Multiscattering and Absolutely Continuous Spectrum

本研究的主要目的是研究含有稠密纯点谱(对应于局域态)和绝对连续谱(对应于扩展态)的混合谱的随机和确定性微分算子。根据经典的Anderson猜想，对于具有“小”无序的随机薛定谔算符，这两种类型的谱共存。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，当无序的规模不一定很小，而是稀疏分布时。获奖者还将在适当的稀疏函数类中研究相关的非线性演化问题。正在考虑的问题涉及量子粒子或波在局部和随机分布的障碍物云中的行为。粒子与每个单独障碍物的相互作用可以用经典散射理论来描述，而全局图像是多重散射分析的对象。有两种不同的情况不一定是相互排斥的：要么粒子或波被障碍物系统困住，要么它们可以传播到无限远。这项研究将调查在什么情况下，在障碍分布非常稀疏的情况下，哪种情景成立。这一理论对于理解波在无序介质中的传播是至关重要的，例如具有随机分布的微缺陷的晶体。一个具体的应用是使用激光控制硅表面的质量。

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.