{
  "id": "2306.12157",
  "version": "v1",
  "published": "2023-06-21T10:11:28.000Z",
  "updated": "2023-06-21T10:11:28.000Z",
  "title": "Rigorous derivation of the Efimov effect in a simple model",
  "authors": [
    "Davide Fermi",
    "Daniele Ferretti",
    "Alessandro Teta"
  ],
  "comment": "27 pages",
  "categories": [
    "math-ph",
    "cond-mat.quant-gas",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We consider a system of three identical bosons in $\\mathbb{R}^3$ with two-body zero-range interactions and a three-body hard-core repulsion of a given radius $a>0$. Using a quadratic form approach we prove that the corresponding Hamiltonian is self-adjoint and bounded from below for any value of $a$. In particular this means that the hard-core repulsion is sufficient to prevent the fall to the center phenomenon found by Minlos and Faddeev in their seminal work on the three-body problem in 1961. Furthermore, in the case of infinite two-body scattering length, also known as unitary limit, we prove the Efimov effect, \\emph{i.e.}, we show that the Hamiltonian has an infinite sequence of negative eigenvalues $E_n$ accumulating at zero and fulfilling the asymptotic geometrical law $\\;E_{n+1} / E_n \\; \\to \\; e^{-\\frac{2\\pi}{s_0}}\\,\\; \\,\\text{for} \\,\\; n\\to +\\infty$ holds, where $s_0\\approx 1.00624$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-21T10:11:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q10",
      "81Q15",
      "70F07",
      "46N50"
    ],
    "keywords": [
      "efimov effect",
      "simple model",
      "rigorous derivation",
      "three-body hard-core repulsion",
      "quadratic form approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}