{
  "id": "1612.06591",
  "version": "v1",
  "published": "2016-12-20T10:26:54.000Z",
  "updated": "2016-12-20T10:26:54.000Z",
  "title": "Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications",
  "authors": [
    "Sergey Morozov",
    "David Müller"
  ],
  "comment": "28 pages, 1 figure",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "For $\\nu\\in[0, 1]$ let $D^\\nu$ be the distinguished self-adjoint realisation of the three-dimensional Coulomb-Dirac operator $-\\mathrm i\\boldsymbol\\alpha\\cdot\\nabla -\\nu|\\cdot|^{-1}$. For $\\nu\\in[0, 1)$ we prove the lower bound of the form $|D^\\nu| \\geqslant C_\\nu\\sqrt{-\\Delta}$, where $C_\\nu$ is found explicitly and is better then in all previous works on the topic. In the critical case $\\nu =1$ we prove that for every $\\lambda\\in [0, 1)$ there exists $K_\\lambda >0$ such that the estimate $|D^{1}| \\geqslant K_\\lambda a^{\\lambda -1}(-\\Delta)^{\\lambda/2} -a^{-1}$ holds for all $a >0$. As applications we extend the range of coupling constants in the proof of the stability of the relativistic electron-positron field and obtain Cwickel-Lieb-Rozenblum and Lieb-Thirring type estimates on the negative eigenvalues of perturbed projected massless Coulomb-Dirac operators in the Furry picture. We also study the existence of a virtual level at zero for such projected operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-20T10:26:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A63",
      "35P15"
    ],
    "keywords": [
      "three-dimensional coulomb-dirac operator",
      "lower bound",
      "fractional laplacians",
      "applications",
      "relativistic electron-positron field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}