{
  "id": "1506.07431",
  "version": "v1",
  "published": "2015-06-24T15:43:48.000Z",
  "updated": "2015-06-24T15:43:48.000Z",
  "title": "Manifold decompositions and indices of Schrödinger operators",
  "authors": [
    "Graham Cox",
    "Christoper K. R. T. Jones",
    "Jeremy L. Marzuola"
  ],
  "comment": "19 pages, 4 figures",
  "categories": [
    "math.AP",
    "math.SP"
  ],
  "abstract": "We use the Maslov index to relate the spectra of different boundary value problems for Schr\\\"{o}dinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold $M$ divided into two connected components, $\\Omega_1$ and $\\Omega_2$, by a separating hypersurface $\\Sigma$. We use a homotopy argument to relate the spectrum of a second-order elliptic operator on $M$ to its Dirichlet and Neumann spectra on $\\Omega_1$ and $\\Omega_2$, with the difference given by the Maslov index of a path of Lagrangian subspaces. We then use a homotopy argument to relate the Maslov index to the Morse indices of the Dirichlet-to-Neumann maps on $\\Sigma$. Applications are given to doubling constructions, periodic boundary conditions and the counting of nodal domains. In particular, we give a new proof of Courant's nodal domain theorem, with an explicit formula for the nodal deficiency.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-24T15:43:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schrödinger operators",
      "manifold decompositions",
      "maslov index",
      "courants nodal domain theorem",
      "homotopy argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150607431C"
    }
  }
}