We prove that the integrated density of states of random Schrödinger operators with Anderson-type potentials on L ), for d ≥ 1, is locally Hölder continuous at all energies. The single-site potential u must be nonnegative and compactly supported, and the distribution of the random variable must be absolutely continuous with a bounded, compactly supported density. We also prove this result for random Anderson-type perturbations of the Landau Hamiltonian in two-dimensions under a rational flux condition.

Many wave propagation phenomena in classical physics are governed by equations that can be recast in Schrodinger form. In this approach the classical wave equation (e.g., Maxwell's equations, acoustic equation, elastic equation) is rewritten in Schrodinger form, leading to the study of the spectral theory of its classical wave operator, a self-adjoint, partial differential operator on a Hilbert space of vectorvalued, square integrable functions. Physically interesting inhomogeneous media give rise to nonsmooth coefficients. We construct a generalized eigenfunction expansion for classical wave operators with nonsmooth coefficients. Our construction yields polynomially bounded generalized eigenfunctions, the set of generalized eigenvalues forming a subset of the operator's spectrum with full spectral measure. 1 Introduction A self-adjoint operator in a finite dimensional Hilbert spaces can always be diagonalized in an orthonormal basis of eigenvectors (i.e., it has a complete set of eig...

The integrated density of states (IDS) N(E) is the distribution function of a nonnegative measure #, the density of states measure (DOS). This measure

. We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum. If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corresponding eigenvalues in the gap. We use a modified Birman-Schwinger method to derive equations for these eigenmodes and corresponding eigenvalues in the gap, in terms of the spectral attributes of an auxiliary Hilbert-Schmidt operator. We prove that in three dimensions, under some natural conditions on the periodic background, the number of eigenvalues generated in a gap of the periodic operator is finite, and give an estimate on the number of these midgap eigenvalues. In particular, we show that if the defect is weak there are no midgap eigenvalues. Key words. photonic crystal, photonic bandgap, periodic acoustic medium, periodic dielectric medium, midgap states, defect modes, localization of light AMS subject...

. We outline some recent developments in the theory of random operators concentrating on the spectral properties of additive and multiplicative random perturbations. These models describe the propagation of electrons and of classical waves in randomly perturbed media, respectively. The results presented concern the localization of states and the behavior of the integrated density of states at energies in intervals near the band edges of the spectrum of the unperturbed Hamiltonian. 1. Introduction This is an overview of some recent results in the spectral theory of random operators obtained in collaborations with Jean-Marie Barbaroux, Jean Michel Combes, Eric Mourre, and Adriaan Tip. One of the goals of the study of randomly perturbed operators is to describe the propagation of quantum and classical waves in media which is randomly perturbed. One is particularly interested in the conditions under which random perturbations cause localization. Often, the background or unperturbed system...