From Localization to Extended States in Anderson-type Models

PI将研究anderson型随机schr \ odinger算子中本地化-非本地化转换的几个方面。一个目标是发展新的方法，用局部波函数来表征能量状态。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统无序系统。PI和他的合作者将进一步提高该方法的适用性，并将其用于研究几个开放问题，如连续体表面模型和连续体条带上的安德森模型的局部化性质。另一个目标是研究无序介质中扩展状态存在的数学机制。例如，这将通过一维随机聚合物进行研究，其中单一能量的扩展状态可能导致显著的电子输运。PI和合作者还将开发微扰理论方法来研究具有准周期性势的多维Schr\ odinger算子的高能谱，特别是建立该模型的扩展状态的存在性。用安德森模型来描述无序介质的电导率特性。这包括含有嵌入或取代杂质的晶体、合金、非晶介质、高温下平面波动的影响和准晶体。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。所观察到的现象并不局限于量子机械波，而是扩展到声波、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研究量子波导中的随机缺陷、光子晶体和准晶体中的异常输运。PI的研究旨在找到安德森跃迁的数学上严格的证明，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.