu })\in {\mathbb{R}}^{\nu }$ and exponentially decaying Fourier coefficients $|c(m)|\le \epsilon \exp (-{\kappa }_0|m|)$. In the regime of small $\epsilon >0$ we show that the spectrum of the operator $H$ is homogeneous in the sense of Carleson.