{
  "id": "1301.5268",
  "version": "v2",
  "published": "2013-01-22T18:38:44.000Z",
  "updated": "2013-03-16T19:14:28.000Z",
  "title": "Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models",
  "authors": [
    "Alexander Elgart",
    "Abel Klein"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider discrete Schr\\\"odinger operators of the form $H=-\\Delta +V$ on $\\ell^2(\\Z^d)$, where $\\Delta$ is the discrete Laplacian and $V$ is a bounded potential. Given $\\Gamma \\subset \\Z^d$, the $\\Gamma$-trimming of $H$ is the restriction of $H$ to $\\ell^2(\\Z^d\\setminus\\Gamma)$, denoted by $H_\\Gamma$. We investigate the dependence of the ground state energy $E_\\Gamma(H)=\\inf \\sigma (H_\\Gamma)$ on $\\Gamma$. We show that for relatively dense proper subsets $\\Gamma$ of $\\Z^d$ we always have $E_\\Gamma(H)>E_\\emptyset(H)$. We use this lifting of the ground state energy to establish Wegner estimates and localization at the bottom of the spectrum for $\\Gamma$-trimmed Anderson models, i.e., Anderson models with the random potential supported by the set $\\Gamma$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-16T19:14:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B44",
      "47B80",
      "60H25"
    ],
    "keywords": [
      "ground state energy",
      "trimmed discrete schrödinger operators",
      "trimmed anderson models",
      "localization",
      "relatively dense proper subsets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.5268E"
    }
  }
}