{
  "id": "1101.2145",
  "version": "v2",
  "published": "2011-01-11T15:45:57.000Z",
  "updated": "2011-09-09T06:36:12.000Z",
  "title": "Scattering theory for Klein-Gordon equations with non-positive energy",
  "authors": [
    "Christian Gérard"
  ],
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP",
    "math.SP"
  ],
  "abstract": "We study the scattering theory for charged Klein-Gordon equations: \\[\\{{array}{l} (\\p_{t}- \\i v(x))^{2}\\phi(t,x) \\epsilon^{2}(x, D_{x})\\phi(t,x)=0,[2mm] \\phi(0, x)= f_{0}, [2mm] \\i^{-1} \\p_{t}\\phi(0, x)= f_{1}, {array}. \\] where: \\[\\epsilon^{2}(x, D_{x})= \\sum_{1\\leq j, k\\leq n}(\\p_{x_{j}} \\i b_{j}(x))A^{jk}(x)(\\p_{x_{k}} \\i b_{k}(x))+ m^{2}(x),\\] describing a Klein-Gordon field minimally coupled to an external electromagnetic field described by the electric potential $v(x)$ and magnetic potential $\\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the energy: \\[ h[f, f]:= \\int_{\\rr^{n}}\\bar{f}_{1}(x) f_{1}(x)+ \\bar{f}_{0}(x)\\epsilon^{2}(x, D_{x})f_{0}(x) - \\bar{f}_{0}(x) v^{2}(x) f_{0}(x) \\d x. \\] We consider the situation when the energy is not positive. In this case the flow cannot be written as a unitary group on a Hilbert space, and the Klein-Gordon equation may have complex eigenfrequencies. Using the theory of definitizable operators on Krein spaces and time-dependent methods, we prove the existence and completeness of wave operators, both in the short- and long-range cases. The range of the wave operators are characterized in terms of the spectral theory of the generator, as in the usual Hilbert space case.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-09-09T06:36:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "scattering theory",
      "non-positive energy",
      "usual hilbert space case",
      "wave operators",
      "klein-gordon equation preserves"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00023-011-0138-8"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 884105,
      "adsabs": "2012AnHP...13..883G"
    }
  }
}