Cantor spectrum for multidimensional quasi-periodic Schrödinger operators
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909636694638592 |
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| author | Helffer, Bernard Liu, Qinghui Qu, Yanhui Zhou, Qi |
| author_facet | Helffer, Bernard Liu, Qinghui Qu, Yanhui Zhou, Qi |
| contents | In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}. Additionally, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cantor spectrum for multidimensional quasi-periodic Schrödinger operators Helffer, Bernard Liu, Qinghui Qu, Yanhui Zhou, Qi Spectral Theory Mathematical Physics Dynamical Systems Metric Geometry In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}. Additionally, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small. |
| title | Cantor spectrum for multidimensional quasi-periodic Schrödinger operators |
| topic | Spectral Theory Mathematical Physics Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/2506.03577 |