{
  "id": "2301.13129",
  "version": "v1",
  "published": "2023-01-30T18:07:31.000Z",
  "updated": "2023-01-30T18:07:31.000Z",
  "title": "Resolvent bounds for Lipschitz potentials in dimension two and higher with singularities at the origin",
  "authors": [
    "Donnell Obovu"
  ],
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "We consider, for $h,E>0$, the semiclassical Schr\\\"odinger operator $-h^2\\Delta + V - E$ in dimension two. The potential $V$, and its radial derivative $\\partial_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\\delta}$ near the origin while $\\partial_{r}V$ is bounded by $r^{-1-\\delta}$, where $0\\leq\\delta\\leq 4(\\sqrt{2}-1)$. In this setting, we show that the resolvent bound is exponential in $h^{-1}$, while the exterior resolvent bound is linear in $h^{-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-30T18:07:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lipschitz potentials",
      "singularities",
      "exterior resolvent bound",
      "long-range decay",
      "bounded away"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}