Localization of Classical Waves and Phenomena in Disordered Systems

提案：DMS-9800883主要研究者：Abel Klein 摘要：本研究提案的主题是研究经典波和相关现象的局部化，以及无序系统中发生的现象。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，这种类比才被逆转，经典波局域化的可能性才开始被研究。局域化是一种普遍的波现象，并不是电子波所特有的。 这个建议的主要目的是建立一个坚实的数学基础，为调查的现象，本地化的经典波。本课程将研究以下几个主题：复杂介质中的局域化效应、光子晶体中的非线性效应、精确可解模型、经典波局域化的一般框架、经典波的动力学局域化、内部Lifshitz尾、周期性麦克斯韦算子谱的绝对连续性。 其他要研究的课题是随机薛定谔算子（一种电子在含有杂质的晶体中运动的模型）和长程自旋玻璃。 新的光学材料和设备将导致“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。光电子学的发展将产生巨大的技术影响。电子设备（例如，晶体管）基于量子力学现象。电子束缚态是局域电子波最简单的例子，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，即使在理论上是可能的。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域化必须制造相应的材料。这需要对经典波（例如，光）可以表现出通常与量子力学电子波相关联的现象。其中一项是本㆞化问题，而本建议正是针对这个问题。这是基础科学和应用科学的一个新的研究领域，可能会给社会带来巨大的回报。

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.