Conditional Wegner estimate for the standard random breather potential

TL;DR

This paper establishes a conditional Wegner estimate for Schrödinger operators with standard random breather potentials, linking it to a scale free unique continuation principle, and broadening the class of applicable models.

Contribution

It introduces a new approach to proving Wegner estimates for breather-type potentials by reducing the problem to a unique continuation principle, applicable to a wider class of models.

Findings

Wegner estimate is proven for standard breather potentials.

Method connects Wegner estimates to unique continuation principles.

Results extend to larger classes of breather potentials.

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.