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.

该项目致力于研究随机薛定谔算符中的定域、离域和其他现象，随机薛定谔算符描述了电子在具有随机杂质的介质中的运动。 在被广泛接受的图像中，在三维或更多维中存在从以局域态为特征的绝缘体区域到以扩展态为特征的非常不同的金属区域的过渡，而在一维或二维中只有局域态，没有金属-绝缘体过渡。该项目旨在进一步对这幅图的数学理解。 将研究具有任意单点概率分布的连续安德森哈密顿量，其目的是 证明本地化的底部的频谱，并证明一个匡威的多尺度分析，显示存在一个非零的最小传输率在互补区域的动态本地化的区域的特点。 如果单站点概率分布有界密度，则局部Wegner估计和将被证明在局部化区域中获得Minami估计（以及特征值的泊松统计）。 PI将通过研究条带上的安德森模型来研究二维（离散）安德森模型中的局部化;将使用基于超对称复制技巧的传递矩阵方法。 PI将研究多粒子安德森模型， 相互作用的电子在随机杂质的介质中运动，并调查在Fock空间的本地化。PI将搜索安德森模型的局部化证明，其中单点势是二维或多维的伯努利随机变量，这是连续安德森哈密顿量的已知结果。将研究安德森模型的Mott公式中对数校正的正确指数。 PI还将研究随机经典波算子特征值的Minami估计和泊松统计（例如，随机声学和麦克斯韦算子），它们描述了随机介质中的经典波。随机薛定谔算子描述了电子在具有随机杂质的介质中运动。 在存在杂质的情况下，通常表现得像金属的材料，即，它传导电流，将表现出局部化并且表现得像电流的绝缘体。杂质产生金属-绝缘体转变，对电流产生重要影响。这一研究将有助于理解凝聚态物理中的电子现象，如安德森局域化和量子霍尔效应。 一些研究课题适合博士论文，并将用于未来研究人员的培训。