Bethe-Sommerfeld Conjecture and absolutely continuous spectrum of multi-dimensional quasi-periodic Schro¨dinger operators

• 成果类型: Article
• 署名作者: Karpeshina, Yulia; Parnovski, Leonid; Shterenberg, Roman
• 署名单位: University of Alabama System; University of Alabama Birmingham; University College London; University of London; University College London
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2026.203.2.1
• 发表日期: 2026-03
• 页码: 349-470
• 关键词: quasi-periodic operators; Bethe-Sommerfeld Conjecture; Absolutely continuous spectrum; SCHRODINGER-OPERATORS; anderson localization; Z(D); NU

摘要

We consider Schro & uml;dinger operators H = -triangle + V (x) in Rd, d >= 2, with quasi-periodic potentials V (x). We prove that the absolutely continuous spectrum of a generic H contains a semi-axis [lambda & lowast;, +infinity). 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.