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。杂质通过随机过程建模，并且可观察量的期望值，例如扩散率、电导率和状态密度，表示在许多测量中这些量的平均值。研究了几种模型：杂质可以位于代表合金型模型的晶体的晶格点处，或者它们可以位于间隙位置处，间隙位置的位置由泊松过程给出。在一个完美的晶体中，电子自由地传播，只要它们的能量位于由晶体结构决定的固定能带中。光谱是绝对连续的，电导率是无限的。随机杂质的添加改变了这一描述。如果无序足够大，所有电子态都在空间中局域化，电导率为零。在弱无序和大于2的维度中，人们认为在某些能量下存在有限的电导率，而在带边缘附近的其他能量下存在局域态。这是一个开放的猜想，也是提出研究的主要动机之一。 该建议的主要项目涉及随机单体和多体随机薛定谔算子的光谱和运输方面。态密度和高阶关联函数的规律性，包括电导率测量和带间光吸收系数，将被彻底研究。这些量在弱无序区的行为特别有趣。即使是最简单的相关函数，如态密度，也知之甚少。随机薛定谔算子的动力学性质也表现在各种模型的谱统计中。对于许多单粒子薛定谔算符，在无序区的局部谱统计是泊松的。这是多粒子薛定谔算子的预期，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标志。半导体中的随机杂质会影响这些器件的电子特性。虽然单电子模型是许多现象的一个很好的描述，一个完整的图片需要使用多粒子薛定谔算子。这些随机模型的行为才刚刚开始研究，是本研究提案的组成部分之一。该研究员将与肯塔基州大学的几名博士生一起研究几种随机模型的传输和光谱现象。主要目标之一是了解各种粒子之间的相关性以及无序如何影响它们。