Correlations and Transport for Random Schrodinger Operators

本文研究了具有随机势的单粒子和多粒子量子哈密顿量的谱和输运性质。目标是了解这些模型在局域和传输区域中的关联函数的行为。一阶关联函数，即态密度，已经得到了广泛的研究。该项目的一个部分是证明它的光滑性和密度的下界。特别令人感兴趣的是称为电流-电流相关度量的二阶相关函数。该项目的一个组成部分是证明该度量的密度的存在性，并研究其性质。这一点很重要，因为密度的对角值提供了传导性。在研究随机薛定谔算子的本征值统计时，还出现了其他的二阶和高阶相关函数。还将探讨某些随机矩阵模型的相关函数和相关函数。整数量子霍尔效应与这些想法密切相关，该项目涉及到使用非平衡稳态态来研究量子霍尔系统。电子在固体中的传播一直是凝聚态物理的一个重要研究方向。量子力学已经成功地提供了对金属、半导体和绝缘体的理解。有限导电性的本质仍然难以捉摸。人们认为，由于在自然界中观察到的位错和缺陷，晶体的自然无序是晶体输运性质的部分原因。这项研究的重点是通过将晶体建模为含有随机杂质的有序阵列来理解有限电导率的数学和物理基础。在这些模型中，可以估计实验上可观察到的量，如态密度和电导率。

This research concerns the spectral and transport properties on one-and many-particle quantum Hamiltonians with random potentials. The goal is to understand the behavior of the correlation functions for these models in the localized and transport regimes. The first-order correlation function, the density of states, has been extensively studied. One part of the project is to prove its smoothness and a lower bound on the density. Of special interest is the second-order correlation function called the current-current correlation measure. One component of the project is to prove the existence of a density for this measure and to study its properties. This is important as the diagonal values of the density give the conductivity. Other second- and higher-order correlation functions arise in the study of eigenvalue statistics for random Schrodinger operators. These correlation functions and related ones for certain random matrix models will also be explored. The integer quantum Hall effect is closely related to these ideas and the project involves a study of quantum Hall systems using nonequilibrium stationary states. The propagation of electrons in solids has long been an important focus of condensed matter physics. Quantum mechanics has been successful in providing an understanding of metals, semiconductors, and insulators. The nature of finite conductivity remains elusive. It is believed that the natural disorder of crystals, due to dislocations and defects that are observed in nature, is partially responsible for the transport properties of crystals. The focus of this research is to understand the mathematical and physical basis for finite conductivity by modeling crystals as an ordered array with random impurities. Experimentally observable quantities, such as the density of states and the conductivity, can be estimated in these models.