{
  "id": "1512.05940",
  "version": "v1",
  "published": "2015-12-18T13:21:55.000Z",
  "updated": "2015-12-18T13:21:55.000Z",
  "title": "Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation",
  "authors": [
    "F. Ali Mehmeti",
    "F. Dewez"
  ],
  "comment": "This paper is a unified version of the two articles \"Explicit error estimates for the stationary phase method I: The influence of amplitude singularities\" (arXiv:1412.5789) and \"Explicit error estimates for the stationary phase method II: Interaction of amplitude singularities with stationary points\" (arXiv:1412.5792). Results and presentation have been slightly improved",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a version of the stationary phase method in one dimension of A. Erd\\'elyi, allowing the phase to have stationary points of non-integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut-off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time-asymptotic behaviour of the solution of the free Schr\\\"odinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space-time cones as well as uniform and optimal estimates in curved regions which are asymptotically larger than any space-time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-18T13:21:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A80",
      "41A60",
      "35B40",
      "35B30",
      "35Q41"
    ],
    "keywords": [
      "stationary phase method",
      "lossless error estimates",
      "propagation features",
      "schrödinger equation",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}