Spectral Transitions and Critical Phenomena

研究将集中在准周期（几乎但不完全周期）结构的异常光谱特性。准周期算符为整数量子霍尔效应、实验准晶和量子混沌理论提供了中心或重要的模型。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性（不规则）结构也将受到关注。严格理论的计划发展预计将有助于理解所有上述现象，并可能导致找到具有所需物理特性的新材料。主题将包括研究具有强无序和弱无序（分别为许多杂质和很少杂质）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教育研究生和其他年轻研究人员。该项目由几个部分组成，包括将对偶李雅普诺夫指数与光谱和光谱成分的特征化联系起来，证明算术光谱跃迁的普遍性和解析准周期算子本征函数的普遍层次结构，证明多维准周期算子的扩展状态，临界现象和“两粒子交错”效应的研究。其他重要目标将是研究与某些量子混沌模型有关的问题。该项目将涉及非微扰方法的持续发展，以证明局域化类型的影响，以及研究绝对连续的spectrum.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.