Localization of Classical Waves and Phenomena in Disordered Systems

摘要：本课题的课题是研究经典波及其相关现象的局域化，以及发生在无序系统中的现象。经典波可以表现出通常与量子力学电子波相关的现象。电子的量子力学波动性质是基于与经典波的散射和干涉的类比。但直到最近，这种类比才被推翻，人们开始研究经典波局域化的可能性。局域化是一种普遍的波现象，不是电子波所特有的。本建议的主要目的是为研究经典波的局域化现象建立一个坚实的数学基础。将研究以下子主题：复杂介质中的局部化效应，光子晶体中的非线性效应，精确可解模型，经典波局部化的一般框架，经典波的动态局部化，内部Lifshitz尾，周期Maxwell算子谱的绝对连续性。其他要研究的课题是随机薛定谔算符（含杂质晶体中电子运动的模型）和远程自旋玻璃。新的光学材料和器件将导致“光电子学”（光电子学），目前是一个深入研究的课题。光学晶体管就是这种新型光学器件的例子，它们的制造将需要新的光学材料。光电子技术的发展将产生巨大的技术影响。电子器件（如晶体管）是以量子力学现象为基础的。电子束缚态是局域电子波的最简单的例子，在自然界中无处不在。但是光的类似现象在自然界中并不常见，即使理论上是可能的。大自然为电子定位提供了各种现成的材料。但是光和其他经典波的局部化需要制造相应的材料。这需要对经典波（例如光）如何表现出通常与量子力学电子波相关的现象有一个基本的理解。其中之一是本地化，这是本提案要讨论的主题。这是基础科学和应用科学研究的一个新领域，对社会有巨大的潜在回报。

Proposal:DMS-9800883 Principal Investigator: Abel Klein Abstract: The subject of this research proposal is the study of localization of classical waves and related phenomena, and of phenomena that occur in disordered systems. Classical waves can exhibit phenomena normally associated with quantum mechanical electron waves. The quantum mechanical wave nature of electrons is based on an analogy with the scattering and interference of classical waves. But only recently has the analogy been reversed and the possibility of the localization of classical waves started to be investigated. Localization is a generic wave phenomena, not specific to electron waves. The primary objective of this proposal is to establish a solid mathematical basis for the investigation of the phenomena of localization of classical waves. The following subtopics will be studied: localization effects in complex media, nonlinear effects in photonic crystals, exactly solvable models, a general framework for the localization of classical waves, dynamical localization of classical waves, internal Lifshitz tails, absolute continuity of the spectrum of periodic Maxwell operators. Other topics to be investigated are random Schroedinger operators (a model for the motion of an electron in a crystal with impurities) and long range spin glasses. New optical materials and devices will lead to "lightronics" (electronics for light), presently a subject of intensive research. Optical transistors are examples of such new optical devices, and their fabrication will require new optical materials. The development of lightronics will have tremendous technological implications. Electronic devices (e.g., transistors) are based on quantum mechanical phenomena. Electronic bound states, the simplest example of a localized electron wave, are ubiquitous in nature. But the analogous phenomena for light are not commonly observed in nature, even if theoretically possible. Nature provides a variety of readily available materials for electronic locali zation. But the corresponding materials for the localization of light and other classical waves have to be fabricated. This requires a fundamental understanding of how classical waves (e.g., light) can exhibit phenomena normally associated with quantum mechanical electron waves. One of these is localization, a subject that the present proposal addresses. This is a new area of research in both basic and applied sciences, with a potentially huge payoff to society.