Multiscattering and Absolutely Continuous Spectrum

本文的主要目的是研究具有混合谱的随机和确定性微分算子，混合谱包含一个稠密纯点谱（对应于局域态）和一个绝对连续谱（对应于扩展态）.根据经典的安德森猜想，这两种类型的光谱共存的随机薛定谔算子与“小”的障碍。这个猜想还没有得到严格的证明。该奖项将支持对这一猜想的严格证明的研究，即当无序不一定是小的，而是稀疏分布时。获奖者还将研究适当类别的稀疏函数中的相关非线性演化问题。所考虑的问题涉及量子粒子或波在局部和随机分布的障碍云中的行为。一个粒子与每一个单独的障碍物的相互作用可以用经典散射理论来描述，而全局图像是一个多散射分析的主题。有两种不同的情况，它们不一定相互排斥：粒子或波被障碍物系统捕获，或者它们可以传播到无限远。本研究将调查在什么情况下，在非常稀疏的分布障碍的情况下，哪种情况下成立。这一理论对于理解波在无序介质中的传播是必不可少的，例如具有随机分布的微缺陷的晶体。 一个具体的应用是用激光对硅表面进行质量控制。

The main objective of this research is to study random anddeterministic differential operators with mixed spectrum containing a dense pure point spectrum (corresponding to localized states) and an absolutely continuous spectrum (corresponding to extended states). According to the classical Anderson conjecture, both types of spectra coexist for random Schrodinger operators with "small" disorder. This conjecture hasnot been proved yet in a rigorous fashion. The award will support research on a rigorous proof of this conjecture when the disorder is not necessarily small in magnitude, but instead sparsely distributed.The awardees will also study related non-linear evolution problems in suitable classes of sparse functions.The problems under consideration concern the behavior of quantumparticles or waves in a cloud of localized and randomly distributed obstacles. The interaction of a particle with each individual obstacle can bedescribed in terms of classical scattering theory, while the globalpicture is a subject of multiscattering analysis. There are twodifferent scenarios which are not necessarily mutually exclusive: either particles or waves are trapped by the system of obstacles,or they can propagate to infinity. This research will investigate which scenario holds under what circumstances, in the case of very thinly distributed obstacles. This theory is essential to an understanding of wave propagation in disordered media, such as crystals with randomly distributed microdefects. A concrete application is inquality control of silicon surfaces using lasers.