From Localization to Extended States in Anderson-type Models

PI将研究Anderson型随机Schr odinger算子的定域-离域转变的几个方面。其中一个目标是发展新的方法来表征局域波函数的能量状态。主要的焦点将放在分数矩（或Aizenman-Molchanov）方法上，该方法最近被证明适用于连续无序系统。PI和他的合作者将进一步增加这种方法的适用性，并使用它来研究几个开放的问题，如连续表面模型的局部化性质和连续带上的Anderson模型。另一个目标是研究无序介质中扩展态存在的数学机制。例如，这将通过一维无规聚合物来研究，在一维无规聚合物中，单一能量下的扩展态可以导致显著的电子传输。PI和合作者还将发展微扰理论方法来研究具有准双势的多维Schr\“odinger算子的高能谱，特别是建立该模型的扩展态的存在性。安德森模型用于描述无序介质的导电特性。这包括有嵌入或替代杂质的晶体、合金、非晶介质、高温下的晶格涨落效应和准晶体。中心目标是了解无序对电子输运的影响，从而区分导体和绝缘体。所观察到的现象不仅限于量子力学波，而且还扩展到声波、电磁波和弹性波在无序介质中的传播，在工程上的应用包括量子波导、光子晶体和准晶体中的反常输运的研究。PI的研究旨在找到安德森转变的数学上严格的理由，规定无序介质可以在足够高的能量下经历从绝缘体到导体的相变。

The PI will investigate several aspects of thelocalization-delocalization transition in Anderson-type randomSchr\"odinger operators. One goal is the development of newmethods to characterize energy regimes with localized wavefunctions. The main focus will be on the fractional moment(or Aizenman-Molchanov) method, which has recently been shown toapply to continuum disordered systems. The PI and hiscollaborators will further increase the applicability of thismethod and use it to study several open problems such as thelocalization properties of continuum surface models and theAnderson model on continuum strips. Another objective is to studymathematical mechanisms which show existence ofextended states in disordered media. For example, this will bestudied through one-dimensional random polymers, where an extendedstate at a single energy can lead to significant electrontransport. The PI and collaborators will also develop perturbationtheoretic methods to study the high energy spectrum ofmulti-dimensional Schr\"odinger operators with quasi-periodicpotentials, and, in particular, establish the existence ofextended states for this model.The Anderson model is used to describe the conductivity propertiesof disordered media. This includes crystals with imbedded orsubstitutional impurities, alloys, amorphous media, the effects oflattice fluctuations at high temperature, and quasi crystals. Thecentral goal is to understand the effects of disorder on electrontransport and therefore to distinguish between conductors andinsulators. The phenomena which are observed are not restricted toquantum mechanical waves, but extend to the propagation ofacoustic, electro-magnetic and elastic waves in disordered media.Applications in engineering include the study of randomimperfections in quantum wave guides, photonic crystals, andanomalous transport in quasi crystals. The PI's research is aimedat finding mathematically rigorous justifications of the Andersontransition, stipulating that disordered media can undergo a phasetransition from insulator to conductor at sufficiently highenergy.