{
  "id": "math-ph/0511045",
  "version": "v1",
  "published": "2005-11-11T21:19:26.000Z",
  "updated": "2005-11-11T21:19:26.000Z",
  "title": "A second eigenvalue bound for the Dirichlet Laplacian in hyperbolic space",
  "authors": [
    "Rafael D. Benguria",
    "Helmut Linde"
  ],
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $\\Omega$ be some domain in the hyperbolic space $\\Hn$ (with $n\\ge 2$) and $S_1$ the geodesic ball that has the same first Dirichlet eigenvalue as $\\Omega$. We prove the Payne-P\\'olya-Weinberger conjecture for $\\Hn$, i.e., that the second Dirichlet eigenvalue on $\\Omega$ is smaller or equal than the second Dirichlet eigenvalue on $S_1$. We also prove that the ratio of the first two eigenvalues on geodesic balls is a decreasing function of the radius.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-11-11T21:19:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P15",
      "49Rxx",
      "58Jxx"
    ],
    "keywords": [
      "second eigenvalue bound",
      "hyperbolic space",
      "dirichlet laplacian",
      "second dirichlet eigenvalue",
      "geodesic ball"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.ph..11045B"
    }
  }
}