Spectral and Transport Properties of Random Media

随机介质的光谱和输运特性[来源：美国肯塔基大学]Peter D. Hislop摘要：本提案涉及首席研究员对随机介质的光谱和输运特性的持续研究。基本问题是：背景介质（例如完美晶体）的随机扰动如何影响量子波和经典波在介质中的传播？提出的工作集中于与量子霍尔效应相关的态的综合密度和模型的性质。讨论了其他几种模型和相关的技术问题。该建议的主要内容包括：1)国家综合密度的连续性和规律性；2)量子霍尔效应中的光谱和输运问题；3)定位、能级统计和随机磁场的Aizenman-Molchanov方法。这项工作的总体目标是描述无序对系统可测量特性的影响，如状态密度、电导率和边缘电流。作者还讨论了一些以前未处理过的系统，如随机磁场和非局部电位的局部化。所提出的方法将有助于理解波在随机介质中的传播。我们对固体的许多基本电子特性的理解是基于电子在周期性原子阵列中传播的简单单电子模型。这个简单的想法解释了许多基本现象，如金属、绝缘体和半导体。这个模型的一个限制是它预测了无限的电导率。1958年，p·w·安德森提出，无序是限制原子周期性排列中电子的弹道行为的一种机制。他认为，由于随机分布的量的多重和非相干散射，电子波函数应该在空间中定位，概率为1。对于许多模型来说，无序引起的状态局部化已经得到了证明。这个提议中的一些工作涉及到将这些想法应用到具有有趣几何形状的系统的研究中，比如表现出量子哈雷效应的条形或环形。首席研究员感兴趣的是量子霍尔边电流的性质，以及具有两条边的区域（如条带）的电子态的性质。研究这些系统的工具之一是态密度，它提供了单位体积内电子态数量的度量。态密度测量的连续性提供了一种衡量无序在打破有序系统简并方面有效性的度量。我们对这个函数所知甚少。主要研究者还对探索磁场中随机性对电子传播的影响以及随机性对某些具有非局部相互作用的系统的影响感兴趣。

Spectral and Transport Properties of Random MediaPI: Peter D. Hislop, University of KentuckyDMS-0202656Abstract:This proposal concerns continuing investigations of the principalinvestigator into the spectral and transport properties of random media.The basic question is: How do random perturbations of a backgroundmedium, for example, a perfect crystal, influence the propagationof quantum and classical waves in the medium?The proposed work concentrates on properties of the integrated density ofstates and models related to the quantum Hall effect.Several other models and related technical problems are discussed.The main aspects of the proposal include:1) Continuity and Regularity of the Integrated Density of States;2) Spectral and Transport Problems in the Quantum Hall Effect;3) the Aizenman-Molchanov Method for Localization, Energy-Level Statistics,and Random Magnetic Fields.The overall goal of this work is to describe the effect ofdisorder on the measurable properties of the system,such as the density of states, the conductivity,and the edge currents. The proposer also discusseslocalization for some previously untreated systemslike random magnetic fields and nonlocal potentials.The proposed methods will contribute to theunderstanding of wave propagation in random media.Our understanding of many basic electronic properties of solids isbased on the simple one-electron model of an electron propagatingin a periodic array of atoms. This simple idea explains many fundamentalphenomena such as metals, insulators, and semiconductors. One of thelimitations of this model is that it predicts infinite conductivity. In1958, P. W. Anderson proposed disorder as a mechanism for limiting theballistic behavior of an electron in a periodic array of atoms. He arguedthat the electron wave function should be localized in spacedue to multiple and incoherent scattering from the randomly distributedimpurities, with probability one. Disorder-induced localization of statesfor many models has now been proved. Some of the work in this proposalinvolves applying these ideas to the study of systems with interestinggeometries, such as a strip or a torus, that exhibit the quantum Halleffect. The principal investigator is interested in the nature of the quantum Hall edge-currents, and the nature of the electron states for regions with two-edges, such as a strip. One of the tools for studying these systems is the density of states that provides a measure of the number of electronicstates per unit volume. The continuity of the density of states measureprovides a measure of the effectiveness of the disorder in breaking degeneracy of the ordered system. Very little is known about this function. The principal investigator is also interested in exploring the effect on electron propagation of randomness in the magnetic field, and the influence of randomness on certain systems with nonlocal interactions.