Spectral Theory of Schrodinger Operators and Localization Type Effects in Disordered Environments

无序环境中薛定谔算子的谱理论和局域型效应

• 批准号: 9706443
• 负责人: Svetlana Jitomirskaya
• 金额: $ 8.01万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706443&HistoricalAwards=false
• 关键词: Spectral; Theory; Schrodinger; Operators; Localization

Proposal: DMS-9706443 PI: Svetlana Jitomirskaya This research involves localization type effects for Schrodinger operators and for quantum spin systems, percolation, and contact processes in disordered environments, and the general study of spectra and wave functions of Schrodinger operators. The main objective is to develop methods of proving localization type effects for Schrodinger operators with deterministic potentials. The other goal is to study the relation between spectral and quantum-dynamical properties and long time behavior as they relate to generalized eigenfunctions, particularly in the multidimensional case. Another goal of the proposed research is to study singular continuous spectrum that exhibits critical behavior, particularly for models where it appears for critical values (or intervals) of the parameters. The proposed research is centered around the fundamental properties of disordered systems that serve as models of systems with impurities. Impurities can substantially change various properties of the media and can thus be used to create materials with desired properties. In just this way transistors are made of materials with impurities. Disordered systems are also used in modeling many micro and macro effects: from quantum localization to earthquakes. The proposed topics include studying properties of highly disordered systems of Quantum Mechanics and of systems at critical levels of disorder that demonstrate highly unstable behavior.

PI: Svetlana Jitomirskaya本研究涉及薛定谔算子和量子自旋系统的局域化效应，无序环境中的渗透和接触过程，以及薛定谔算子的谱和波函数的一般研究。主要目的是发展具有确定性势的薛定谔算子的局部化效应的证明方法。另一个目标是研究谱和量子动力学性质与长时间行为之间的关系，因为它们与广义特征函数有关，特别是在多维情况下。提出的研究的另一个目标是研究表现出临界行为的奇异连续谱，特别是对于参数的临界值（或区间）出现的模型。提出的研究是围绕无序系统的基本性质，作为系统的模型与杂质。杂质可以实质上改变介质的各种性质，因此可以用来制造具有所需性质的材料。就这样，晶体管是由含有杂质的材料制成的。无序系统也用于模拟许多微观和宏观效应：从量子局域化到地震。提议的主题包括研究量子力学的高度无序系统和系统在无序的临界水平，表现出高度不稳定的行为的性质。