Abstract-We prove localization for Anderson-type random perturbations of periodic Schr dinger operators on R 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 Schr dinger 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.

Abstract. Perturbations ofa Dirichlet form 0 by measures/~ are studied. The perturbed form 0 -#-+ /z+ is defined for/~_ in a suitable Kato class and #+ absolutely continuous with respect to capacity. Lp-properties of the corresponding semigroups are derived by approximating #_ by functions. For treating #+, a criterion for domination of positive semigroups is proved. If the unperturbed semigroup has Lp -Lq-smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ~ the same is shown to be true for the perturbed semigroup, for a large class of measures.Mathematics Subject Classifications (1991): Primary 47D07; Secondary 31 C25,

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Abstract. We prove that a strong form of dynamical localization follows from a variable energy multi-scale analysis. This abstract result is applied to a number of models for wave propagation in disordered media.

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