A linear Wegner estimate for alloy type Schroedinger operators on metric graphs

Abstract

We study spectra of alloy-type random Schr\"odinger operators on metric graphs. For finite edge subsets of general graphs we prove a Wegner estimate which is linear in the volume (i.e. the number of edges) and the length of the considered energy interval. The single site potential of the alloy-type model needs to have fixed sign, but the considered metric graph does not need to have a periodic structure. The second result we obtain is an exhaustion construction of the integrated density of states for ergodic random Schr\"odinger operators on metric graphs with a $\ZZ^{\nu}$-structure. For certain models the two above results together imply the Lipschitz continuity of the integrated density of states.