Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918213781028864 |
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| author | Ge, Lingrui You, Jiangong Zhou, Qi |
| author_facet | Ge, Lingrui You, Jiangong Zhou, Qi |
| contents | This paper establishes several sharp spectral results for analytic quasiperiodic Schrodinger operators. Key contributions include: (1) exact exponential decay rates for spectral gaps of the almost Mathieu operator, addressing a question raised by Goldstein; (2) sharp arithmetic Anderson localization for a class of quasiperiodic operators on higher-dimensional lattices, which in particular resolves and generalizes Jitomirskaya's phase transition conjecture; and (3) stratified growth patterns for extended eigenfunctions revealing universal partial hierarchical structures for subcritical quasiperiodic Schrodinger operators.
The proofs are based on novel frameworks-structured quantitative almost reducibility and sharp quantitative duality-to overcome the longstanding challenge of taming infinitely many rotation-number resonances, which enables us to obtain optimal arithmetic reducibility results for analytic SL(2,R)-cocycles, thereby solving another Jitomirskaya's conjecture. These methods enable a first comprehensive treatment of resonance-driven dynamical asymptotics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys Ge, Lingrui You, Jiangong Zhou, Qi Dynamical Systems Mathematical Physics Spectral Theory This paper establishes several sharp spectral results for analytic quasiperiodic Schrodinger operators. Key contributions include: (1) exact exponential decay rates for spectral gaps of the almost Mathieu operator, addressing a question raised by Goldstein; (2) sharp arithmetic Anderson localization for a class of quasiperiodic operators on higher-dimensional lattices, which in particular resolves and generalizes Jitomirskaya's phase transition conjecture; and (3) stratified growth patterns for extended eigenfunctions revealing universal partial hierarchical structures for subcritical quasiperiodic Schrodinger operators. The proofs are based on novel frameworks-structured quantitative almost reducibility and sharp quantitative duality-to overcome the longstanding challenge of taming infinitely many rotation-number resonances, which enables us to obtain optimal arithmetic reducibility results for analytic SL(2,R)-cocycles, thereby solving another Jitomirskaya's conjecture. These methods enable a first comprehensive treatment of resonance-driven dynamical asymptotics. |
| title | Sharp Spectral Gaps, Arithmetic Localization, and Reducibility via Resonance Analys |
| topic | Dynamical Systems Mathematical Physics Spectral Theory |
| url | https://arxiv.org/abs/2407.05490 |