Mathematical Sciences: Spectral Properties of Random Media

数学科学：随机介质的谱特性

• 批准号: 9707049
• 负责人: Peter Hislop
• 金额: $ 7.91万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9707049&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Spectral; Properties; Random

9707049 Hislop Localization at band-edge energies for Schroedinger operators is now reasonably well understood for simple models of disorder, like Anderson-type potentials. A part of this project is directed towards extending these results to other interesting systems for which disorder strongly affects wave propagation: acoustic, electromagnetic, and elastic waves. Another goal is to extend the types of disorder allowed to more realistic models such as the Poisson and random displacement models. This will require an improvement of the Wegner estimate obtained in previous works. These techniques will be useful in studying the question of correlated systems and high-energy localization for two-dimensional disordered system. The propagation of classical acoustic and electromagnetic waves and of a single electron in perfectly ordered media, like a crystal, is well understood. There are allowed and forbidden energies of propagation for the waves. At the allowed energies, the waves propagate out to infinity behaving almost as if they were free waves. The effects of randomly distributed impurities in solids, or randomly distributed defects in an otherwise perfect material, is profound: the particles and waves no longer propagate out to infinity. Rather, at many energies, the transport of energy will be completely suppressed. This is effect is referred to as localization. The goal of this research is to better understand the mechanisms of localization for classical and quantum mechanical systems for many models of disorder. These results have applications to many engineering devices such as the effect of random imperfections in wave guides, photonic crystals, and superlattices.

9707049 HISLOP在施罗辛格运营商的带边缘能量的HISLOP定位现在已经对简单的混乱模型（例如Anderson型潜力）进行了相当理解。 该项目的一部分是针对将这些结果扩展到其他有趣的系统的，这些系统强烈影响波传播：声学，电磁波和弹性波。另一个目标是将允许的疾病类型扩展到更现实的模型，例如泊松和随机位移模型。这将需要改进以前的作品中获得的韦格纳估计。 这些技术将有助于研究二维无序系统的相关系统和高能定位问题。 可以很好地理解经典声学和电磁波以及在完美有序的介质中的单个电子（如晶体）的传播。允许使用波浪的传播能量。在允许的能量下，波浪传播到无穷大的行为几乎好像是自由波。固体中随机分布的杂质或原本完美材料中随机分布的缺陷的影响是深刻的：颗粒和波不再传播到无穷大。相反，在许多能量上，能量的运输将被完全抑制。这就是效果称为本地化。这项研究的目的是更好地了解许多疾病模型的经典和量子机械系统的定位机制。 这些结果在许多工程设备上都有应用，例如波导指南，光子晶体和超晶格中随机缺陷的影响。