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Bethe-Sommerfeld Conjecture and absolutely continuous spectrum of multi-dimensional quasi-periodic Schro¨dinger operators
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DOI：10.4007/annals.2026.203.2.1.png)
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Abstract
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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.
Keywords:
Quasi-periodic operators
Bethe-Sommerfeld Conjecture
Absolutely continuous spectrum
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