{
  "id": "0904.4386",
  "version": "v1",
  "published": "2009-04-28T13:06:12.000Z",
  "updated": "2009-04-28T13:06:12.000Z",
  "title": "Intrinsic ultracontractivity for Schrodinger operators based on fractional Laplacians",
  "authors": [
    "Kamil Kaleta",
    "Tadeusz Kulczycki"
  ],
  "categories": [
    "math.PR",
    "math.SP"
  ],
  "abstract": "We study the Feynman-Kac semigroup generated by the Schr{\\\"o}dinger operator based on the fractional Laplacian $-(-\\Delta)^{\\alpha/2} - q$ in $\\Rd$, for $q \\ge 0$, $\\alpha \\in (0,2)$. We obtain sharp estimates of the first eigenfunction $\\phi_1$ of the Schr{\\\"o}dinger operator and conditions equivalent to intrinsic ultracontractivity of the Feynman-Kac semigroup. For potentials $q$ such that $\\lim_{|x| \\to \\infty} q(x) = \\infty$ and comparable on unit balls we obtain that $\\phi_1(x)$ is comparable to $(|x| + 1)^{-d - \\alpha} (q(x) + 1)^{-1}$ and intrinsic ultracontractivity holds iff $\\lim_{|x| \\to \\infty} q(x)/\\log|x| = \\infty$. Proofs are based on uniform estimates of $q$-harmonic functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-04-28T13:06:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G30",
      "60G52"
    ],
    "keywords": [
      "fractional laplacian",
      "schrodinger operators",
      "feynman-kac semigroup",
      "intrinsic ultracontractivity holds",
      "harmonic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}