{
  "id": "2304.01493",
  "version": "v1",
  "published": "2023-04-04T03:13:16.000Z",
  "updated": "2023-04-04T03:13:16.000Z",
  "title": "Bounds on the number of scattering poles of half-Laplacian in odd dimensions, $d\\geq 3$",
  "authors": [
    "Ebru Toprak"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the scattering poles of $\\sqrt{-\\Delta} + V$, where $V$ is a compactly supported, bounded and complex valued potential. We show that the resolvent operator $ \\chi R_V \\chi$ has a meromorphic continuation to the whole Riemannian surface of $\\Lambda$ of $ \\log z $ as an operator $L^2 \\to L^2 $. We then obtain the upper bound on the counting function $N(r,a)= \\# \\{ z_j \\in \\Lambda: 0 \\leq |z_j| \\leq r, |\\arg z_j| \\leq a \\}$, $r >1$, $ |a| >1$ as $C \\langle a \\rangle ( \\langle r \\rangle^{d} + (\\log \\langle a \\rangle)^d) $, where $z_j$ are the poles of $ \\chi R_V \\chi$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-04T03:13:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47A10",
      "35P25"
    ],
    "keywords": [
      "scattering poles",
      "odd dimensions",
      "half-laplacian",
      "resolvent operator",
      "meromorphic continuation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}