{
  "id": "math/0608343",
  "version": "v1",
  "published": "2006-08-14T13:45:48.000Z",
  "updated": "2006-08-14T13:45:48.000Z",
  "title": "On a spectral representation for correlation measures in configuration space analysis",
  "authors": [
    "Yu. M. Berezansky",
    "Yu. G. Kondratiev",
    "T. Kuna",
    "E. Lytvynov"
  ],
  "journal": "Meth. Funct. Anal. Topol. 5 (1999), no.4, 87-100",
  "categories": [
    "math.PR"
  ],
  "abstract": "The paper is devoted to the study of configuration space analysis by using the projective spectral theorem. For a manifold $X$, let $\\Gamma_X$, resp.\\ $\\Gamma_{X,0}$ denote the space of all, resp. finite configurations in $X$. The so-called $K$-transform, introduced by A. Lenard, maps functions on $\\Gamma_{X,0}$ into functions on $\\Gamma_{X}$ and its adjoint $K^*$ maps probability measures on $\\Gamma_X$ into $\\sigma$-finite measures on $\\Gamma_{X,0}$. For a probability measure $\\mu$ on $\\Gamma_X$, $\\rho_\\mu:=K^*\\mu$ is called the correlation measure of $\\mu$. We consider the inverse problem of existence of a probability measure $\\mu$ whose correlation measure $\\rho_\\mu$ is equal to a given measure $\\rho$. We introduce an operation of $\\star$-convolution of two functions on $\\Gamma_{X,0}$ and suppose that the measure $\\rho$ is $\\star$-positive definite, which enables us to introduce the Hilbert space ${\\cal H}_\\rho$ of functions on $\\Gamma_{X,0}$ with the scalar product $(G^{(1)},G^{(2)})_{{\\cal H}_{\\rho}}= \\int_{\\Gamma_{X,0}}(G^{(1)}\\star\\bar G{}^{(2)})(\\eta) \\rho(d\\eta)$. Under a condition on the growth of the measure $\\rho$ on the $n$-point configuration spaces, we construct the Fourier transform in generalized joint eigenvectors of some special family $A=(A_\\phi)_{\\phi\\in\\D}$, $\\D:=C_0^\\infty(X)$, of commuting selfadjoint operators in ${\\cal H}_\\rho$. We show that this Fourier transform is a unitary between ${\\cal H}_{\\rho}$ and the $L^2$-space $L^2(\\Gamma_X,d\\mu)$, where $\\mu$ is the spectral measure of $A$. Moreover, this unitary coincides with the $K$-transform, while the measure $\\rho$ is the correlation measure of $\\mu$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-14T13:45:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G57",
      "47A75",
      "60K35"
    ],
    "keywords": [
      "configuration space analysis",
      "correlation measure",
      "spectral representation",
      "fourier transform",
      "point configuration spaces"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8343B"
    }
  }
}