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.

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of Mathematical Physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

We prove localization for random perturbations of periodic divergence form operators of the form-V 9 a ~- V near the band edges. Here a ~ is a matrix function which results from an Anderson type perturbation of a periodic matrix function.

We use scattering theoretic methods to prove exponential localization for random displacement models in one dimension. The operators we consider model both quantum and classical wave propagation. Our main tools are the reflection and transmission coefficients for compactly supported single site perturbations. We show that randomly displaced, non-reflectionless single sites lead to localization.

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

Abstract not found

Abstract. We show absence of energy levels repulsion for the eigenvalues of random Schrödinger operators in the continuum. We prove that, in the localization region at the bottom of the spectrum, the properly rescaled eigenvalues of a continuum Anderson Hamiltonian are distributed as a Poisson point process with intensity measure given by the density of states. We derive a Minami estimate for continuum Anderson Hamiltonians. We also obtain simplicity of the eigenvalues, 1.

Dedicated to the memory of Klaus Floret Abstract We present an eigenfunction expansion theorem for generators of strongly local, regular Dirichlet forms. Conditions are phrased in terms of the intrinsic metric. The result covers many cases of Hamiltonians which appear in Mathematical Physics and Geometry.

-periodic function V per on IR d , which in addition satisfies the technical requirement V per 2 L p loc (IR d ), where p = 2 if d 3, and p ? d=2 if d 4. The impurities are thought to be alloy like, i.e. fixed to the period lattice, and described by an Anderson type random potential V ! (x) = X k2ZZ d q k (!)f(x \Gamma k): (1.2) Our assumptions on the single site potential f and the random coupling constants q k are as follows: Let f 2 L p (IR d ) with p as above have compact s

We apply nite volume criteria for localization to random Schrodinger operators. These provide explicit conditions, depending on the parameters of the model, for starting the bootstrap multiscale analysis. The criteria are used to study localization of Anderson Hamiltonians on the continuum at the bottom of the spectrum at high disorder.