{
  "id": "1209.5822",
  "version": "v3",
  "published": "2012-09-26T03:23:46.000Z",
  "updated": "2014-04-09T22:47:13.000Z",
  "title": "Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator",
  "authors": [
    "Blair Davey"
  ],
  "comment": "Final version as it appears in Communications in Partial Differential Equations",
  "journal": "Communications in Partial Differential Equations. 39(5), 876-945",
  "doi": "10.1080/03605302.2013.796380",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove quantitative unique continuation results for solutions of $-\\Delta u + W\\cdot \\nabla u + Vu = \\lambda u$, where $\\lambda \\in \\mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \\lesssim \\langle x\\rangle^{-N}$ and $|W(x)| \\lesssim \\langle x\\rangle^{-P}$. For $M(R) = \\inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, we show that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \\gtrsim \\exp(-C R^{\\beta_0}(\\log R)^{A( R)})$, where $\\beta_0 = \\max\\{2 - 2P, \\frac{4-2N}{3}, 1\\}$. Under certain conditions on $N$, $P$ and $\\lambda$, we construct examples (some of which are in the style of Meshkov) to prove that this estimate for $M(R)$ is sharp. That is, we construct functions $u, V$ and $W$ such that $-\\Delta u + W\\cdot \\nabla u + Vu = \\lambda u$, $|V(x)| \\lesssim \\langle x\\rangle^{-N}$, $|W(x)| \\lesssim \\langle x\\rangle^{-P}$ and $|u(x)| \\lesssim \\exp(-c|x|^{\\beta_0}(\\log |x|)^C)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-04-09T22:47:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "35J15",
      "35B60"
    ],
    "keywords": [
      "quantitative unique continuation results",
      "magnetic schrödinger operator",
      "eigenfunctions",
      "construct examples",
      "construct functions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.5822D"
    }
  }
}