Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators

Abstract

We prove a scale-free quantitative unique continuation estimate for the gradient of eigenfunctions of divergence-type operators, i.e., operators of the form −divA∇, where the matrix function A is uniformly elliptic. The proof uses a unique continuation principle for elliptic second-order operators and a lower bound on the L2-norm of the gradient of eigenfunctions corresponding to strictly positive eigenvalues.As an application, we prove an eigenvalue lifting estimate that allows us to prove a Wegner estimate for random divergence-type operators. Here our approach allows us to get rid of a restrictive covering condition that was essential in previous proofs of Wegner estimates for such models.