{
  "id": "2005.03790",
  "version": "v1",
  "published": "2020-05-07T22:36:04.000Z",
  "updated": "2020-05-07T22:36:04.000Z",
  "title": "The semiclassical limit on a star-graph with Kirchhoff conditions",
  "authors": [
    "Claudio Cacciapuoti",
    "Davide Fermi",
    "Andrea Posilicano"
  ],
  "comment": "31 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the dynamics of a quantum particle of mass $m$ on a $n$-edges star-graph with Hamiltonian $H_K=-(2m)^{-1}\\hbar^2 \\Delta$ and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\\u{\\i}n's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple $(H_K,H_{D}^{\\oplus})$, where $H_{D}^{\\oplus}$ is the free Hamiltonian with Dirichlet conditions in the vertex.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-07T22:36:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q20",
      "81Q35",
      "47A40"
    ],
    "keywords": [
      "semiclassical limit",
      "kirchhoff conditions",
      "initial state",
      "gaussian coherent state",
      "self-adjoint operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}