Spectral theory of ergodic Schrodinger operators and related models

拟议的研究涉及对无序系统的基本性质的研究。无序系统是含杂质系统的模型。无序系统的严格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。无序系统还用于模拟许多其他微观和宏观效应，从量子局部化（量子计算中的一个重要主题）到地震。该项目的一个组成部分是教育研究生和其他年轻研究人员。PI还将继续努力吸引K-12学生学习数学。这个项目由几个部分组成。一是证明了多维拟周期算子的扩展态。二是研究紧结合准周期模型中相互作用的影响。另一个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。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，特别是扩展哈珀模型。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。

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.