{
  "id": "2012.03045",
  "version": "v1",
  "published": "2020-12-05T15:01:31.000Z",
  "updated": "2020-12-05T15:01:31.000Z",
  "title": "Finite reflection groups and symmetric extensions of Laplacian",
  "authors": [
    "Krzysztof Stempak"
  ],
  "comment": "30 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $W$ be a finite reflection group associated with a root system $R$ in $\\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $\\Omega$ of $\\mathbb R^d$, symmetric with respect to reflections from $W$. Let $\\Omega_+=\\Omega\\cap C_+$ be the positive part of $\\Omega$. We define a family $\\{-\\Delta_{\\eta}^+\\}$ of self-adjoint extensions of the Laplacian $-\\Delta_{\\Omega_+}$, labeled by homomorphisms $\\eta\\colon W\\to \\{1,-1\\}$. In the construction of these $\\eta$-Laplacians $\\eta$-symmetrization of functions on $\\Omega$ is involved. The Neumann Laplacian $-\\Delta_{N,\\Omega_+}$ is included and corresponds to $\\eta\\equiv1$. If $H^{1}(\\Omega)=H^{1}_0(\\Omega)$, then the Dirichlet Laplacian $-\\Delta_{D,\\Omega_+}$ is either included and corresponds to $\\eta={\\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $\\Psi(-\\Delta_{N,\\Omega})$ and $\\Psi(-\\Delta_{\\eta}^+)$, or $\\Psi(-\\Delta_{D,\\Omega})$ and $\\Psi(-\\Delta_{D,\\Omega_+})$, where $\\Psi$ is a Borel function on $[0,\\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-05T15:01:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite reflection group",
      "symmetric extensions",
      "dirichlet laplacian",
      "spectral functional calculus",
      "open subset"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}