{
  "id": "2310.01192",
  "version": "v1",
  "published": "2023-10-02T13:29:01.000Z",
  "updated": "2023-10-02T13:29:01.000Z",
  "title": "Eigenvalues and resonances of dissipative acoustic operator for strictly convex obstacles",
  "authors": [
    "Vesselin Petkov"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We examine the wave equation in the exterior of a strictly convex bounded domain $K$ with dissipative boundary condition $\\partial_{\\nu} u - \\gamma(x) \\partial_t u = 0$ on the boundary $\\Gamma$ and $0 < \\gamma(x) <1, \\:\\forall x \\in \\Gamma.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, \\: t \\geq 0.$ The poles $\\lambda$ of the meromorphic incoming resolvent $(G - \\lambda)^{-1} $ with ${\\rm Re}\\: \\lambda > 0$ are called incoming resonances. We obtain sharper results for the location of the eigenvalues of $G$ and incoming resonances in $\\Lambda = \\{\\lambda \\in {\\mathbb C}:\\: |{\\rm Re}\\: \\lambda| \\leq C_2(1 + |{\\rm Im}\\: \\lambda|)^{-2},\\: |{\\rm Im}\\: \\lambda| \\geq A_2 > 1\\}$ and we prove a Weyl formula for the asymptotic of these eigenvalues and incoming resonances. For $K = \\{x \\in {\\mathbb R}^3:\\:|x| \\leq 1\\}$ and $\\gamma$ constant we show that $G$ has no eigenvalues so the Weyl formula concerns only the incoming resonances.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-02T13:29:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P20",
      "35P25",
      "47A40",
      "58J50"
    ],
    "keywords": [
      "strictly convex obstacles",
      "dissipative acoustic operator",
      "eigenvalues",
      "incoming resonances",
      "weyl formula concerns"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}