{
  "id": "1404.4325",
  "version": "v1",
  "published": "2014-04-16T17:53:47.000Z",
  "updated": "2014-04-16T17:53:47.000Z",
  "title": "On the spectrum of the discrete $1d$ Schrödinger operator with an arbitrary even potential",
  "authors": [
    "Sergei B. Rutkevich"
  ],
  "comment": "14 pages, no figures",
  "categories": [
    "math-ph",
    "cond-mat.dis-nn",
    "math.MP"
  ],
  "abstract": "The discrete one-dimensional Schr\\\"odinger operator is studied in the finite interval of length $N=2 M$ with the Dirichlet boundary conditions and an arbitrary potential even with respect to the spacial reflections. It is shown, that the eigenvalues of such a discrete Schr\\\"odinger operator (Hamiltonian), which is represented by the $2M\\times2M$ tridiagonal matrix, satisfy a set of polynomial constrains. The most interesting constrain, which is explicitly obtained, leads to the effective Coulomb interaction between the Hamiltonian eigenvalues. In the limit $M\\to\\infty$, this constrain induces the requirement, which should satisfy the scattering date in the scattering problem for the discrete Schr\\\"odinger operator in the half-line. We obtain such a requirement in the simplest case of the Schr\\\"odinger operator, which does not have bound and semi-bound states, and which potential has a compact support.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-16T17:53:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schrödinger operator",
      "dirichlet boundary conditions",
      "compact support",
      "arbitrary potential",
      "spacial reflections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4325R"
    }
  }
}