{
  "id": "0709.2171",
  "version": "v1",
  "published": "2007-09-13T20:44:51.000Z",
  "updated": "2007-09-13T20:44:51.000Z",
  "title": "Inverse spectral problems on a closed manifold",
  "authors": [
    "Katsiaryna Krupchyk",
    "Yaroslav Kurylev",
    "Matti Lassas"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "In this paper we consider two inverse problems on a closed connected Riemannian manifold $(M,g)$. The first one is a direct analog of the Gel'fand inverse boundary spectral problem. To formulate it, assume that $M$ is divided by a hypersurface $\\Sigma$ into two components and we know the eigenvalues $\\lambda_j$ of the Laplace operator on $(M,g)$ and also the Cauchy data, on $\\Sigma$, of the corresponding eigenfunctions $\\phi_j$, i.e. $\\phi_j|_{\\Sigma},\\partial_\\nu\\phi_j|_{\\Sigma}$, where $\\nu$ is the normal to $\\Sigma$. We prove that these data determine $(M,g)$ uniquely, i.e. up to an isometry. In the second problem we are given much less data, namely, $\\lambda_j$ and $\\phi_j|_{\\Sigma}$ only. However, if $\\Sigma$ consists of at least two components, $\\Sigma_1, \\Sigma_2$, we are still able to determine $(M,g)$ assuming some conditions on $M$ and $\\Sigma$. These conditions are formulated in terms of the spectra of the manifolds with boundary obtained by cutting $M$ along $\\Sigma_i$, $i=1,2$, and are of a generic nature. We consider also some other inverse problems on $M$ related to the above with data which is easier to obtain from measurements than the spectral data described.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-13T20:44:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "58J50"
    ],
    "keywords": [
      "inverse spectral problems",
      "closed manifold",
      "gelfand inverse boundary spectral problem",
      "inverse problems",
      "closed connected riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.2171K"
    }
  }
}