{
  "id": "0801.3976",
  "version": "v2",
  "published": "2008-01-25T16:51:01.000Z",
  "updated": "2008-09-17T19:06:57.000Z",
  "title": "Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations",
  "authors": [
    "Enno Lenzmann"
  ],
  "comment": "27 pages. Revised version. Statement of Theorem 2 changed",
  "journal": "Anal. PDE 2 (2009), no. 1, 1-27",
  "doi": "10.2140/apde.2009.2.1",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove uniqueness of ground states $Q$ in $H^{1/2}$ for pseudo-relativistic Hartree equations in three dimensions, provided that $Q$ has sufficiently small $L^2$-mass. This result shows that a uniqueness conjecture by Lieb and Yau in [CMP 112 (1987),147--174] holds true at least under a smallness condition. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation (also known as the Choquard-Pekard or Schroedinger-Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the so-called nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2008-09-17T19:06:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55"
    ],
    "keywords": [
      "pseudo-relativistic hartree equations",
      "ground states",
      "nonrelativistic hartree equations",
      "effective solitary wave motion",
      "limiting hartree equation"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.3976L"
    }
  }
}