{
  "id": "0808.0588",
  "version": "v1",
  "published": "2008-08-05T09:15:28.000Z",
  "updated": "2008-08-05T09:15:28.000Z",
  "title": "Spectral estimates for periodic fourth order operators",
  "authors": [
    "Andrey Badanin",
    "Evgeny Korotyaev"
  ],
  "comment": "26 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the operator $H={d^4dt^4}+{ddt}p{ddt}+q$ with 1-periodic coefficients on the real line. The spectrum of $H$ is absolutely continuous and consists of intervals separated by gaps. We describe the spectrum of this operator in terms of the Lyapunov function, which is analytic on a two-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. We describe the spectrum of $H$ in terms of periodic, antiperiodic eigenvalues, and so-called resonances. We prove that 1) the spectrum of $H$ at high energy has multiplicity two, 2) the asymptotics of the periodic, antiperiodic eigenvalues and of the resonances are determined at high energy, 3) for some specific $p$ the spectrum of $H$ has an infinite number of gaps, 4) the spectrum of $H$ has small spectral band (near the beginner of the spectrum) with multiplicity 4 and its asymptotics are determined as $p\\to 0, q=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-08-05T09:15:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34L20",
      "34L40"
    ],
    "keywords": [
      "periodic fourth order operators",
      "spectral estimates",
      "lyapunov function",
      "high energy",
      "antiperiodic eigenvalues"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.0588B"
    }
  }
}