{
  "id": "1211.3336",
  "version": "v2",
  "published": "2012-11-14T15:36:57.000Z",
  "updated": "2015-07-24T07:06:26.000Z",
  "title": "On spectral stability of the nonlinear Dirac equation",
  "authors": [
    "Nabile Boussaid",
    "Andrew Comech"
  ],
  "comment": "42 pages. The spectral stability of solitary waves in a charge-subcritical NLD in the nonrelativistic limit ($\\omega\\lesssim m$) will be proved in a forthcoming paper",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "nlin.PS"
  ],
  "abstract": "We study the point spectrum of the nonlinear Dirac equation in any spatial dimension, linearized at one of the solitary wave solutions. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds. We then prove that the birth of point eigenvalues with nonzero real part (the ones which lead to linear instability) from the essential spectrum is only possible from the embedded eigenvalues or thresholds, and therefore can not take place beyond the embedded thresholds. We also prove that \"in the nonrelativistic limit\" $\\omega\\to m$, the point eigenvalues can only accumulate to $0$ and $\\pm 2 m i$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-14T15:36:57.000Z",
      "abstract": "We study the point spectrum of the nonlinear Dirac equation linearized at one of the solitary wave solutions $\\phi_\\omega(x)e^{-i \\omega t}$. We prove that, in any dimension, the linearized equation has no embedded eigenvalues in the part of the essential spectrum beyond the embedded thresholds (located at $\\lambda=\\pm i (m+|\\omega|)$). We then prove that the birth of point eigenvalues with nonzero real part from the essential spectrum is only possible from the embedded eigenvalues, and therefore can not take place beyond the embedded thresholds. We also study the birth of point eigenvalues in the nonrelativistic limit, $\\omega\\to m$. We apply our results to show that small amplitude solitary waves ($\\omega\\lesssim m$) of cubic nonlinear Dirac equation in one dimension are spectrally stable.",
      "comment": "34 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-07-24T07:06:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35C08",
      "35Q41",
      "37K40",
      "47A13",
      "81Q05"
    ],
    "keywords": [
      "spectral stability",
      "small amplitude solitary waves",
      "point eigenvalues",
      "essential spectrum",
      "cubic nonlinear dirac equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1203226,
      "adsabs": "2012arXiv1211.3336B"
    }
  }
}