Topics in the theory of random Schrodinger operators

单粒子和多粒子随机算符描述了电子和声波与随机扰动介质的相互作用。主要的例子是电子在含有杂质的晶体中的传播。杂质是由随机过程模拟的，可观测量的预期，如扩散系数、电导率和态密度，代表了这些量在许多测量中的平均值。研究了几种模型：杂质可能位于代表合金型模型的晶体的晶格点，也可能定域在由泊松过程给出位置的间隙位置。在完美的晶体中，只要电子的能量位于由晶体结构决定的固定能带内，电子就可以自由传播。光谱是绝对连续的，导电性是无穷大的。随机杂质的加入改变了这种描述。如果无序足够大，所有电子态都在空间中定域，电导率为零。在弱无序和维度大于2的情况下，人们认为在某些能量处存在有限导电性，而在靠近带边缘的其他能量处存在局域态。这是一个公开的猜想，也是这项研究的主要动机之一。本方案的主要工作涉及随机单体和多体随机薛定谔算符的谱和输运问题。将深入研究态密度和高阶关联函数的规律性，包括电导测量和带间光吸收系数。这些量在弱无序状态下的行为特别有趣。即使是最简单的关联函数，比如态密度，也知之甚少。随机薛定谔算符的动力学性质也体现在各种模型的谱统计中。对于许多单粒子薛定谔算符，无序区的局域谱统计是泊松的。这对于多粒子薛定谔算符是预期的，也是拟议研究的一部分。电子和波在随机介质中的传播是物理世界的一个标志。半导体中的随机杂质会影响这些器件的电子性能。尽管单电子模型很好地描述了许多现象，但完整的图像需要使用多粒子薛定谔算符。这些随机模型的行为才刚刚开始被研究，并且是本研究方案的组成部分之一。这位研究人员将与肯塔基大学的几名博士生一起研究几个随机模型的输运和光谱现象。主要目标之一是了解各种粒子之间的相互关系以及无序如何影响它们。

One- and many-particle random operators describe the interaction of electrons and acoustic waves with randomly perturbed media. The primary example is the propagation of electrons in a crystal with impurities. The impurities are modeled by a random process and the expectations of observable quantities, such as diffusivity, conductivity, and the density of states, represent the average over these quantities over many measurements. Several models are studied: the impurities may be located at the lattice points of the crystal representing alloy-type models, or they may be localized at interstitial sites whose location is given by a Poisson process. In a perfect crystal, the electrons propagate freely provided their energies lay in fixed energy bands determined by the crystalline structure. The spectrum is absolutely continuous and the conductivity is infinite. The addition of random impurities changes this description. If the disorder is sufficiently large, all electron states are localized in space and the conductivity is zero. At weak disorder, and in dimensions greater than two, it is believed that there is finite conductivity at certain energies, whereas there are localized states at other energies near the edges of the bands. This is an open conjecture and one of the main motivations for the proposed research. The main projects of this proposal concern spectral and transport aspects of random one- and many-body random Schrodinger operators. The regularity of the density of states and higher-order correlation functions, including the conductivity measure and the inter-band light absorption coefficients, will be thoroughly investigated. The behavior of these quantities in the weak disorder regime is particularly interesting. Very little is known for even the simplest correlation functions, such as the density of states. The dynamical properties of random Schrodinger operators are also manifest in the spectral statistics of various models. For many one-particle Schrodinger operators, the local spectral statistics in the disorder regime is Poissonian. This is expected for many-particle Schrodinger operators and is part of the proposed research. The propagation of electrons and waves in random media is a hallmark of the physical world. Random impurities in semiconductors impact the electronic properties of these devices. Although the one-electron model is a good description of many phenomena, a complete picture requires the use of multi-particle Schrodinger operators. The behavior of these random models is just beginning to be studied and is one of the components of this research proposal. The investigator, together with several doctoral students at the University of Kentucky, will investigate transport and spectral phenomena of several random models. One of the main goals is to understand correlations between various particles and how the disorder influences them.