Algebras of Random Operators Associated to Delone Dynamical Systems

We carry out a careful study of operator algebras associated with Delone dynamical systems. A von Neumann algebra is defined using noncommutative integration theory. Features of these algebras and the operators they contain are discussed. We restrict our attention to a certain C ^*-subalgebra to discuss a Shubin trace formula.     

Trace norm estimates for products of integral operators and diffusion semigroups

We give trace norm estimates for products of integral operators and for diffusion semigroups. These are applied to differences of heat semigroups. A natural example of an integral operator with finite trace which is not trace class is given.     

Localization for random perturbations of anisotropic periodic media

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

An ergodic theorem for Delone dynamical systems and existence of the integrated density of states

We study strictly ergodic Delone dynamical systems and prove an ergodic theorem for Banach space valued functions on the associated set of pattern classes. As an application, we prove existence of the integrated density of states in the sense of uniform convergence in distribution for the associated random operators. Dedicated to J. Voigt on the occasion of his 60th birthday Research partly supported by the DFG in the priority program Quasicrystals.     

Dynamical localization for continuum random surface models

We prove Anderson localization and strong dynamical localization for random surface models in \mathbbR^d \mathbb{R}^d .     

Absence of Continuous Spectral Types for Certain Non-Stationary Random Schrödinger Operators

We consider continuum random Schrödinger operators of the type H = – + V₀ + V with a deterministic background potential V₀. We establish criteria for the absence of continuous and absolutely continuous spectrum, respectively, outside the spectrum of – + V₀. The models we treat include random surface potentials as well as sparse or slowly decaying random potentials. In particular, we establish absence of absolutely continuous surface spectrum for random potentials supported near a one-dimensional surface (random tube) in arbitrary dimension.submitted 07/04/04, accepted 19/08/04     

From Uncertainty Principles to Wegner Estimates

We give a shortcut from simple uncertainty principles to Wegner estimates with correct volume term.     

Perturbation of Dirichlet forms by measures

Perturbations of a Dirichlet form by measures are studied. The perturbed form –_–+₊ is defined for _– in a suitable Kato class and ₊ absolutely continuous with respect to capacity. L _p-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 L _p -L _q -smoothing properties the same is shown to hold for the perturbed semigroup. If the unperturbed semigroup is holomorphic on L ₁ the same is shown to be true for the perturbed semigroup, for a large class of measures.     

An uncertainty principle, Wegner estimates and localization near fluctuation boundaries

We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven.