{
  "id": "0906.5478",
  "version": "v1",
  "published": "2009-06-30T11:11:47.000Z",
  "updated": "2009-06-30T11:11:47.000Z",
  "title": "Spectral and scattering theory of charged $P(\\varphi)_2$ models",
  "authors": [
    "Christian Gérard"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider in this paper space-cutoff charged $P(\\varphi)_{2}$ models arising from the quantization of the non-linear charged Klein-Gordon equation: \\[ (\\p_{t}+\\i V(x))^{2}\\phi(t, x)+ (-\\Delta_{x}+ m^{2})\\phi(t,x)+ g(x)\\p_{\\overline{z}}P(\\phi(t,x), \\overline{\\phi}(t,x))=0, \\] where $V(x)$ is an electrostatic potential, $g(x)\\geq 0$ a space-cutoff and $P(\\lambda, \\overline{\\lambda})$ a real bounded below polynomial. We discuss various ways to quantize this equation, starting from different CCR representations. After describing the construction of the interacting Hamiltonian $H$ we study its spectral and scattering theory. We describe the essential spectrum of $H$, prove the existence of asymptotic fields and of wave operators, and finally prove the {\\em asymptotic completeness} of wave operators. These results are similar to the case when V=0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-30T11:11:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scattering theory",
      "wave operators",
      "non-linear charged klein-gordon equation",
      "asymptotic completeness",
      "electrostatic potential"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s11005-010-0392-6",
      "journal": "Letters in Mathematical Physics",
      "year": 2010,
      "month": "Jun",
      "volume": 92,
      "number": 3,
      "pages": 197
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010LMaPh..92..197G"
    }
  }
}