{
  "id": "1009.4039",
  "version": "v2",
  "published": "2010-09-21T10:12:34.000Z",
  "updated": "2011-06-18T07:33:09.000Z",
  "title": "Spectral Properties of Grain Boundaries at Small Angles of Rotation",
  "authors": [
    "Rainer Hempel",
    "Martin Kohlmann"
  ],
  "comment": "22 pages, 3 figures",
  "journal": "J. Spectr. Theory 1 (2011) 197-219",
  "doi": "10.4171/JST/9",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We study some spectral properties of a simple two-dimensional model for small angle defects in crystals and alloys. Starting from a periodic potential $V \\colon \\R^2 \\to \\R$, we let $V_\\theta(x,y) = V(x,y)$ in the right half-plane $\\{x \\ge 0\\}$ and $V_\\theta = V \\circ M_{-\\theta}$ in the left half-plane $\\{x < 0\\}$, where $M_\\theta \\in \\R^{2 \\times 2}$ is the usual matrix describing rotation of the coordinates in $\\R^2$ by an angle $\\theta$. As a main result, it is shown that spectral gaps of the periodic Schr\\\"odinger operator $H_0 = -\\Delta + V$ fill with spectrum of $R_\\theta = -\\Delta + V_\\theta$ as $0 \\ne \\theta \\to 0$. Moreover, we obtain upper and lower bounds for a quantity pertaining to an integrated density of states measure for the surface states.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-06-18T07:33:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral properties",
      "grain boundaries",
      "simple two-dimensional model",
      "small angle defects",
      "usual matrix describing rotation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.4039H"
    }
  }
}