{
  "id": "1303.2459",
  "version": "v3",
  "published": "2013-03-11T09:31:45.000Z",
  "updated": "2014-11-02T08:00:34.000Z",
  "title": "A probabilistic proof of the fundamental gap conjecture via the coupling by reflection",
  "authors": [
    "Fuzhou Gong",
    "Huaiqian Li",
    "Dejun Luo"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\Omega\\subset\\R^n$ be a strictly convex domain with smooth boundary and diameter $D$. The fundamental gap conjecture claims that if $V:\\bar\\Omega\\to\\R$ is convex, then the spectral gap of the Schr\\\"odinger operator $-\\Delta+V$ with Dirichlet boundary condition is greater than $\\frac{3\\pi^2}{D^2}$. Using analytic methods, Andrews and Clutterbuck recently proved in [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916] a more general spectral gap comparison theorem which implies this conjecture. In the first part of the current work, we shall give an independent probabilistic proof of their result via the coupling by reflection of the diffusion processes. Moreover, we also present in the second part a simpler probabilistic proof of the original conjecture.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-04-23T09:00:13.000Z",
      "title": "The fundamental gap conjecture: a probabilistic approach via the coupling by reflection",
      "abstract": "The fundamental gap conjecture asserts that the spectral gap of the Schr\\\"odinger operator $-\\Delta+V$ with Dirichlet boundary condition on the bounded convex domain $\\Omega\\subset \\R^n$ is greater than $\\frac{3\\pi^2}{D^2}$, provided that the potential $V:\\bar\\Omega\\ra\\R$ is convex. Here $D>0$ is the diameter of $\\Omega$. Using analytic methods, Andrews and Clutterbuck proved recently a more general spectral gap comparison result which implies the conjecture. In this work we shall give a simple probabilistic proof via the coupling by reflection of the diffusion processes.",
      "comment": "13 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-11-02T08:00:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probabilistic approach",
      "general spectral gap comparison result",
      "reflection",
      "fundamental gap conjecture asserts",
      "dirichlet boundary condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1303.2459G"
    }
  }
}