Phenomena in random Schrodinger operators

随机薛定谔算子中的现象

• 批准号: 1301641
• 负责人: Abel Klein
• 金额: $ 78.3万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301641&HistoricalAwards=false
• 关键词: Phenomena; random; Schrodinger; operators

Random Schrödinger operators 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. This proposal aims to further the mathematical understanding of this picture and of related topics. Arguably the most important open question is the existence of delocalization in three or more dimensions. Since localization has only been proved by a multiscale analysis (or by the fractional moment method, if applicable), in practice the region of localization is the spectral region where the multiscale analysis can be performed. We propose to make progress on delocalization by proving the existence of a spectral region where the multiscale analysis breaks down, i.e., some consequence of the multiscale analysis is violated, and by showing delocalization properties (e.g., transport) in this region. We will continue our study of the ac-conductivity in linear response theory for the Anderson model to obtain insight on localization and delocalization. We will investigate the continuity of the density of states for Schrödinger operators in four or more dimensions, and for Schrödinger operators with a magnetic field in two or more dimensions. We will extend the bootstrap multiscale analysis to the multi-particle continuous Anderson Hamiltonian, an interacting multi-particle random Schrödinger operator, obtaining Anderson localization with finite multiplicity of eigenvalues, dynamical localization, decay of eigenfunction correlations, etc. We will also investigate localization for multi-particle continuous Anderson Hamiltonians with no assumptions on the single site probability distribution except for compact support, allowing for Bernoulli and other singular single site probability distributions. We will investigate localization for the (discrete) Anderson model in two or more dimensions when the single-site potential is a Bernoulli random variable, a longstanding open problem.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 practical consequences. 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.

随机薛定谔算符描述了电子在随机杂质介质中的运动。 在广泛接受的图像中，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。这个建议的目的是进一步的数学理解这幅图和相关的主题。可以说，最重要的悬而未决的问题是在三维或三维以上的离域的存在。 由于局部化只能通过多尺度分析（或分数阶矩法，如果适用的话）来证明，因此在实践中，局部化的区域是可以进行多尺度分析的谱区域。 我们建议通过证明多尺度分析崩溃的谱区域的存在，即，违反了多尺度分析的某些结果，并且通过显示离域特性（例如，（交通）在这一地区。我们将继续研究安德森模型的线性响应理论中的交流电导率，以获得关于局域化和离域化的见解。我们将研究四维或更多维中薛定谔算符的态密度连续性，以及二维或更多维中具有磁场的薛定谔算符的态密度连续性。 我们将把Bootstrap多尺度分析方法推广到多粒子连续安德森哈密顿量，一个相互作用的多粒子随机薛定谔算子，得到了具有有限重本征值的安德森局域化，动力学局域化，本征函数相关的衰减等。我们还将研究多粒子连续安德森哈密顿量的局域化，除了紧支撑外，没有对单点概率分布的假设，考虑到伯努利和其他奇异单点概率分布。我们将研究二维或多维（离散）安德森模型的局部化，当单格点势是伯努利随机变量时，这是一个长期的开放问题。随机薛定谔算子描述了电子在随机杂质介质中的运动。 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。杂质产生了具有重要实际后果的金属-绝缘体转变。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。