{
  "id": "1308.4883",
  "version": "v2",
  "published": "2013-08-22T14:46:37.000Z",
  "updated": "2013-10-29T18:40:51.000Z",
  "title": "On the spectrum of the hierarchical Laplacian",
  "authors": [
    "Alexander Bendikov",
    "Paweł Krupski"
  ],
  "comment": "27 pages",
  "doi": "10.1007/s11118-014-9409-6",
  "categories": [
    "math.PR",
    "math.GN",
    "math.SP"
  ],
  "abstract": "Let $(X,d)$ be a locally compact separable ultrametric space. We assume that $(X,d)$ is proper, that is, any closed ball $B$ in $X$ is a compact set. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of balls (the choice function), we define the hierarchical Laplacian $L_C$ which is closely related to the concept of the hierarchical lattice of F.J. Dyson. $L_C$ is a non-negative definite, self-adjoint operator in $L^2(X,m)$. We address in this paper to the following question: How general can be the spectrum $\\mathsf{Spec}(L_C)$ as a subset of the non-negative reals? When $(X,d)$ is compact, $\\mathsf{Spec}(L_C)$ is an increasing sequence of eigenvalues of finite multiplicity which contains $0$. Assuming that $(X,d)$ is not compact we show that, under some natural conditions concerning the structure of the hierarchical lattice (= the tree of $d$-balls), any given closed subset $S$ of $[0,\\infty)$, which contains $0$ as an accumulation point and is unbounded if $X$ is non-discrete, may appear as $\\mathsf{Spec}(L_C)$ for some appropriately chosen function $C(B)$. The operator $-L_C$ extends to $L^q(X,m)$, $0 < q < \\infty$, as Markov generator and its spectrum does not depend on $q$. As an example, we consider the operator $\\mathfrak{D}^{\\alpha}$ of fractional derivative defined on the field $\\mathbb{Q}_p$ of $p$-adic numbers.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-29T18:40:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47S10",
      "60J25",
      "81Q10",
      "54E45",
      "05C05"
    ],
    "keywords": [
      "hierarchical laplacian",
      "hierarchical lattice",
      "locally compact separable ultrametric space",
      "choice function",
      "self-adjoint operator"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1308.4883B"
    }
  }
}