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

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 backward difference , 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 divergencetype 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.& COPY; 2023 Elsevier Inc. All rights reserved.