{
  "id": "1807.09653",
  "version": "v1",
  "published": "2018-07-25T15:11:06.000Z",
  "updated": "2018-07-25T15:11:06.000Z",
  "title": "Spectral Theory for Systems of Ordinary Differential Equations with Distributional Coefficients",
  "authors": [
    "Ahmed Ghatasheh",
    "Rudi Weikard"
  ],
  "categories": [
    "math.CA",
    "math.SP"
  ],
  "abstract": "We study the spectral theory for the first-order system $Ju'+qu=wf$ of differential equations on the real interval $(a,b)$ when $J$ is a constant, invertible skew-Hermitian matrix and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. Also, we do not pose the definiteness condition customarily required for the coefficients of the equation. Specifically, we construct minimal and maximal relations, and study self-adjoint restrictions of the maximal relation. For these we determine Green's function and prove the existence of a spectral (or generalized Fourier) transformation. We have a closer look at the special cases when the endpoints of the interval $(a,b)$ are regular as well as the case of a $2\\times2$ system. Two appendices provide necessary details on distributions of order zero and the abstract spectral theory for relations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-25T15:11:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ordinary differential equations",
      "distributional coefficients",
      "maximal relation",
      "order zero",
      "determine greens function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}