Localization, delocalization, and other phenomena in random Schrodinger operators

随机薛定谔算子中的定位、离域和其他现象

• 批准号: 1001509
• 负责人: Abel Klein
• 金额: $ 15.92万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001509&HistoricalAwards=false
• 关键词: Localization; delocalization; other; phenomena; random

This project is devoted to the study of localization, delocalization, and other phenomena in random Schrödinger operators, which describe an electron moving in a medium with random impurities. In the widely accepted picture, in three or more dimensions there exists a transition from an insulator region, characterized by localized states, to a very different metallic region, characterized by extended states, while in one or two dimensions there are only localized states and no metal-insulator transition. This project aims to further the mathematical understanding of this picture. The continuum Anderson Hamiltonian with arbitrary single-site probability distribution will be studied, with the objective of proving localization at the bottom of the spectrum, and to characterize the region of dynamical localization by proving a converse to the multiscale analysis, showing existence of a nonzero minimal rate of transport in the complementary region. If single-site probability distribution has a bounded density, a local Wegner estimate and will be proved to obtain Minami's estimate (and hence Poisson statistics for eigenvalues) in the region of localization. The PI will investigate localization in the two-dimensional (discrete) Anderson model by studying the Anderson model on the strip; a transfer matrix approach based on the supersymmetric replica trick will be used. The PI will study a multi-particle Anderson model describing interacting electrons moving in a medium with random impurities, and investigate localization in Fock space. The PI will search for a proof of localization for the Anderson model where the single-site potential is a Bernoulli random variable in two or more dimensions, a known result for the continuum Anderson Hamiltonian. The correct exponent for the logarithmic correction in Mott's formula for the Anderson model will be investigated. The PI will also study Minami's estimate and Poisson statistics for eigenvalues of random classical wave operators (e.g., random acoustic and Maxwell operators), which describe classical waves in random media.Random Schrödinger operators describe an electron moving in a medium with random impurities. In the presence of impurities, a material that normally acts like a metal, i.e., it conducts electric current, will exhibit localization and behave like an insulator for electric currents. The impurities create a metal-insulator transition with important consequences for electric currents. This research will contribute to the understanding of electronic phenomena in condensed matter physics, such as Anderson localization and the quantum Hall effect. Some of the topics of research are suitable for PhD theses, and will be used for the training of future researchers.

该项目致力于随机Schrödinger运算符中的定位，离域和其他现象的研究，该项目描述了一种在具有随机杂质的介质中移动的电子。在公认的图像中，在三个或多个维度中存在从以局部状态为特征的绝缘体区域过渡到以扩展状态为特征的非常不同的金属区域，而在一个或两个维度中，只有局部状态和金属绝缘体过渡。该项目旨在进一步了解这张图片的数学理解。将研究具有任意单位概率分布的持续安德森汉密尔顿人，目的是证明在频谱底部的定位，并通过证明与多尺度分析的交谈来表征动态定位区域，并显示存在整个区域中非零最小运输速率的存在。如果单位概率分布的密度有限，则是局部韦格纳估计值，并将被证明可以在定位区域获得Minami的估计值（以及特征值的泊松统计）。 PI将通过研究脱带上的Anderson模型来研究在二维（离散）Anderson模型中的定位。将使用基于超对称复制技巧的转移矩阵方法。 PI将研究一个多粒子的安德森模型，描述了在具有随机杂质的介质中移动的相互作用电子设备，并研究了在Fock空间中的定位。 PI将为安德森模型寻找本地化的证明，其中单位点电位是两个或多个维度的伯努利随机变量，这是继续安德森·汉密尔顿（Anderson Hamiltonian）的已知结果。将研究Mott公式中对数校正的正确指数。 PI还将研究Minami的估计值和泊松统计数据，用于随机经典波浪运算符（例如随机声学和麦克斯韦操作员）的特征值，这些电波描述了随机介质中的经典波。在存在杂质的情况下，通常像金属一样起作用的材料，即进行电流，将执行定位，并像电流的绝缘体一样表现。这些杂质会产生金属 - 绝缘体过渡，对电流产生重要影响。这项研究将有助于理解凝结物理学中的电子现象，例如安德森本地化和量子厅效应。研究的一些主题适用于博士学位论文，将用于培训未来的研究人员。