{
  "id": "1501.00816",
  "version": "v1",
  "published": "2015-01-05T11:06:26.000Z",
  "updated": "2015-01-05T11:06:26.000Z",
  "title": "Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms",
  "authors": [
    "Anna Canale",
    "Abdelaziz Rhandi",
    "Cristian Tacelli"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that the heat kernel associated to the Schr\\\"odinger type operator $(1+|x|^\\alpha)\\Delta-|x|^\\beta$ satisfies the estimate $$k(t,x,y)\\leq c_1e^{\\lambda_0t}e^{c_2t^{-b}}\\frac{(|x||y|)^{-\\frac{N-1}{2}-\\frac{\\beta-\\alpha}{4}}}{1+|y|^\\alpha} e^{-\\int_1^{|x|}\\frac{s^{\\beta/2}}{\\sqrt{1+s^\\alpha}}\\,ds} e^{-\\int_1^{|y|}\\frac{s^{\\beta/2}}{\\sqrt{1+s^\\alpha}}\\,ds} $$ for $t>0,|x|,|y|\\ge 1$, where $c_1,c_2$ are positive constants and $b=\\frac{\\beta-\\alpha+2}{\\beta+\\alpha-2}$ provided that for $N>2,\\,\\alpha\\geq 2$ and $\\beta>\\alpha-2$. We also obtain an estimate of the eigenfunctions of $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-05T11:06:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K08",
      "35K10",
      "35J10",
      "47D07"
    ],
    "keywords": [
      "schrödinger type operators",
      "kernel estimates",
      "potential terms",
      "unbounded diffusion",
      "heat kernel"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150100816C"
    }
  }
}