{
  "id": "1011.1086",
  "version": "v1",
  "published": "2010-11-04T09:14:15.000Z",
  "updated": "2010-11-04T09:14:15.000Z",
  "title": "The Schrödinger-Poisson system on the sphere",
  "authors": [
    "Patrick Gérard",
    "Florian Méhats"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Schr\\\"odinger-Poisson system on the unit sphere $\\SS^2$ of $\\RR^3$, modeling the quantum transport of charged particles confined on a sphere by an external potential. Our first results concern the Cauchy problem for this system. We prove that this problem is regularly well-posed on every $H^s(\\SS ^2)$ with $s>0$, and not uniformly well-posed on $L^2(\\SS ^2)$. The proof of well-posedness relies on multilinear Strichartz estimates, the proof of ill-posedness relies on the construction of a counterexample which concentrates exponentially on a closed geodesic. In a second part of the paper, we prove that this model can be obtained as the limit of the three dimensional Schr\\\"odinger-Poisson system, singularly perturbed by an external potential that confines the particles in the vicinity of the sphere.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-11-04T09:14:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schrödinger-poisson system",
      "external potential",
      "multilinear strichartz estimates",
      "unit sphere",
      "ill-posedness relies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.1086G"
    }
  }
}