Conditional Wegner estimate for the standard random breather potential

Abstract

We prove a conditional Wegner estimate for Schr\"odinger operators with random potentials of breather type. More precisely, we reduce the proof of the Wegner estimate to a scale free unique continuation principle. The relevance of such unique continuation principles has been emphasized in previous papers, in particular in recent years. We consider the standard breather model, meaning that the single site potential is the characteristic function of a ball or a cube. While our methods work for a substantially larger class of random breather potentials, we discuss in this particular paper only the standard model in order to make the arguments and ideas easily accessible.