From Localization to Extended States in Anderson-type Models

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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.