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Algebra i Analiz, 2010, Volume 22, Issue 5, Pages 1–48 (Mi aa1203)  

This article is cited in 13 scientific papers (total in 13 papers)

Research Papers

Spectral estimates for a periodic fourth-order operator

A. V. Badanina, E. L. Korotyaevb

a Arkhangelsk State Technical University, Arkhangelsk, Russia
b School of Mathematics, Cardiff University, Cardiff, UK

Abstract: The operator $H=\frac{d^4}{dt^4}+\frac d{dt}p\frac d{dt}+q$ 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 $p\to0$, 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 bands, spectral asymptotics.

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English version:
St. Petersburg Mathematical Journal, 2011, 22:5, 703–736

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Received: 11.03.2009

Citation: A. V. Badanin, E. L. Korotyaev, “Spectral estimates for a periodic fourth-order operator”, Algebra i Analiz, 22:5 (2010), 1–48; St. Petersburg Math. J., 22:5 (2011), 703–736

Citation in format AMSBIB
\by A.~V.~Badanin, E.~L.~Korotyaev
\paper Spectral estimates for a~periodic fourth-order operator
\jour Algebra i Analiz
\yr 2010
\vol 22
\issue 5
\pages 1--48
\jour St. Petersburg Math. J.
\yr 2011
\vol 22
\issue 5
\pages 703--736

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    This publication is cited in the following articles:
    1. Badanin A., Korotyaev E.L., “Even order periodic operators on the real line”, Int. Math. Res. Not. IMRN, 2012:5 (2012), 1143–1194  crossref  mathscinet  zmath  isi  elib  scopus
    2. Badanin A., Korotyaev E., “Spectral asymptotics for the third order operator with periodic coefficients”, J. Differential Equations, 253:11 (2012), 3113–3146  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. A. Badanin, E. Korotyaev, “Third order operator with periodic coefficients on the real axis”, St. Petersburg Math. J., 25:5 (2014), 713–734  mathnet  crossref  mathscinet  zmath  isi  elib
    4. Badanin A., Korotyaev E., “Trace Formula for Fourth Order Operators on the Circle”, Dyn. Partial Differ. Equ., 10:4 (2013), 343–352  crossref  mathscinet  zmath  isi  elib  scopus
    5. Badanin A., Korotyaev E., “Sharp Eigenvalue Asymptotics For Fourth Order Operators on the Circle”, J. Math. Anal. Appl., 417:2 (2014), 804–818  crossref  mathscinet  zmath  isi  elib  scopus
    6. D. M. Polyakov, “Spectral analysis of a fourth-order nonselfadjoint operator with nonsmooth coefficients”, Siberian Math. J., 56:1 (2015), 138–154  mathnet  crossref  mathscinet  isi  elib  elib
    7. D. M. Polyakov, “On spectral properties of fourth order differential operator with periodic and semiperiodic boundary conditions”, Russian Math. (Iz. VUZ), 59:5 (2015), 64–68  mathnet  crossref
    8. D. M. Polyakov, “Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions”, St. Petersburg Math. J., 27:5 (2016), 789–811  mathnet  crossref  mathscinet  isi  elib
    9. Badanin A., Korotyaev E., “Trace Formulas For Fourth Order Operators on Unit Interval, II”, Dyn. Partial Differ. Equ., 12:3 (2015), 217–239  crossref  mathscinet  zmath  isi  elib  scopus
    10. Badanin A., Korotyaev E., “Inverse Problems and Sharp Eigenvalue Asymptotics For Euler-Bernoulli Operators”, Inverse Probl., 31:5 (2015), 055004  crossref  mathscinet  zmath  isi  elib  scopus
    11. Polyakov D.M., “Method of Similar Operators in Spectral Analysis of a Fourth-Order Nonself-Adjoint Operator”, Differ. Equ., 51:3 (2015), 421–425  crossref  mathscinet  zmath  isi  elib  scopus
    12. D. M. Polyakov, “O spektralnykh kharakteristikakh nesamosopryazhennogo operatora chetvertogo poryadka s matrichnymi koeffitsientami”, Matem. zametki, 105:4 (2019), 637–642  mathnet  crossref  elib
    13. S. I. Mitrokhin, “Asimptotika spektra periodicheskoi kraevoi zadachi dlya differentsialnogo operatora s summiruemym potentsialom”, Tr. IMM UrO RAN, 25, no. 1, 2019, 136–149  mathnet  crossref  elib
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