{
  "id": "2003.09849",
  "version": "v1",
  "published": "2020-03-22T10:06:10.000Z",
  "updated": "2020-03-22T10:06:10.000Z",
  "title": "Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators",
  "authors": [
    "Alexander Dicke",
    "Ivan Veselic"
  ],
  "comment": "25 pages",
  "categories": [
    "math.FA",
    "math.AP",
    "math.SP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-22T10:06:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J15",
      "47B80",
      "35R60",
      "35R45",
      "35P15"
    ],
    "keywords": [
      "random divergence-type operators",
      "wegner estimate",
      "eigenfunctions",
      "elliptic second order operators",
      "scale-free quantitative unique continuation estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}