Abstract. We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

A random polymer model is a one-dimensional Jacobi matrix randomly composed of two finite building blocks. If the two associated transfer matrices commute, the corresponding energy is called critical. Such critical energies appear in physical models, an example being the widely studied random dimer model. It is proven that the Lyapunov exponent vanishes quadratically at a generic critical energy and that the density of states is positive there. Large deviation estimates around these asymptotics allow to prove optimal lower bounds on quantum transport, showing that it is almost surely overdiffusive even though the models are known to have pure-point spectrum with exponentially localized eigenstates for almost every configuration of the polymers. Furthermore, the level spacing is shown to be regular at the critical energy.

We say that a quantum spin system is dynamically localized if the time-evolution of local observables satisfies a zero-velocity Lieb-Robinson bound. In terms of this definition we have the following main results: First, for general systems with short range interactions, dynamical localization implies exponential decay of ground state correlations, up to an explicit correction. Second, the dynamical localization of random xy spin chains can be reduced to dynamical localization of an effective one-particle Hamiltonian. In particular, the isotropic xy chain in random exterior magnetic field is dynamically localized.Comment: final version with minor revisions, accepted for Comm. Math. Phy

Abstract. 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.

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.