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

这项研究涉及薛定谔算符和量子自旋系统的局域型效应，无序环境中的渗流和接触过程，以及薛定谔算符的谱和波函数的一般研究。主要目的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方法。另一个目标是研究光谱和量子动力学性质与长时间行为之间的关系，因为它们与广义本征函数有关，特别是在多维情况下。这项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。这项拟议的研究是围绕无序系统的基本性质展开的，无序系统是含有杂质的系统的模型。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。就是这样，晶体管是由含有杂质的材料制成的。无序系统也被用来模拟许多微观和宏观效应：从量子局域化到地震。拟议的主题包括研究高度无序的量子力学系统的性质，以及在无序的临界水平上表现出高度不稳定行为的系统的性质。

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.