{
  "id": "1706.06357",
  "version": "v1",
  "published": "2017-06-20T10:30:50.000Z",
  "updated": "2017-06-20T10:30:50.000Z",
  "title": "Harmonic Approximation of Difference Operators",
  "authors": [
    "Markus Klein",
    "Elke Rosenberger"
  ],
  "comment": "30 pages",
  "journal": "Journal of Functional Analysis 257 (2009) 3409-3453",
  "doi": "10.1016/j.jfa.2009.09.004",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "For a general class of difference operators $H_\\varepsilon = T_\\varepsilon + V_\\varepsilon$ on $\\ell^2(\\varepsilon\\mathbb{Z}^d)$, where $V_\\varepsilon$ is a multi-well potential and $\\varepsilon$ is a small parameter, we analyze the asymptotic behavior as $\\varepsilon\\to 0$ of the (low-lying) eigenvalues and eigenfunctions. We show that the first $n$ eigenvalues of $H_\\varepsilon$ converge to the first $n$ eigenvalues of the direct sum of harmonic oscillators on $\\mathbb{R}^d$ located at the several wells. Our proof is microlocal.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-20T10:30:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "difference operators",
      "harmonic approximation",
      "eigenvalues",
      "harmonic oscillators",
      "general class"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}