Spectral theory of ergodic Schrodinger operators and related models

建议的研究涉及无序系统的基本性质的研究。无序系统是含有杂质的系统的模型。严格的无序系统理论的发展有望有助于理解许多类型的物理现象，特别是可能导致发现具有所需物理性质的新材料。无序系统也被用于模拟从量子局域化到地震的许多其他微观和宏观效应。量子局域化是量子计算中的一个重要主题。该项目的一个组成部分是培养研究生和其他年轻的研究人员。该小组亦会继续外展工作，吸引幼稚园至幼稚园的学生学习数学。一是证明了多维拟周期算子的扩张态。二是研究紧束缚准周期模型中相互作用的影响。一是研究离散遍历薛定谔算子局部化状态下的局部特征值统计问题。它还计划研究几个与恒定磁场中的布洛赫电子有关的模型，特别是扩展的哈珀模型。其他重要目标是研究准周期算子的Cantor/非Cantor谱的相关问题。该项目包括继续发展非微扰方法，以证明薛定谔算符和量子自旋系统中的局域型效应，无序环境中的渗流和接触过程，以及绝对连续谱的研究。

The proposed research concerns the study of the fundamental properties of disordered systems. Disordered systems are models of systems with impurities. The development of a rigorous theory of disordered systems is expected to contribute to the understanding of many types of physical 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- an important topic in quantum computation- to earthquakes. An integral part of the project concerns educating graduate students and other young researchers. The PI will also continue her outreach efforts to entice K-12 students to the study of mathematics.The project consists of several parts. One is to prove the extended states for multidimensional quasiperiodic operators. Another is to study the effects of interaction in tight-binding quasi-periodic models. One more is to study local eigenvalue statistics in the regime of localization for discrete ergodic Schroedinger operators. It is also planned to study several models related to Bloch electrons in constant magnetic fields, notably the Extended Harper Model. 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.