Spectral theory of ergodic Schrodinger operators and related models

遍历薛定谔算子的谱论及相关模型

• 批准号: 1401204
• 负责人: Svetlana Jitomirskaya
• 金额: $ 50万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1401204&HistoricalAwards=false
• 关键词: Spectral; theory; ergodic; Schrodinger; operators

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.

拟议的研究涉及无序系统的基本性质的研究。 无序系统是具有杂质的系统的模型。无序系统的严格理论的发展预计将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特性的新材料。 无序系统也被用于模拟许多其他微观和宏观效应，从量子局域化（量子计算中的一个重要课题）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还将继续她的推广工作，以吸引K-12学生学习数学。该项目包括几个部分。一是证明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。另一个是研究离散遍历Schroedinger算子局部化区域的局部特征值统计。 还计划研究与恒定磁场中布洛赫电子有关的几个模型，特别是扩展的哈珀模型。其他重要目标是研究与准周期算子的康托/非康托谱有关的问题。该项目涉及继续发展非微扰方法，用于证明薛定谔算子和量子自旋系统中的局域化类型效应，无序环境中的渗透和接触过程，以及绝对连续光谱的研究。