Spectral and Transport Theory of Schrodinger Operators

这项研究涉及遍历薛定谔算符的局域型效应，多维薛定谔算符的谱和波函数的一般研究，以及准周期算符和二维磁性算符在整数量子霍尔区的输运现象。一个重要的目标是发展证明具有确定性位势的薛定谔算子的局部化类型效应的非摄动方法。另一个目标是研究谱和量子动力学性质之间的关系以及广义本征函数的行为，特别是在局域范围外和多维情况下。这项研究的另一个目的是研究表现出临界行为和/或异常输运的奇异连续谱，特别是对于参数出现临界值(或区间)的模型。所提出的研究集中在无序系统的基本性质上，无序系统是含有杂质的系统的模型。定义势，特别是准周期势最常被用来模拟准晶。为了能够理解大量的准晶实验数据，研究微观模型的电导和热导等输运系数尤为重要。这样的认识有助于寻找具有所需物理性能的新材料。这可能会导致各种工业应用(目前的第一个应用是用平底锅覆盖来取代传统的特法尔薄膜)。无序系统也被用来模拟其他许多微观和宏观效应：从量子局部化到地震。我们的研究涉及准周期结构和其他确定性结构的反常谱和扩散性质。自1985年以来，美国国家标准局一直使用量子霍尔效应来定义精细结构常数(以及电子的电荷)。人们仍然不能很好地理解为什么这个实验可以重现，相对误差只有10^-8美元。我们的研究涉及到量子霍尔效应的微观理论，旨在更深入地了解这一现象。

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.