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We give a new proof of exponential localization in the Anderson tight binding model which uses many ideas of the Frohlich, Martinelli, Scoppola and Spencer proof, but is technically simpler-particularly the probabilistic estimates.

We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Holder continuous distributions and for bounded potentials whose distribution is a convex combination of a Holder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions.We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.

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

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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 consider classical acoustic waves in a medium described by a position dependent mass density ρ(x). We assume that ρ(x) is a random perturbation of a periodic function ρo(*) and that the periodic acoustic operator A o = -V ^jV has a gap in the spectrum We prove the existence of localized waves, i e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one Localization of acoustic waves is a consequence of Anderson localization for the self-adjoint operators A = -V γ^V on L 2 (ΊR. d ) We prove that, in the random medium described by ρ(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A o This is shown even in situations when the gap is totally filled by the spectrum of the random operator, we can prescribe random environments that ensure localization in almost the whole gap. Contents

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.