The Spectrum of a Schr\"odinger Operator With Small Quasi-Periodic Potential is Homogeneous

We consider the quasi-periodic Schr"odinger operator $$[H\psi ](x)=-\psi \mathrm{"}(x)+V(x)\psi (x)$$ in $L^2(\mathbb{R})$, where the potential is given by $$V(x)=\sum _{m\in {\mathbb{Z}}^{\nu }\setminus \{0\}}c(m)\exp (2\pi im\omega x)$$ with a Diophantine frequency vector $\omega = (\omega_1, \dots, \omega_\nu) \in $\mathbb{R}$^\nu$andexponentiallydecayingFouriercoefficients$|c(m)| \le \varepsilon \exp(-\kappa_0|m|)$\mathrm{.}Intheregimeofsmall$\varepsilon > 0$weshowthatthespectrumoftheoperator$H$ is homogeneous in the sense of Carleson.