{
  "id": "1506.08079",
  "version": "v1",
  "published": "2015-06-26T13:57:13.000Z",
  "updated": "2015-06-26T13:57:13.000Z",
  "title": "Time delay for the Dirac equation",
  "authors": [
    "Ivan Naumkin",
    "Ricardo Weder"
  ],
  "categories": [
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator $\\int\\limits_{0} ^{\\infty}e^{iH_{0}t}\\zeta\\left( \\frac{\\left\\vert x\\right\\vert }{R}\\right) e^{-iH_{0}t}dt,$ as $R\\rightarrow\\infty,$ is presented. Here $H_{0}$ is the free Dirac operator and $\\zeta\\left( t\\right) $ is such that $\\zeta\\left( t\\right) =1$ for $0\\leq t\\leq1$ and $\\zeta\\left( t\\right) =0$ for $t>1.$ This approach allows us to obtain the time delay operator $\\delta \\mathcal{T}\\left( f\\right) $ for initial states $f$ in $\\mathcal{H} _{2}^{3/2+\\varepsilon}\\left( \\mathbb{R}^{3};\\mathbb{C}^{4}\\right) ,$ $\\varepsilon>0,$ the Sobolev space of order $3/2+\\varepsilon$ and weight $2.$ The relation between the time delay operator $\\delta\\mathcal{T}\\left( f\\right) $ and the Eisenbud-Wigner time delay operator is given. Also, the relation between the averaged time delay and the spectral shift function is presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-26T13:57:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q40",
      "35P25",
      "35Q41",
      "81U99"
    ],
    "keywords": [
      "dirac equation",
      "eisenbud-wigner time delay operator",
      "spectral shift function",
      "free dirac operator",
      "expectation values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150608079N"
    }
  }
}