Spectral theory of ergodic Schrodinger operators and related models

该项目由两个主要部分组成。一是研究紧结合准周期模型中相互作用的影响。二是研究离散遍历薛定谔算子在局部域的局部特征值统计量。一个相关的项目是证明或否定安德森-伯努利模型的状态积分密度的奇异性。还计划研究几个与恒定或随机磁场中的布洛赫电子相关的模型。其他重要的目标是研究与准周期算子的康托/非康托谱有关的问题。该项目包括继续发展非摄动方法，用于证明薛定谔算符和量子自旋系统中的局域型效应，无序环境中的渗透和接触过程，以及绝对连续谱的研究。拟研究准周期结构和其他确定性和随机结构的异常光谱和扩散特性。因此，这是对作为含杂质系统模型的无序系统的基本性质的研究。准周期算符为整数量子霍尔效应、实验准晶体、量子混沌理论和石墨烯理论提供了中心或重要的模型。严格理论的发展有望有助于理解上述所有现象，特别是可能导致发现具有理想物理性质的新材料。无序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。提出的主题包括研究量子力学中表现出某些异常行为的高度和弱无序系统的性质。该项目的一个组成部分涉及研究生教育。还计划继续开展有关的外联活动。

The project consists of two main parts. One is to study the effects of interaction in tight-binding quasi-periodic models. The other is to study local eigenvalue statistics in the regime of localization for discrete ergodic Schrodinger operators. A related project is to prove or disprove singularity of the integrated density of states for the Anderson-Bernoulli model. It is also planned to study several models related to Bloch electrons in constant or random magnetic fields. Other important objectives are the study of issues related to Cantor/non-Cantor spectra of quasiperiodic operators. The project involves the continuing development of non-perturbative methods for the proofs of localization type effects both in Schrodinger operators and in quantum spin systems, percolation and contact processes in disordered environments, as well as for the study of absolutely continuous spectrum.The proposed research concerns the anomalous spectral and diffusive properties of quasiperiodic and other deterministic and random structures. This is therefore research on the fundamental properties of disordered systems that serve as models of systems with impurities. Quasiperiodic operators provide central or important models for integer quantum Hall effect, experimental quasicrystals, quantum chaos theory, and the theory of graphene. The development of the rigorous theory is expected to contribute to the understanding of all the above phenomena, and in particular, may lead to finding new materials with desired physical properties. Disordered systems are also used in modeling many other micro and macro effects: from quantum localization to earthquakes. The proposed topics include studying properties of both highly and weakly disordered systems of Quantum Mechanics that demonstrate certain anomalous behavior. An integral part of the project concerns educating graduate students. It is also planned to continue the related outreach activities.