Spectral Properties of Ergodic Schroedinger Operators

Svetlana Jitomirskaya这个项目是研究遍历薛定谔算符的谱和局域型效应，以及准周期和Anderson模型类型背景下的反常绝对连续谱和反常量子输运。它还计划研究与恒定和/或随机磁场中的Bloch电子有关的几个模型。该项目包括继续发展非微扰方法，用于证明局部化和其他相关性质，以及研究绝对连续谱。其他重要目标是研究准周期算子的Cantor谱的相关问题，它的出现、流行和标度性质，以及光滑(而不是解析)方法的发展。这是对无序系统的基本性质的基础研究，无序系统是含有杂质的系统的模型。准周期算符为整数量子霍尔效应、实验准晶和量子混沌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所有这三种现象，特别是可能导致找到具有所需物理性质的新材料。无序系统也被用来模拟其他许多微观和宏观效应：从量子局域化到地震。建议的主题包括研究表现出某些反常行为的高度无序和弱无序量子力学系统的性质。

Spectral Properties of Ergodic Schroedinger OperatorsAbstract of Proposed ResearchSvetlana Jitomirskaya This project is to study spectra and localization type effects for ergodic Schroedinger operators and the study of anomalous absolutely continuous spectrum and anomalous quantum transport in the quasiperiodic and Anderson model-type settings. It is also planned to study several models related to Bloch electrons in constant and/or random magnetic fields. The project involves the continuing development of non-perturbative methods both for the proofs of localization and other related properties as well as for the study of absolutely continuous spectrum. Other important objectives are the study of issues related to Cantor spectra of quasiperiodic operators, it's occurrence, prevalence, and scaling properties, and the development of smooth (rather than analytic) methods.The proposed research investigates the anomalous spectral and diffusive properties of quasiperiodic and other deterministic and random structures. This is basic 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, and quantum chaos theory. The development of the rigorous theory is expected to contribute to the understanding of all three 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.