Spectral Transitions and Critical Phenomena

研究将集中在准周期(几乎但不完全周期)结构的反常谱特性上。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。准周期系统也被用来模拟其他许多微观和宏观效应：从量子局域化到地震。其他确定性的非周期(不规则)结构也将是感兴趣的。严格理论的有计划的发展有望有助于理解所有上述现象，并可能导致找到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(分别是多杂质和少杂质)的量子力学系统的性质，这些系统表现出一定的反常行为。该项目的一个组成部分将包括培养研究生和其他年轻的研究人员。该项目由几个部分组成，包括对偶Lyapunov指数与光谱和光谱分量的表征的联系，证明解析准周期算子的算术光谱跃迁和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，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某些量子混沌模型相关的问题。该项目将包括继续开发非微扰方法来证明局部化类型的效应，以及研究绝对连续的光谱。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.