{
  "id": "math/0501049",
  "version": "v2",
  "published": "2005-01-04T14:44:48.000Z",
  "updated": "2006-10-23T05:53:55.000Z",
  "title": "Inverse Problems and Index Formulae for Dirac Operators",
  "authors": [
    "Yaroslav Kurylev",
    "Matti Lassas"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We consider a Dirac-type operator $D_P$ on a vector bundle $V$ over a compact Riemannian manifold $(M,g)$ with a nonempty boundary. The operator $D_P$ is specified by a boundary condition $P(u|_{\\p M})=0$ where $P$ is a projector which may be a non-local, i.e. a pseudodifferential operator. We assume the existence of a chirality operator which decomposes $L^2(M, V)$ into two orthogonal subspaces $X_+ \\oplus X_-$. Under certain conditions, the operator $D_P$ restricted to $X_+$ and $ X_-$ defines a pair of Fredholm operators which maps $X_+\\to X_-$ and $X_-\\to X_+$ correspondingly, giving rise to a superstructure on $V$. In this paper we consider the questions of determining the index of $D_P$ and the reconstruction of $(M, g), V$ and $D_P$ from the boundary data on $\\p M$. The data used is either the Cauchy data, i.e. the restrictions to $\\p M \\times \\R_+$ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e. the set of the eigenvalues and the boundary values of the eigenfunctions of $D_P$. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in $M\\times \\C^4$, $M \\subset \\R^3$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-10-23T05:53:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J25",
      "58J45"
    ],
    "keywords": [
      "inverse problems",
      "index formulae",
      "dirac operators",
      "inverse boundary value problems",
      "compact riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......1049K"
    }
  }
}