Peter Stollmann

Perturbation of Dirichlet forms by measures

AbstractPerturbations of a Dirichlet form

Localization for random perturbations of periodic Schrödinger operators

\mathfrak{h}

Bounds on the Spectral Shift Function and the Density of States

by measures μ are studied. The perturbed form

Lifshitz Asymptotics via Linear Coupling of Disorder

\mathfrak{h}

LOCALIZATION ON A QUANTUM GRAPH WITH A RANDOM POTENTIAL ON THE EDGES

−μ−+μ+ is defined for μ− in a suitable Kato class and μ+ absolutely continuous with respect to capacity. Lp-properties of the corresponding semigroups are derived by approximating μ− by functions. For treating μ+, 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 L1 the same is shown to be true for the perturbed semigroup, for a large class of measures.

LIFSHITZ ASYMPTOTICS AND LOCALIZATION FOR RANDOM QUANTUM WAVEGUIDES

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.