{
  "id": "2009.06909",
  "version": "v1",
  "published": "2020-09-15T08:01:22.000Z",
  "updated": "2020-09-15T08:01:22.000Z",
  "title": "Large $|k|$ behavior of complex geometric optics solutions to d-bar problems",
  "authors": [
    "C. Klein",
    "J. Sjöstrand",
    "N. Stoilov"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter $k$. For potentials \\( q\\in \\langle \\cdot \\rangle^{-2} H^{s}(\\mathbb{C}) \\) for some $s \\in]1,2]$, it is shown that the solution converges as the geometric series in $1/|k|^{s-1}$. For potentials $q$ being the characteristic function of a strictly convex open set with smooth boundary, this still holds with $s=3/2$ i.e., with $1/\\sqrt{|k|}$ instead of $1/|k|^{s-1}$. The leading order controbutions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-15T08:01:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex geometric optics solutions",
      "d-bar problems",
      "characteristic function",
      "strictly convex open set",
      "large values"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}