### Abstract

The operator H = d^{4}/dt^{4} + d/dtpd/dt +q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy is found; 2) the spectrum of H at high energy has multiplicity two; 3) if p belongs to a certain class, then for any q the spectrum of H has infinitely many gaps, and all branch points of the Lyapunov function, except for a finite number of them, are real and negative; 4) if q = 0 and p → 0, then at the beginning of the spectrum there is a small spectral band of multiplicity 4, and its asymptotic behavior is found; the remaining spectrum has multiplicity 2.

Original language | English |
---|---|

Pages (from-to) | 703-736 |

Number of pages | 34 |

Journal | St. Petersburg Mathematical Journal |

Volume | 22 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Dec 2011 |

### Fingerprint

### Scopus subject areas

- Analysis
- Algebra and Number Theory
- Applied Mathematics

### Cite this

}

*St. Petersburg Mathematical Journal*, vol. 22, no. 5, pp. 703-736. https://doi.org/10.1090/S1061-0022-2011-01164-1

**Spectral estimates for a periodic fourth-order operator.** / Badanin, A. V.; Korotyaev, E. L.

Research output › › peer-review

TY - JOUR

T1 - Spectral estimates for a periodic fourth-order operator

AU - Badanin, A. V.

AU - Korotyaev, E. L.

PY - 2011/12/1

Y1 - 2011/12/1

N2 - The operator H = d4/dt4 + d/dtpd/dt +q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy is found; 2) the spectrum of H at high energy has multiplicity two; 3) if p belongs to a certain class, then for any q the spectrum of H has infinitely many gaps, and all branch points of the Lyapunov function, except for a finite number of them, are real and negative; 4) if q = 0 and p → 0, then at the beginning of the spectrum there is a small spectral band of multiplicity 4, and its asymptotic behavior is found; the remaining spectrum has multiplicity 2.

AB - The operator H = d4/dt4 + d/dtpd/dt +q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy is found; 2) the spectrum of H at high energy has multiplicity two; 3) if p belongs to a certain class, then for any q the spectrum of H has infinitely many gaps, and all branch points of the Lyapunov function, except for a finite number of them, are real and negative; 4) if q = 0 and p → 0, then at the beginning of the spectrum there is a small spectral band of multiplicity 4, and its asymptotic behavior is found; the remaining spectrum has multiplicity 2.

KW - Periodic differential operator

KW - Spectral asymptotics

KW - Spectral bands

UR - http://www.scopus.com/inward/record.url?scp=84858263427&partnerID=8YFLogxK

U2 - 10.1090/S1061-0022-2011-01164-1

DO - 10.1090/S1061-0022-2011-01164-1

M3 - Article

VL - 22

SP - 703

EP - 736

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 5

ER -