Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators

Abstract

We prove a scale-free, quantitative unique continuation principle for functions in the range of the spectral projector ${\chi }_{(-\infty ,E]}(H_L)$ of a Schrödinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. We apply it to (i) prove a Wegner estimate for random Schrödinger operators with non-linear parameter-dependence and to (ii) exhibit the dependence of the control cost on geometric model parameters for the heat equation in a multi-scale domain.