Scale-free unique continuation principle for spectral projectors, 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 ${\chi }_{\left(-\infty ,E\right]}\left(H_L\right)$ of a Schrödinger operator $H_L$ on a cube of side $L\in \mathbb{N}$, with bounded potential. Previously, such estimates were known only for individual eigenfunctions and for spectral projectors ${\chi }_{\left(E-\gamma ,E\right]}\left(H_L\right)$ with small $\gamma$. Such estimates are also called, depending on the context, uncertainty principles, observability estimates, or spectral inequalities. Our main application of such an estimate is to find lower bounds for the lifting of eigenvalues under semidefinite positive perturbations, which in turn can be applied to derive a Wegner estimate for random Schrödinger operators with nonlinear parameter-dependence. Another application is an estimate of the control cost for the heat equation in a multiscale domain in terms of geometric model parameters. Let us emphasize that previous uncertainty principles for individual eigenfunctions or spectral projectors onto small intervals were not sufficient to study such applications.