{
  "id": "1107.2800",
  "version": "v1",
  "published": "2011-07-14T12:26:15.000Z",
  "updated": "2011-07-14T12:26:15.000Z",
  "title": "Discrete Schrödinger operators with random alloy-type potential",
  "authors": [
    "Alexander Elgart",
    "Helge Krüger",
    "Martin Tautenhahn",
    "Ivan Veselić"
  ],
  "comment": "Proceedings of the Spectral Days 2010, Pontificia Universidad Cat\\'olica de Chile, Santiago",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schr\\\"odinger operators $H_\\omega = - \\Delta + V_\\omega$ on $\\ell^2 (\\ZZ^d)$ where $\\Delta$ is the discrete Laplacian and $V_\\omega$ the multiplication by the function $V_\\omega (x) = \\sum_{k \\in \\ZZ^d} \\omega_k u(x-k)$. Here $\\omega_k$, $k \\in \\ZZ^d$, are i.i.d. random variables and $u \\in \\ell^1 (\\ZZ^d ; \\RR)$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $H_\\omega$ depend in a non-monotone way on the random parameters $\\omega_k$. This requires new methods at certain stages of the localization proof.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-14T12:26:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B44",
      "60H25",
      "35J10"
    ],
    "keywords": [
      "random alloy-type potential",
      "discrete schrödinger operators",
      "discrete alloy-type model",
      "fractional moment method",
      "non-monotone way"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.2800E"
    }
  }
}