{
  "id": "1708.03509",
  "version": "v1",
  "published": "2017-08-11T11:53:34.000Z",
  "updated": "2017-08-11T11:53:34.000Z",
  "title": "Asymptotics of resonances induced by point interactions",
  "authors": [
    "Jiri Lipovsky",
    "Vladimir Lotoreichik"
  ],
  "comment": "13 pages, 1 figure, submission to the proceedings of the 8th Workshop on Quantum Chaos and Localisation Phenomena, Warsaw, May 2017",
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We consider the resonances of the self-adjoint three-dimensional Schr\\\"odinger operator with point interactions of constant strength supported on the set $X = \\{ x_n \\}_{n=1}^N$. The size of $X$ is defined by $V_X = \\max_{\\pi\\in\\Pi_N} \\sum_{n=1}^N |x_n - x_{\\pi(n)}|$, where $\\Pi_N$ is the family of all the permutations of the set $\\{1,2,\\dots,N\\}$. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius $R$ asymptotically behaves as $\\frac{W_X}{\\pi} R + O(1)$ as $R \\to \\infty$, where $W_X \\in [0,V_X]$ is the effective size of $X$. Moreover, we show that there exist configurations of any number of points such that $W_X = V_X$. Finally, we construct an example for $N = 4$ with $W_X < V_X$, which can be viewed as an analogue of a non-Weyl quantum graph.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-11T11:53:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "point interactions",
      "resonances",
      "asymptotics",
      "non-weyl quantum graph",
      "self-adjoint three-dimensional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}