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.

Abstract not found

Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current

We use scattering theoretic methods to prove strong dynamical and exponential localization for one-dimensional, continuum, Anderson-type models with singular distributions; in particular, the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single-site perturbations of a periodic background which we use to verify the necessary hypotheses of multi-scale analysis. We show that nonreflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs. 1.

We give finite volume criteria for localization of quantum or classical waves in continuous random media. We provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. A simple application yields localization for Anderson Hamiltonians on the continuum at the bottom of the spectrum in an interval of size O() for large , where stands for the disorder parameter. A more sophisticated application proves localization for two-dimensional random Schrödinger operators in a constant magnetic field (random Landau Hamiltonians) up to a distance O( B ) from the Landau levels, where B is the strength of the magnetic field.

The purpose of this paper is to prove that, for a 2-dimensional continuous Anderson model, at an open band edge, the density of states exhibits a Lifshitz behavior. The Lifshitz exponent need not be d/2 = 1 but is determined by the behaviour of the Floquet eigenvalues of a well chosen background periodic Schrödinger operator.

. We present new results on Lifshitz tails at internal band edges for the density of states of random Schrodinger operators on R 2 . In particular, we show the existence and compute the value of the Lifshitz exponent in the case when the band edge is simple. 0. Introduction In the early 60's (see [16, 18]), I.M. Lifshitz produced a heuristic showing that, at the uctuational band edges of the spectrum of a random Schrodinger operator, the density of states decays exponentially fast. This diers dramatically from the behavior of the density of states of a periodic Schrodinger operator : in this case, the band edge decay is polynomial. One of the major consequences of this dierence is that the band edge spectral behaviors for these two classes of operators are radically dierent: in the random case, the spectrum is localized and, in the periodic case, the spectrum is extended. To be more specic, let us turn to the random model we will study. As our results will only be valid in d...

We outline a differential inequality method for proving estimates on expectations of products of random variables occurring in the theory of random Schrödinger operators. This method is applied to prove correlated Wegner estimates for local Hamiltonians with Anderson-type potentials constructed with either long-range, single-site potentials and/or correlated coupling constants. These estimates on the joint probability of correlated random variables are a key ingredient in the proof of the existence of intervals of exponentially localized states for the corresponding families of random Hamiltonians. This application completes the proof of localization for long-range, single-site potentials described in our earlier paper [6]. This work also extends the results of [3] to Anderson--type potentials with correlated coupling constants. In particular, these results establish exponential localization in energy intervals near the band edges of the spectrum of the unperturbed Hamiltonian for ge...

. We use scattering theoretic methods to prove strong dynamical and exponential localization for one dimensional, continuum, Anderson-type models with singular distributions; in particular the case of a Bernoulli distribution is covered. The operators we consider model alloys composed of at least two distinct types of randomly dispersed atoms. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations of a periodic background which we use to verify the necessary hypotheses of multi-scale analysis. We show that non-reflectionless single sites lead to a discrete set of exceptional energies away from which localization occurs. 1. Introduction Perhaps the most studied and best understood type of random operators which are used to describe spectral and transport properties of disordered media are the Anderson models, whether one considers their original discrete version in ` 2 (Z d ) or their continuum analogs. They describe mater...