Research on the Anderson metal-insulator transport transition and otherphenomena in disordered systems

University Of California-Irvine, Irvine CA

Investigators

Abstract

The subject of this research proposal is the Anderson metal-insulator transport transition and other phenomena in disordered systems. A new approach to the Anderson metal-insulator transition based on transport instead of spectral properties will be investigated. In addition, several related topics will be investigated. Constructive criteria for localization in random media will be developed; an application is planned for the Landau Hamiltonian with a random potential. Local Poisson statistics for the strong insulator spectrum of Anderson-type Hamiltonians in the continuum will be studied. The spectrum of the Anderson model on the Bethe lattice will be studied. Localization at low disorder in one or two dimensions will be investigated. Fortysome years have passed since P. Anderson's seminal article on localization of electrons in random media, but our mathematical understanding of the metal-insulator transition is still very unsatisfactory. In three or more dimensions a transition is believed to occur from an insulator regime, characterized by localized states, to a very different metallic regime characterized by extended states. The energy at which this metal insulator transition occurs is called the mobility edge. The standard mathematical interpretation of this picture is that there should be a transition in the spectrum of the random SchrAdinger from pure point spectrum (localized states) to absolutely continuous spectrum (extended states). But up to now there are no mathematical results on the existence of continuous spectrum and a metal-insulator transition (except for the special case of the Anderson model on the Bethe lattice). A new approach to the Anderson metal-insulator transition is proposed based on transport instead of spectral properties. It is motivated by the fact that the intuitive physical notion of localization has a dynamical interpretation: an initially localized wave packet should remain localized under time evolution, and delocalization may be interpreted as nontrivial transport. The main goal of this proposal is to show the existence of such a transport transition.