{
  "id": "1709.09200",
  "version": "v1",
  "published": "2017-09-26T18:09:53.000Z",
  "updated": "2017-09-26T18:09:53.000Z",
  "title": "Bounds on the Pure Point Spectrum of Lattice Schrödinger Operators",
  "authors": [
    "Volker Bach",
    "Walter de Siqueira Pedra",
    "Saidakhmat Lakaev"
  ],
  "comment": "24 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In dimension $d\\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schr\\\"odinger operators $H(\\mathfrak{e},V)$ on the hypercubic lattice $\\mathbb{Z}^{d}$, with dispersion relation $\\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $|x|^{-2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of $H(\\mathfrak{e},V)$ for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator $B$ and positive number $\\lambda >0$, the number of negative eigenvalues of $\\lambda B$ is independent of $\\lambda$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-26T18:09:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pure point spectrum",
      "lattice schrödinger operators",
      "variational principle",
      "dispersion relation",
      "class ot potentials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}