{
  "id": "1510.06485",
  "version": "v1",
  "published": "2015-10-22T03:45:13.000Z",
  "updated": "2015-10-22T03:45:13.000Z",
  "title": "Remark on Asymptotic Completeness for Nonlinear Klein-Gordon Equations with Metastable States",
  "authors": [
    "Xinliang An",
    "Avy Soffer"
  ],
  "comment": "42 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "In this paper, we are in a further exploration of metastable states constructed by Soffer and Weinstein in [15]. These metastable states are the outcome of instabilities of excited states for Klein-Gordon equations under nonlinear Fermi golden rule. In [15], Soffer and Weinstein derived an upper bound ($1/(1+t)^{1/4}$) for an anomalously slow decay rate of these states. We derive the asymptotic behavior (including a lower bound $1/(1+t)^{1/4}$) and prove asymptotic completeness for current cases. This means that at very late stage ($t$ goes to $+\\infty$) the solutions to this nonlinear Klein-Gordon equation could be written as the sum of free waves and error terms; and the $L^2_x$ norm of error terms will vanish eventually ($t$ goes to $+\\infty$). Therefore, even though there is a lower bound for the anomalously slow decay rate for these states, asymptotic completeness still holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-22T03:45:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear klein-gordon equation",
      "asymptotic completeness",
      "metastable states",
      "anomalously slow decay rate",
      "lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151006485A"
    }
  }
}