{
  "id": "1712.03925",
  "version": "v1",
  "published": "2017-12-11T18:11:32.000Z",
  "updated": "2017-12-11T18:11:32.000Z",
  "title": "Level spacing for continuum random Schrödinger operators with applications",
  "authors": [
    "Adrian Dietlein",
    "Alexander Elgart"
  ],
  "comment": "35 pages; comments welcome",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "For continuum alloy-type random Schr\\\"odinger operators with sign-definite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size $L$, we prove that with high probability the eigenvalues below some threshold energy keep a distance of at least $e^{-(\\log L)^\\beta}$ for sufficiently large $\\beta>1$. This implies simplicity of the spectrum of the infinite-volume operator below the threshold energy. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy $E$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-11T18:11:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuum random schrödinger operators",
      "level spacing",
      "threshold energy",
      "sign-definite single-site bump functions",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}