{
  "id": "2007.04661",
  "version": "v1",
  "published": "2020-07-09T09:40:18.000Z",
  "updated": "2020-07-09T09:40:18.000Z",
  "title": "Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains",
  "authors": [
    "Bruno Colbois",
    "Alessandro Savo"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential $1$-form (hence, with zero magnetic field) acting on complex functions of a planar domain $\\Omega$, with magnetic Neumann boundary conditions. It is well-known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $\\epsilon$-net. In the last part we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-09T09:40:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J50",
      "35P15"
    ],
    "keywords": [
      "upper bound",
      "zero magnetic field",
      "ground state energy",
      "planar domain",
      "first eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}