RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Algebra i Analiz: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (568 kB)
References: PDF file   HTML file

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

Bibliographic databases:

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
\Bibitem{BadKor10} \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 \mathnet{http://mi.mathnet.ru/aa1203} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2828825} \zmath{https://zbmath.org/?q=an:1230.34071} \transl \jour St. Petersburg Math. J. \yr 2011 \vol 22 \issue 5 \pages 703--736 \crossref{https://doi.org/10.1090/S1061-0022-2011-01164-1} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000295022600001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84858263427} 

• http://mi.mathnet.ru/eng/aa1203
• http://mi.mathnet.ru/eng/aa/v22/i5/p1

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Badanin A., Korotyaev E., “Spectral asymptotics for the third order operator with periodic coefficients”, J. Differential Equations, 253:11 (2012), 3113–3146
3. A. Badanin, E. Korotyaev, “Third order operator with periodic coefficients on the real axis”, St. Petersburg Math. J., 25:5 (2014), 713–734
4. Badanin A., Korotyaev E., “Trace Formula for Fourth Order Operators on the Circle”, Dyn. Partial Differ. Equ., 10:4 (2013), 343–352
5. Badanin A., Korotyaev E., “Sharp Eigenvalue Asymptotics For Fourth Order Operators on the Circle”, J. Math. Anal. Appl., 417:2 (2014), 804–818
6. D. M. Polyakov, “Spectral analysis of a fourth-order nonselfadjoint operator with nonsmooth coefficients”, Siberian Math. J., 56:1 (2015), 138–154
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
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
9. Badanin A., Korotyaev E., “Trace Formulas For Fourth Order Operators on Unit Interval, II”, Dyn. Partial Differ. Equ., 12:3 (2015), 217–239
10. Badanin A., Korotyaev E., “Inverse Problems and Sharp Eigenvalue Asymptotics For Euler-Bernoulli Operators”, Inverse Probl., 31:5 (2015), 055004
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
12. D. M. Polyakov, “O spektralnykh kharakteristikakh nesamosopryazhennogo operatora chetvertogo poryadka s matrichnymi koeffitsientami”, Matem. zametki, 105:4 (2019), 637–642
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
•  Number of views: This page: 415 Full text: 107 References: 73 First page: 15