From Localization to Extended States in Anderson-type Models

University Of Alabama At Birmingham, Birmingham AL

Investigators

Abstract

The PI will investigate several aspects of the localization-delocalization transition in Anderson-type random Schr\"odinger operators. One goal is the development of new methods to characterize energy regimes with localized wave functions. The main focus will be on the fractional moment (or Aizenman-Molchanov) method, which has recently been shown to apply to continuum disordered systems. The PI and his collaborators will further increase the applicability of this method and use it to study several open problems such as the localization properties of continuum surface models and the Anderson model on continuum strips. Another objective is to study mathematical mechanisms which show existence of extended states in disordered media. For example, this will be studied through one-dimensional random polymers, where an extended state at a single energy can lead to significant electron transport. The PI and collaborators will also develop perturbation theoretic methods to study the high energy spectrum of multi-dimensional Schr\"odinger operators with quasi-periodic potentials, and, in particular, establish the existence of extended states for this model. The Anderson model is used to describe the conductivity properties of disordered media. This includes crystals with imbedded or substitutional impurities, alloys, amorphous media, the effects of lattice fluctuations at high temperature, and quasi crystals. The central goal is to understand the effects of disorder on electron transport and therefore to distinguish between conductors and insulators. The phenomena which are observed are not restricted to quantum mechanical waves, but extend to the propagation of acoustic, electro-magnetic and elastic waves in disordered media. Applications in engineering include the study of random imperfections in quantum wave guides, photonic crystals, and anomalous transport in quasi crystals. The PI's research is aimed at finding mathematically rigorous justifications of the Anderson transition, stipulating that disordered media can undergo a phase transition from insulator to conductor at sufficiently high energy.