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Spectral Transitions and Critical Phenomena

光谱跃迁和临界现象

基本信息

批准号：
2155211
负责人：
Svetlana Jitomirskaya
金额：
$ 72.5万
依托单位：
University of California-Irvine
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2027-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2155211&HistoricalAwards=false
关键词：
Spectral Transitions Critical Phenomena

项目摘要

The research will focus on the anomalous spectral properties of quasiperiodic (almost but not quite periodic) structures. Quasiperiodic operators provide central or important models for integer quantum Hall effect, experimental quasicrystals, and the quantum chaos theory. Quasiperiodic systems are also used in modeling many other micro and macro effects: from quantum localization to earthquakes. Other deterministic aperiodic (irregular) structures will be of interest as well. The planned development of the rigorous theory is expected to contribute to the understanding of all the above phenomena and may lead to finding new materials with desired physical properties. The topics will include studying properties of quantum mechanical systems with both strong and weak disorder (many and few impurities, respectively), which demonstrate certain anomalous behavior. An integral part of the project will consist of educating graduate students and other young researchers. Related outreach activities will take place.The project consists of several parts, including the connection of dual Lyapunov exponents to characterization of spectra and spectral components, proof of the ubiquity of arithmetic spectral transitions and universal hierarchical structures of eigenfunctions for analytic quasiperiodic operators, proof of extended states for multidimensional quasiperiodic operators, studies of the critical phenomena and of the ‘two interlacing particles’ effect. Other important objectives will be the study of issues related to certain models of quantum chaos. The project will involve the continuing development of non-perturbative methods for the proofs of localization-type effects, as well as for the study of absolutely continuous spectrum.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

研究将集中于准周期（几乎但不完全是周期）结构的异常光谱性质。准周期算符是整数量子霍尔效应、实验准晶体和量子混沌理论的中心或重要模型。准周期系统也用于模拟许多其他微观和宏观效应：从量子局域化到地震。其他确定性的非周期（不规则）结构也会引起我们的兴趣。严格理论的计划发展有望有助于理解上述所有现象，并可能导致发现具有所需物理性质的新材料。主题将包括研究具有强无序和弱无序（分别为多杂质和少杂质）的量子力学系统的性质，这些系统表现出某些异常行为。该项目的一个组成部分将包括研究生和其他年轻研究人员的教育。将举行有关的外联活动。该项目由几个部分组成，包括对偶Lyapunov指数与谱和谱分量表征的联系，证明算术谱跃迁的普遍性和解析拟周期算子的特征函数的普遍层次结构，证明多维拟周期算子的扩展状态，研究临界现象和“两个交错粒子”效应。其他重要的目标将是研究与量子混沌的某些模型有关的问题。该项目将包括继续发展非摄动方法，以证明局域型效应，以及研究绝对连续光谱。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。