Erklärung Ich erkläre, daß die vorliegende Habilitationsschrift von mir selbst und ohne andere als die darin angegebenen Hilfsmittel angefertigt wurde. Abstract. We give a survey on results on spectral properties of random Schrödinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a… (More)
We survey recent results on spectral properties of random Schrö-dinger operators. The focus is set on the integrated density of states (IDS). First we present a proof of the existence of a self-averaging IDS which is general enough to be applicable to random Schrödinger and Laplace-Beltrami operators on manifolds. Subsequently we study more specific models… (More)
We study spectra of Schrödinger operators on R d. First we consider a pair of operators which differ by a compactly supported potential, as well as the corresponding semigroups. We prove almost exponential decay of the singular values µ n of the difference of the semigroups as n → ∞ and deduce bounds on the spectral shift function of the pair of operators.… (More)
We analyze Schrödinger operators whose potential is given by a singular interaction supported on a sub-manifold of the ambient space. Under the assumption that the operator has at least two eigenvalues below its essential spectrum we derive estimates on the lowest spectral gap. In the case where the sub-manifold is a finite curve in two dimensional… (More)
We consider the Riemannian universal covering of a compact manifold M = X/Γ and assume that Γ is amenable. We show for an er-godic random family of Schrödinger operators on X the existence of a (non-random) integrated density of states.
We review recent and give some new results on the spectral properties of Schrödinger operators with a random potential of alloy type. Our point of interest is the so called Wegner estimate in the case where the single site potentials change sign. The indefinitness of the single site potential poses certain difficulties for the proof of the Wegner estimate… (More)
We study discrete alloy type random Schrödinger operators on 2 (Z d). Wegner estimates are bounds on the average number of eigenvalues in an energy interval of finite box restrictions of these types of operators. If the single site potential is compactly supported and the distribution of the coupling constant is of bounded variation a Wegner estimate holds.… (More)
We consider the quantum site percolation model on graphs with an amenable group action. It consists of a random family of Hamiltonians. Basic spectral properties of these operators are derived: non-randomness of the spectrum and its components, existence of an self-averaging integrated density of states and an associated trace-formula.
We prove a localization theorem for continuous ergodic Schrödinger operators H ω := H 0 + V ω , where the random potential V ω is a nonnegative Anderson-type perturbation of the periodic operator H 0. We consider a lower spectral band edge of σ(H 0), say E = 0, at a gap which is preserved by the perturbation V ω. Assuming that all Floquet eigenvalues of H 0… (More)
We study Schrödinger operators with a random potential of alloy type. The single site potentials are allowed to change sign. For a certain class of them we prove a Wegner estimate. This is a key ingredient in an existence proof of pure point spectrum of the considered random Schrödinger operators. Our estimate is valid for all bounded energy intervals and… (More)