In this article, we prove exponential localization for wide classes of Schrödinger operators, including those with magnetic fields, at the edges of unperturbed spectral gaps. We assume that the unperturbed operator H 0 has an open gap I 0 j (B \Gamma ; B+ ). The random potential is assumed to be Andersontype with independent, identically distributed coupling constants. The common density may have either bounded or unbounded support. For either case, we prove that there exists an interval of energies in the unperturbed gap for which the almost sure spectrum of the family H ! j H 0 +V ! is dense pure point with exponentially decaying eigenfunctions. We also prove that the integrated density of states is Lipschitz continuous in the unperturbed spectral gap I 0 .

We prove localization for Anderson-type random perturbations of periodic Schrödinger operators on R d near the band edges. General, possibly unbounded, single site potentials of fixed sign and compact support are allowed in the random perturbation. The proof is based on the following methods: (i) A study of the band shift of periodic Schrodinger operators under linearly coupled periodic perturbations. (ii) A proof of the Wegner estimate using properties of the spatial distribution of eigenfunctions of finite box hamiltonians. (iii) An improved multiscale method together with a result of de Branges on the existence of limiting values for resolvents in the upper half plane, allowing for rather weak disorder assumptions on the random potential. (iv) Results from the theory of generalized eigenfunctions and spectral averaging. The paper aims at high accessibility in providing details for all the main steps in the proof.

This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.

We show for a large class of random Schrodinger operators H! on ` 2 ( ) and on L 2 ( ) that dynamical localization holds, i.e. that, with probability one, for a suitable energy interval I and for q a positive real, sup t r q (t) j sup t ! PI (H! )/ t ; jXj q PI (H! )/ t ? ! 1: Here / is a function of sufficiently rapid decrease, / t = e \GammaiH ! t / and PI (H! ) is the spectral projector of H! corresponding to the interval I. The result is obtained through the control of the decay of the eigenfunctions of H! and covers, in the discrete case, the Anderson tight-binding model with Bernouilli potential (dimension = 1) or singular potential ( ? 1), and in the continuous case Anderson as well as random Landau Hamiltonians. 1 Introduction We show for a large class of random Schrodinger operators H! on ` 2 ( ) and on L 2 ( ) that dynamical localization holds, i.e. that, with probability one, for a 1 suitable energy interval I and q ? 0, sup t r q (t)...

We prove pure point spectrum at all band edges for Schrödinger Operators with a periodic potential plus a random potential of the form V ! (x) = P q i (!)f(x \Gamma i) where f decays at infinity like jxj \Gammam for m ? 3d resp. m ? 2d depending on the regularity of f . Eigenfunctions are shown to decay more rapidly than every inverse polynomial. The random variables q i are supposed to be independent and identically distributed. We suppose that their distribution has a bounded density of compact support.

Mathematical analysis of the Anderson localization has been facilitated by the use of suitable fractional moments of the Green function. Related methods permit now a readily accessible derivation of a number of physical manifestations of localization, in regimes of strong disorder, extreme energies, or weak disorder away from the unperturbed spectrum. The present work establishes on this basis exponential decay for the modulus of the two-point function, at all temperatures as well as in the ground state, for a Fermi gas within the one-particle approximation. Different implications, in particular for the Integral Quantum Hall Effect, are reviewed.

We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r 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 = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . 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 wh...

We investigate the Anderson metal-insulator transition for random Schrödinger operators. We define the strong...

We study the integrated density of states of random Anderson-type additive and multiplicative perturbations of deterministic background operators for which the single-site potential does not have a fixed sign. Our main result states that, under a suitable assumption on the regularity of the random variables, the integrated density of states of such random operators is locally Hölder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder, and at energies in the unperturbed spectral gaps, provided the randomness is sufficiently small. The result is based on a proof of a Wegner estimate with the correct volume dependence. The proof relies upon the L function for p 1 [9], and the vector field methods of [20]. We discuss the application of this result to Schrödinger operators with random magnetic fields and to band-edge localization.

The integrated density of states (IDS) for random operators is an important function describing many physical characteristics of a random system. Properties of the IDS are derived from the Wegner estimate that describes the influence of finite-volume perturbations on a background system. In this paper, we present a simple proof of the Wegner estimate applicable to a wide variety of random perturbations of deterministic background operators. The proof yields the correct volume dependence of the upper bound. This implies the local Hölder continuity of the integrated density of states at energies in the unperturbed spectral gap. The proof depends on the L - theory of the spectral shift function (SSF), for p 1, applicable to pairs of...