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

TL;DR

This paper proves that multi-dimensional quasi-periodic Schrödinger operators generally have an absolutely continuous spectrum extending to infinity, and constructs eigenfunctions as small perturbations of exponentials using advanced multi-scale analysis.

Contribution

It establishes the presence of a semi-infinite absolutely continuous spectrum for generic multi-dimensional quasi-periodic Schrödinger operators and introduces new multi-scale analysis techniques in momentum space.

Findings

Absolutely continuous spectrum contains a semi-axis [λ_*, +∞)

Constructs eigenfunctions as small perturbations of exponentials

Develops new multi-scale analysis methods in momentum space

Abstract

We consider Schr\"odinger 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.