Topics in the theory of random Schrodinger operators

随机薛定谔算子理论的主题

• 批准号: 1103104
• 负责人: Peter Hislop
• 金额: $ 19.91万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-15 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1103104&HistoricalAwards=false
• 关键词: Topics; theory; random; Schrodinger; operators

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.

单粒子随机操作员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具有杂质的晶体中的传播。杂质以随机过程和可观察量的期望为模型，例如扩散率，电导率和状态的密度，代表了这些数量超过许多测量值的平均值。研究了几种型号：杂质可能位于代表合金型型号的晶体的晶格点，或者它们可以定位在泊松过程给定位置的间隙位点。在完美的晶体中，电子传播的能量在由晶体结构确定的固定能带中。频谱绝对连续，电导率是无限的。随机杂质的添加改变了此描述。如果该疾病足够大，所有电子状态都位于空间中，电导率为零。在弱混乱症和大于两个的维度下，人们认为某些能量有限的电导率，而频带边缘附近的其他能量处有局部状态。这是一个开放的猜想，也是拟议研究的主要动机之一。 该提案的主要项目涉及随机单体和多体随机施罗宾格运营商的频谱和运输方面。状态密度和高阶相关函数的规律性（包括电导率度量和带间光吸收系数）将得到彻底研究。这些数量在弱混乱制度中的行为特别有趣。即使是最简单的相关函数，例如状态的密度，也很少闻名。随机Schrodinger操作员的动力学特性也体现在各种模型的光谱统计中。对于许多单粒子Schrodinger操作员来说，疾病制度中的局部光谱统计是泊松人。这是许多粒子施罗宾格运营商预期的，并且是拟议研究的一部分。随机媒体中电子和波浪的传播是物理世界的标志。半导体中的随机杂质会影响这些设备的电子特性。尽管单电子模型是对许多现象的良好描述，但完整的图片需要使用多粒子Schrodinger操作员。这些随机模型的行为才刚刚开始研究，是该研究建议的组成部分之一。研究人员与肯塔基大学的几位博士生一起，将研究几种随机模型的运输和光谱现象。主要目标之一是了解各种粒子与疾病如何影响它们之间的相关性。