{
  "id": "1205.4351",
  "version": "v4",
  "published": "2012-05-19T17:29:43.000Z",
  "updated": "2014-11-05T15:00:16.000Z",
  "title": "Unitary groups and spectral sets",
  "authors": [
    "Dorin Ervin Dutkay",
    "Palle E. T. Jorgensen"
  ],
  "comment": "We improved the paper and partition it into several independent parts",
  "categories": [
    "math.FA"
  ],
  "abstract": "We study spectral theory for bounded Borel subsets of $\\br$ and in particular finite unions of intervals. For Hilbert space, we take $L^2$ of the union of the intervals. This yields a boundary value problem arising from the minimal operator $\\Ds = \\frac1{2\\pi i}\\frac{d}{dx}$ with domain consisting of $C^\\infty$ functions vanishing at the endpoints. We offer a detailed interplay between geometric configurations of unions of intervals and a spectral theory for the corresponding selfadjoint extensions of $\\Ds$ and for the associated unitary groups of local translations. While motivated by scattering theory and quantum graphs, our present focus is on the Fuglede-spectral pair problem. Stated more generally, this problem asks for a determination of those bounded Borel sets $\\Omega$ in $\\br^k$ such that $L^2(\\Omega)$ has an orthogonal basis of Fourier frequencies (spectrum), i.e., a total set of orthogonal complex exponentials restricted to $\\Omega$. In the general case, we characterize Borel sets $\\Omega$ having this spectral property in terms of a unitary representation of $(\\br, +)$ acting by local translations. The case of $k = 1$ is of special interest, hence the interval-configurations. We give a characterization of those geometric interval-configurations which allow Fourier spectra directly in terms of the selfadjoint extensions of the minimal operator $\\Ds$. This allows for a direct and explicit interplay between geometry and spectra. As an application, we offer a new look at the Universal Tiling Conjecture and show that the spectral-implies-tile part of the Fuglede conjecture is equivalent to it and can be reduced to a variant of the Fuglede conjecture for unions of integer intervals.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-09-05T19:10:31.000Z",
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2014-11-05T15:00:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unitary groups",
      "spectral sets",
      "fuglede conjecture",
      "local translations",
      "selfadjoint extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.4351E"
    }
  }
}