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arXiv · 2003.09849

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

Alexander Dicke·Ivan Veselic
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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 $-\mathrm{div}A\nabla$, 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 $L^2$-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.

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BibTeXRIS

Alexander Dicke, Ivan Veselic. 2020-03-22. Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators. https://doi.org/10.1016/j.jfa.2023.110040

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