{
  "id": "1807.03990",
  "version": "v1",
  "published": "2018-07-11T08:20:01.000Z",
  "updated": "2018-07-11T08:20:01.000Z",
  "title": "Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented",
  "authors": [
    "Pierre Bérard",
    "Bernard Helffer"
  ],
  "comment": "Comments: Continues arXiv:1803.00449 on Gelfand's approach to Sturm's theorem, with a small overlap",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "math.SP"
  ],
  "abstract": "In the second section ``Courant-Gelfand theorem'' of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\\, y''(s) + q(x)\\, y(x) = \\lambda\\, y(x) \\mbox{ in } ]0,1[\\,, \\mbox{ with } y(0)=y(1)=0\\,,$$divide the interval into at most $n$ connected components, and concludes that ``the lack of a published formal text with a rigorous proof \\dots is still distressing.''\\\\Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the analysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$ particle operator acting on Fermions.\\\\In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, we refine this strategy, and prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-11T08:20:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sturms theorem",
      "first eigenfunction",
      "arnold recounts gelfands strategy",
      "linear combination",
      "steklov institute math"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}