Bethe–Sommerfeld Conjecture and absolutely continuous spectrum of multi-dimensional quasi-periodic Schrödinger operators

Abstract

We consider Schrödinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb{R}}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[\lambda_*,+\infty)$. 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.