Spectral estimates for a periodic fourth-order operator

Research outputpeer-review

9 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)703-736
Number of pages34
JournalSt. Petersburg Mathematical Journal
Volume22
Issue number5
DOIs
Publication statusPublished - 1 Dec 2011

Fingerprint

Lyapunov functions
Fourth Order
Operator
Multiplicity
Branch Point
Estimate
Lyapunov Function
High Energy
Asymptotic Behavior
Periodic Coefficients
Absolutely Continuous
Real Line
Eigenvalue
Interval

Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics

Cite this

@article{f708f7da07234477ba301a47443b9368,
title = "Spectral estimates for a periodic fourth-order operator",
abstract = "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.",
keywords = "Periodic differential operator, Spectral asymptotics, Spectral bands",
author = "Badanin, {A. V.} and Korotyaev, {E. L.}",
year = "2011",
month = "12",
day = "1",
doi = "10.1090/S1061-0022-2011-01164-1",
language = "English",
volume = "22",
pages = "703--736",
journal = "St. Petersburg Mathematical Journal",
issn = "1061-0022",
publisher = "American Mathematical Society",
number = "5",

}

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

In: St. Petersburg Mathematical Journal, Vol. 22, No. 5, 01.12.2011, p. 703-736.

Research outputpeer-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 -