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arXiv · 2010.05881

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

Yulia Karpeshina·Leonid Parnovski·Roman Shterenberg
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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.

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Yulia Karpeshina, Leonid Parnovski, Roman Shterenberg. 2020-10-12. Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schr\"odinger Operators. https://arxiv.org/abs/2010.05881

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