Random Schrodinger operators

我们将使用广泛的数学方法(从泛函分析和算子理论、常微分方程和偏微分方程式、概率和随机过程、复数和调和分析)来增加对随机薛定谔算子的理解。特别是，这些算符的本征值之间的能级排斥将被研究。对于比以前可能的更大类别的模型，将显示绝缘无序材料的水平斥力的缺失。此外，还将研究一个模型，该模型显示了从没有能级斥力到存在能级斥力的转变。对于随机位移模型，即模拟晶体中晶格涨落的薛定谔算符，将确定低能态。为了更好地理解外力作用下局域化现象的稳定性，我们将随机薛定谔算符作为数学模型来研究无序介质中的电子输运。材料结构中的无序产生的效应是区分导体和绝缘体的核心。所观察到的局域态和扩展态等现象为电子学提供了数学基础，但也适用于无序介质中的声波、电磁波和弹性波。该项目将进一步开发广泛的数学学科的工具，这是严格理解这些现象所必需的。将特别注意电子能量的水平统计，它允许通过小材料样本的性质来表征绝缘体和导体。这项工作将包括多项合作努力，并为培养三名博士生做出贡献。结果将通过出版物以及在国家和国际会议上广泛传播。

A wide array of mathematical methods (from functional analysis and operator theory, ordinary and partial differential equations, probability and stochastic processes, complex and harmonic analysis) will be used to increase the understanding of random Schrodinger operators. In particular, level repulsion between the eigenvalues of these operators will be studied. Absence of level repulsion for insulating disordered materials will be shown for larger classes of models than previously possible. Also, a model will be investigated which shows the transition from absence to presence of level repulsion. For the random displacement model, a Schrodinger operator modelling lattice fluctuations in crystals, the low energy states will be determined. Work will be done to better understand the stability of the localization phenomenon in the presence of exterior deterministic forces, leading to non- stationary random media.Random Schrodinger operators are used as mathematical models to study electron transport of disordered media such as crystals with impurities, alloys, and amorphous materials. The effects arising from the presence of disorder in the structure of a material are central to distinguish between conductors and insulators. The phenomena which are observed, such as localization and extended states, provide mathematical foundations for electronics, but also apply to acoustic, electro-magnetic and elastic waves in disordered media. This project will further develop tools from a broad array of mathematical disciplines which are required for a rigorous understanding of these phenomena. Particular attention will be given to level statistics of electronic energies which allow to characterize insulators and conductors through properties of small material samples. The work will include multiple collaborative efforts and contribute to the training of three PhD students. Results will be widely disseminated through publications and at national and international conferences.