Phenomena in random Schrodinger operators

随机薛定谔算符描述了电子在含有随机杂质的介质中的运动。在被广泛接受的图像中，在三维或更多维上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转变，而在一维或二维中仅存在局域态。这项提议旨在加深对这幅图画和相关主题的数学理解。可以说，最重要的悬而未决的问题是三维或更多维度中非本地化的存在。由于局部化仅通过多尺度分析(或如果适用，则通过分数矩方法)来证明，在实践中，局部化区域是可以执行多尺度分析的光谱区域。我们建议通过证明多尺度分析失效的谱区的存在，即违反了多尺度分析的某些结果，并通过显示该区域的离域性质(例如，输运)来在离域方面取得进展。我们将继续研究Anderson模型中线性响应理论中的交流电导，以获得局域化和离域化的见解。我们将研究四维或更多维薛定谔算符和二维或更多维磁场薛定谔算符的态密度的连续性。我们将把Bootstrap多尺度分析推广到多粒子连续Anderson哈密顿量，一个相互作用的多粒子随机薛定谔算子，得到具有有限多个本征值的Anderson局部化，动态局部化，本征函数相关的衰变等。我们还将研究多粒子连续Anderson哈密顿量的局部化，除了紧支撑外，不假设单点概率分布，允许Bernoulli和其他奇异的单点概率分布。当单位点势是Bernoulli随机变量时，我们将研究(离散)Anderson模型在二维或更多维中的局域化，这是一个长期存在的公开问题。随机薛定谔算符描述了电子在含有随机杂质的介质中运动的情况。在存在杂质的情况下，正常情况下类似金属的材料(即，它传导电流)将表现出局域性，并且其行为类似于电流的绝缘体。这些杂质造成了金属-绝缘体的转变，产生了重要的实际后果。这项研究将有助于理解凝聚态物理中的电子现象，如安德森定域化和量子霍尔效应。一些研究课题适合博士论文，并将用于培养未来的研究人员。

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.