Spectral and Transport Theory of Schrodinger Operators

薛定谔算子的谱与输运理论

基本信息

批准号：
0070755
负责人：
Svetlana Jitomirskaya
金额：
$ 18.6万
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：
2000
资助国家：
美国
起止时间：
2000-07-01 至 2003-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0070755&HistoricalAwards=false
关键词：
Spectral Transport Theory Schrodinger Operators

项目摘要

This research involves localization type effects for ergodic Schrodingeroperators, the general study of spectra and wave functions ofmultidimensional Schrodinger operators, and also of the transportphenomena of quasiperiodic operators and two-dimensional magneticoperators in the integer quantum Hall regime. An important objective is todevelop nonperturbative methods of proving localization type effects forSchrodinger operators with deterministic potentials. The other goal is tostudy the relation between spectral and quantum-dynamical properties andbehavior of the generalized eigenfunctions, particularly outside thelocalization range and in the multidimensional case. Another goal of theproposed research is to study singular continuous spectrum that exhibitscritical behavior and/or anomalous transport, particularly for models whereit appears for critical values (or intervals) of the parameters. The proposed research is centered around the fundamental propertiesof disordered systems that serve as models of systems with impurities.Deterministic, particularly quasiperiodic, potentials are most often usedto model quasicrystals. In order to be able to understand much of theexperimental data on quasicrystals, it is particularly important toinvestigate the transport coefficients like the electrical and heatconductivities of the microscopic models. Such an understanding is mosthelpful for finding new materials with desired physical properties. Thismay lead to various industrial applications (the first one nowadays beingthe covering of pans replacing the conventional Tefal film). Disorderedsystems are also used in modeling many other micro and macro effects: fromquantum localization to earthquakes. Our research concerns the anomalousspectral and diffusive properties of quasiperiodic and other deterministicstructures. The quantum Hall effect is since 1985 used by the NationalBureau of Standards to define the Fine Structure Constant (and hence theelectrical charge of an electron). It is still not well understood why theexperiment can be reproduced with a relative error of only $10^{-8}$. Ourresearch is concerned with a microscopic theory of the quantum Hall effectthat is aimed at getting deeper insights of this phenomena.

本研究涉及遍历薛定谔算子的局域型效应、多维薛定谔算子的谱和波函数的一般研究，以及整数量子霍尔体系中准周期算子和二维磁算子的输运现象。一个重要的目标是开发非微扰方法来证明具有确定性势的薛定谔算子的局域化效应。另一个目标是研究光谱和量子动力学特性以及广义本征函数的行为之间的关系，特别是在局域范围之外和多维情况下。拟议研究的另一个目标是研究表现出临界行为和/或异常传输的奇异连续谱，特别是对于出现参数临界值（或区间）的模型。拟议的研究以无序系统的基本特性为中心，这些无序系统可作为含杂质系统的模型。确定性势，特别是准周期势，最常用于模拟准晶体。为了能够理解准晶体的大量实验数据，研究微观模型的输运系数（例如电导率和热导率）尤为重要。这种理解对于寻找具有所需物理性能的新材料最有帮助。这可能会带来各种工业应用（目前第一个应用是取代传统特福薄膜的锅盖）。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。我们的研究涉及准周期和其他确定性结构的反常光谱和扩散特性。自 1985 年以来，国家标准局使用量子霍尔效应来定义精细结构常数（以及电子的电荷）。目前还不太清楚为什么可以以只有 $10^{-8}$ 的相对误差来重现该实验。我们的研究涉及量子霍尔效应的微观理论，旨在更深入地了解这种现象。