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arXiv · 2503.06918

The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schr\"odinger operators

Lingrui Ge·Yiqian Wang·Jiahao Xu
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Abstract

This paper solves ``The Dry Ten Martini Problem'' for $C^2$ cosine-type quasiperiodic Schr\"odinger operators with large coupling constants and Diophantine frequencies, a model originally introduced by Sinai in 1987 \cite{sinai}. This shows that the analyticity assumption on the potential is not essential for obtaining a dry Cantor spectrum and can be replaced by a certain geometric condition in the low regularity case. In addition, we prove the homogeneity of the spectrum and the absolute continuity of the integrated density of states (IDS).

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Lingrui Ge, Yiqian Wang, Jiahao Xu. 2025-03-10. The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schr\"odinger operators. https://arxiv.org/abs/2503.06918

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