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.

We discuss the absolutely continuous spectrum of H = — d 2 /dx 2 + V(x) with F almost periodic and its discrete analog (hu)(n) = u(n +1) + u(n — 1) + V(ri)u(ri). Especial attention is paid to the set, A, of energies where the Lyaponov exponent vanishes. This set is known to be the essential support of the a.c. part of the spectral measure. We prove for a.e. Fin the hull and a.e. E in A, H and h have continuum eigenfunctions, u9 with \u \ almost periodic. In the discrete case, we prove that |^4|^4 with equality only if V = const. If k is the integrated density of states, we prove that on A, 2kdk/dE^π ~ 2 in the continuum case and that 2πsmπkdk/dE^.l in the discrete case. We also provide a new proof of the Pastur-Ishii theorem and that the multiplicity of the absolutely continuous spectrum is 2.

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.

. Let H be a \Gamma-periodic Schrodinger operator acting on L 2 (R d ) and consider the random Schrodinger operator H! = H + V! where V! (x) = X fl2\Gamma ! fl V (x \Gamma fl) (here V is a positive potential and (! fl ) fl2\Gamma a collection of positive i.i.d random variables). We prove that, at the edge of a gap of H that is not filled in for H! , the integrated density of states of H! has a Lifshits tail behaviour if and only if the integrated density of states of H is non-degenerate. R' esum' e. Soient H un op'erateur de Schrodinger \Gamma-p'eriodique agissant sur L 2 (R d ), V un potentiel positif et (! fl ) fl2Z d une famille de variables al'eatoires i.i.d positives. Consid'erons l'op'erateur de Schrodinger al'eatoire H! = H + V! o`u V! (x) = X fl2Z d ! fl V (x \Gamma fl). On montre que, au bord d'une lacune spectrale de H qui n'est pas combl'e pour H! la densit'e d"etats int'egr'ee de H! a un comportement asymptotique de Lifshits si et seulement si la densit'e d"...

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...

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 consider electromagnetic waves in a medium described by a position dependent dielectric constant "(x). We assume that "(x) is a random perturbation of a periodic function " 0 (x) and that the periodic Maxwell operator M 0 = r \Theta 1 " 0 (x) r \Theta has a gap in the spectrum, were r \Theta \Psi = r\Theta\Psi. We prove the existence of localized waves, i.e., finite energy solutions of Maxwell's equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times. Localization of electromagnetic waves is a consequence of Anderson localization for the self-adjoint operators M = r \Theta 1 "(x) r \Theta . We prove that, in the random medium described by "(x), the random operator M exhibits Anderson localization inside the gap in the spectrum of M 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 almo...

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.

This paper is devoted to the study of band edge localization for continuous random Schrödinger operators with weak random perturbations. We prove that, in the weak disorder regime, small, the spectrum in intervals of size at a non-degenerate simple band edge is exponentially and dynamically localized. Upper bounds on the localization length in these energy regions are also obtained. Our results rely on the analysis of Lifshitz tails when the disorder is small; the single site potential need not be of fixed sign.