No abstract

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E), and set the metallic transport region to be the part of the spectrum with nontrivial transport (i.e., β(E) > 0). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator, and that the local transport exponent β(E) provides a characterization of the metal-insulator transport transition. Moreover, we show that if there is such a transition, then β(E) has to be discontinuous at a transport mobility edge. More precisely, we show that if the transport is nontrivial then β(E) ≥ 1 2d , where d is the space dimension. These results follow from a proof that slow time evolution of quantum waves in random media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge. Contents

We study the region of complete localization in a class of random operators which includes random Schrödinger operators with Anderson-type potentials and classical wave operators in random media, as well as the Anderson tight-binding model. We establish new characterizations or criteria for this region of complete localization, given either by the decay of eigenfunction correlations or by the decay of Fermi projections. (These are necessary and sufficient conditions for the random operator to exhibit complete localization in this energy region.) Using the first type of characterization we prove that in the region of complete localization the random operator has eigenvalues with finite multiplicity.

We justify the linear response theory for an ergodic Schrödinger operator with magnetic field within the noninteracting particle approximation, and derive a Kubo formula for the electric conductivity tensor. To achieve that, we construct suitable normed spaces of measurable covariant operators where the Liouville equation can be solved uniquely. If the Fermi level falls into a region of localization, we recover the well-known Kubo-Stȓeda formula for the quantum Hall conductivity at zero temperature.