{
  "id": "1912.10949",
  "version": "v1",
  "published": "2019-12-23T16:11:16.000Z",
  "updated": "2019-12-23T16:11:16.000Z",
  "title": "The $1$d nonlinear Schrödinger equation with a weighted $L^1$ potential",
  "authors": [
    "Gong Chen",
    "Fabio Pusateri"
  ],
  "comment": "45 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the $1d$ cubic nonlinear Schr\\\"odinger equation with a large external potential $V$ with no bound states. We prove global regularity and quantitative bounds for small solutions under mild assumptions on $V$. In particular, we do not require any differentiability of $V$, and make spatial decay assumptions that are weaker than those found in the literature (see for example \\cite{Del,N,GPR}). We treat both the case of generic and non-generic potentials, with some additional symmetry assumptions in the latter case. Our approach is based on the combination of three main ingredients: the Fourier transform adapted to the Schr\\\"odinger operator, basic bounds on pseudo-differential operators that exploit the structure of the Jost function, and improved local decay and smoothing-type estimates. An interesting aspect of the proof is an \"approximate commutation\" identity for a suitable notion of a vectorfield, which allows us to simplify the previous approaches and extend the known results to a larger class of potentials. Finally, under our weak assumptions we can include the interesting physical case of a barrier potential as well as recover the result of \\cite{MMS} for a delta potential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-23T16:11:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35P25",
      "35J10"
    ],
    "keywords": [
      "nonlinear schrödinger equation",
      "spatial decay assumptions",
      "additional symmetry assumptions",
      "large external potential",
      "bound states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}