{
  "id": "2012.12735",
  "version": "v1",
  "published": "2020-12-23T15:16:09.000Z",
  "updated": "2020-12-23T15:16:09.000Z",
  "title": "The semi-classical limit with a delta-prime potential",
  "authors": [
    "Claudio Cacciapuoti",
    "Davide Fermi",
    "Andrea Posilicano"
  ],
  "comment": "24 pages. arXiv admin note: text overlap with arXiv:1907.05801",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the quantum evolution $e^{-i\\frac{t}{\\hbar}H_{\\beta}} \\psi_{\\xi}^{\\hbar}$ of a Gaussian coherent state $\\psi_{\\xi}^{\\hbar}\\in L^{2}(\\mathbb{R})$ localized close to the classical state $\\xi \\equiv (q,p) \\in \\mathbb{R}^{2}$, where $H_{\\beta}$ denotes a self-adjoint realization of the formal Hamiltonian $-\\frac{\\hbar^{2}}{2m}\\,\\frac{d^{2}\\,}{dx^{2}} + \\beta\\,\\delta'_{0}$, with $\\delta'_{0}$ the derivative of Dirac's delta distribution at $x = 0$ and $\\beta$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the $L^{2}(\\mathbb{R})$-norm, uniformly for any $t \\in \\mathbb{R}$ away from the collision time) by $e^{\\frac{i}{\\hbar} A_{t}} e^{it L_{B}} \\phi^{\\hbar}_{x}$, where $A_{t} = \\frac{p^{2}t}{2m}$, $\\phi_{x}^{\\hbar}(\\xi) := \\psi^{\\hbar}_{\\xi}(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $\\mathcal{C}^{\\infty}_{c}({\\mathscr M}_{0})$, ${\\mathscr M}_{0} := \\{(q,p) \\in \\mathbb{R}^{2}\\,|\\,q \\neq 0\\}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $\\hbar^{7/2-\\lambda}$, $0 < \\lambda < 1/2$, whereas it turns out to be of order $\\hbar^{3/2-\\lambda}$, $0 < \\lambda < 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-23T15:16:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81Q20",
      "81Q10",
      "47A40"
    ],
    "keywords": [
      "semi-classical limit",
      "delta-prime potential",
      "delta potential",
      "quantum evolution",
      "annali di matematica pura"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}