{
  "id": "2306.03936",
  "version": "v1",
  "published": "2023-06-06T18:01:46.000Z",
  "updated": "2023-06-06T18:01:46.000Z",
  "title": "Counting eigenvalues of Schrödinger operators using the landscape function",
  "authors": [
    "Sven Bachmann",
    "Richard Froese",
    "Severin Schraven"
  ],
  "comment": "21 pages. Comments welcome",
  "categories": [
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We prove an upper and a lower bound on the rank of the spectral projections of the Schr\\\"odinger operator $-\\Delta + V$ in terms of the volume of the sublevel sets of an effective potential $\\frac{1}{u}$. Here, $u$ is the `landscape function' of (David, G., Filoche, M., & Mayboroda, S. (2021) Advances in Mathematics, 390, 107946) namely the solution of $(-\\Delta + V)u = 1$ in $\\mathbb{R}^d$. We prove the result for non-negative potentials satisfying a Kato and doubling condition, in all spatial dimensions, in infinite volume, and show that no coarse graining is required. Our result yields in particular a necessary and sufficient condition for discreteness of the spectrum.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-06T18:01:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "35P20",
      "35J10"
    ],
    "keywords": [
      "landscape function",
      "schrödinger operators",
      "counting eigenvalues",
      "lower bound",
      "sublevel sets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}