Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schr\"odinger Operators

We consider Schr"odinger operators $H=-\mathrm{\Delta }+V(\mathbf{x})$ in ${\mathbb R}^d$,$d\geq2$,withquasi-periodicpotentials$V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[{\lambda }_{\ast },+\mathrm{\infty })$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way.