{
  "id": "1302.5012",
  "version": "v2",
  "published": "2013-02-20T16:13:35.000Z",
  "updated": "2017-04-10T15:53:12.000Z",
  "title": "Coulomb scattering in the massless Nelson model II. Regularity of ground states",
  "authors": [
    "Wojciech Dybalski",
    "Alessandro Pizzo"
  ],
  "comment": "88 pages. Problematic intermediate estimate (5.1) of version 1 eliminated. Main results maintained. Analysis of wave functions shifted to part III of this series of papers",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "For the massless Nelson model we provide detailed information about the dependence of the normalized ground states $\\check{\\psi}_{P,\\sigma}$ of the fiber single-electron Hamiltonians $H_{P,\\sigma}$ on the total momentum $P$ and the infrared cut-off $\\sigma$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form \\[ \\|\\partial_{P_i}\\check{\\psi}_{P,\\sigma}\\|,\\ \\| \\partial_{P_i} \\partial_{P_j}\\check{\\psi}_{P,\\sigma} \\|\\leq \\frac{c}{\\sigma^{\\delta_{\\lambda_0}}}, \\] for some constant $c$ and a function of the maximal admissible coupling constant $\\lambda_0\\mapsto \\delta_{\\lambda_0}$ s.t. $\\lim_{\\lambda_0\\to 0}\\delta_{\\lambda_0}=0$. These results hold both in the infrared-regular and infrared-singular case. They are exploited in part I of this series to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of Haag-Ruelle scattering theory. They should also be relevant for the problem of scattering of two infraparticles in the infrared-singular Nelson model, whose solution is the goal of this series of papers. Although a part of a larger investigation, the present work is written in a self-contained fashion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-02-20T16:13:35.000Z",
      "abstract": "For the massless Nelson model we provide detailed information about the dependence of the normalized ground states $\\check{\\psi}_{P,\\sigma}$ of the fiber single-electron Hamiltonians $H_{P,\\sigma}$ on the total momentum $P$ and the infrared cut-off $\\sigma$. This information is obtained with the help of the iterative analytic perturbation theory. In particular, we derive bounds of the form \\[ \\|\\partial_{P}^{\\beta}\\check{\\psi}_{P,\\sigma}\\|\\leq \\frac{c}{\\sigma^{\\delta_{\\lambda_0}}}, \\] for any multiindex $\\beta$, s.t. $0\\leq |\\beta|\\leq 2$ and some function of the maximal admissible coupling constant $\\lambda_0\\mapsto \\delta_{\\lambda_0}$ s.t. $\\lim_{\\lambda_0\\to 0}\\de_{\\lambda_0}=0$. Analogous bounds are obtained for the derivatives w.r.t. $P$ of the $q$-particle momentum wavefunctions $f^q_{P,\\sigma}$ of $\\check{\\psi}_{P,\\sigma}$. These results are exploited in a companion paper to construct the two-electron scattering states in the infrared-regular massless Nelson model (in the absence of an infrared cut-off) along the lines of Haag-Ruelle scattering theory.",
      "comment": "45 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2017-04-10T15:53:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ground states",
      "coulomb scattering",
      "regularity",
      "particle momentum wavefunctions",
      "infrared cut-off"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 88,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.5012D"
    }
  }
}