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: 1101578
• 负责人: Svetlana Jitomirskaya
• 金额: $ 41.99万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101578&HistoricalAwards=false
• 关键词: Spectral; theory; ergodic; Schrodinger; operators

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.

该项目由两个主要部分组成。一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薛定谔算子本地化机制中的局部特征值统计。一个相关的项目是证明或反驳安德森-伯努利模型的积分态密度的奇异性。还计划研究恒定或随机磁场中与布洛赫电子相关的几种模型。其他重要目标是研究与准周期算子的康托/非康托谱相关的问题。该项目涉及持续开发非微扰方法，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。拟议的研究涉及准周期和其他确定性和随机结构的反常光谱和扩散特性。因此，这是对作为含杂质系统模型的无序系统基本性质的研究。准周期算子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。严格理论的发展预计将有助于理解所有上述现象，特别是可能导致寻找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：从量子局域化到地震。拟议的主题包括研究量子力学的高度无序和弱无序系统的特性，这些系统表现出某些异常行为。该项目的一个组成部分涉及研究生教育。还计划继续开展相关外展活动。