{
  "id": "1212.4085",
  "version": "v2",
  "published": "2012-12-17T17:59:03.000Z",
  "updated": "2014-04-10T19:45:48.000Z",
  "title": "Sharp constructions of eigenfunctions of the magnetic Schrödinger operator",
  "authors": [
    "Blair Davey"
  ],
  "comment": "The contents of this paper have been combined with another article, arXiv:1209.5822",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove sharpness of quantitative unique continuation results for solutions of $-\\Delta u + W\\cdot \\nabla u + V u = \\la u$, where $\\la \\in \\C$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \\lesssim <x>^{-N}$ and $|W(x)| \\lesssim <x>^{-P}$. For $M(R) = \\inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, it was shown in a companion paper that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \\gtrsim \\exp(-C R^{\\be_0}(\\log R)^{A(R)})$, where $\\be_0 = max{2 - 2P, (4-2N)/3, 1}$. Under certain conditions on $N$, $P$, $\\la$, and the dimension, 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 $-\\Delta u + W\\cdot \\nabla u + V u = \\la u$, $|V(x)| \\lesssim <x>^{-N},$ $|W(x)| \\lesssim <x>^{-P}$ and $|u(x)| \\lesssim \\exp(-c|x|^{\\be_0}(\\log |x|)^C)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-10T19:45:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "magnetic schrödinger operator",
      "sharp constructions",
      "eigenfunctions",
      "quantitative unique continuation results",
      "companion paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.4085D"
    }
  }
}