Localization, delocalization, and other phenomena in random Schrodinger operators

该项目致力于研究随机Schrödinger算子中的局部化、离局部化和其他现象，这些算子描述了电子在随机杂质介质中的运动。在被广泛接受的图像中，在三维或多维空间中存在着从以局域态为特征的绝缘子区域到以扩展态为特征的非常不同的金属区域的过渡，而在一维或二维空间中只有局域态而没有金属-绝缘子过渡。这个项目旨在进一步加深对这幅图的数学理解。研究具有任意单点概率分布的连续统安德森哈密顿量，目的是证明谱底的局域化，并通过证明多尺度分析的一个逆来表征动态局域化区域，表明在互补区域存在非零最小输运率。如果单点概率分布具有有界密度，则证明了局部Wegner估计，从而得到局部区域内的Minami估计（即特征值的泊松统计量）。PI将通过研究条带上的Anderson模型来研究二维（离散）Anderson模型中的局部化；本文将使用一种基于超对称复制技巧的传输矩阵方法。PI将研究一个多粒子安德森模型，该模型描述了在具有随机杂质的介质中相互作用的电子运动，并研究Fock空间中的定位。PI将寻找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，这是连续体安德森汉密尔顿量的已知结果。对安德森模型的莫特公式中对数修正的正确指数将进行研究。PI还将研究Minami对随机经典波算子（例如随机声学算子和麦克斯韦算子）特征值的估计和泊松统计，它们描述了随机介质中的经典波。随机Schrödinger算符描述电子在含有随机杂质的介质中运动。在杂质存在的情况下，通常像金属一样的材料，也就是说，它传导电流，将表现出局部化，并表现得像电流的绝缘体。杂质产生金属-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聚态物理中的电子现象，如安德森局域化和量子霍尔效应。部分研究课题适合博士论文，并将用于培养未来的研究人员。

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.