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.

该研究将重点关注准周期（几乎但不完全周期）结构的反常光谱特性，准周期算子为整数量子霍尔效应、实验准晶体提供了核心或重要模型，并且量子混沌理论也用于建模许多。其他微观和宏观效应：从量子局域化到地震，其他确定性非周期性（不规则）结构也将引起人们的兴趣。理论预计将有助于理解所有上述现象，并可能导致寻找具有所需物理性质的新材料。这些主题将包括研究具有强无序和弱无序（分别为大量和少量杂质）的量子力学系统的性质，该项目的一个组成部分将包括对研究生和其他年轻研究人员进行教育。该项目由几个部分组成，包括双李雅普诺夫指数与光谱和光谱表征的联系。组件、证明算术谱跃迁的普遍性和解析准周期算子本征函数的通用层次结构的证明，多维准周期算子的扩展状态的证明，临界现象和“两个交错粒子”效应的研究其他重要目标将是研究。该项目将涉及持续开发用于证明局域型效应以及研究绝对连续的非微扰方法。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。