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Spectral estimates for a periodic fourth-order operator

Authors: A. V. Badanin and E. L. Korotyaev
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 22 (2010), nomer 5.
Journal: St. Petersburg Math. J. 22 (2011), 703-736
MSC (2010): Primary 34L15; Secondary 34L40
Published electronically: June 27, 2011
MathSciNet review: 2828825
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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\to 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.

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Additional Information

A. V. Badanin
Affiliation: Northern (Arctic) Federal University, Northern Dvina Quay 17, Archangelsk, Russia

E. L. Korotyaev
Affiliation: St. Petersburg State University, Ul′yanovskaya 3, Petrodvoretz, St. Petersburg 198504, Russia, and Leningrad State University named after A. S. Pushkin, Peterburgskoe Shosse 10, Pushkin, St. Petersburg 196605, Russia

Keywords: Periodic differential operator, spectral bands, spectral asymptotics
Received by editor(s): March 11, 2009
Published electronically: June 27, 2011
Additional Notes: The work of A. V. Badanin was partially supported by a joint grant of DAAD (the program “Mikhail Lomonosov-2007”) and by a grant of the Ministry of Education of RF (the program “Development of the scientific potential of the higher school in 2006–2008”). Part of the paper was written at the Mathematical Institute of Potsdam University, Germany (September–December, 2007). Part of the work was done by E. L. Korotyaev at the Institute of Mathematics of Tsukkuba University, Japan (March, 2010) and at École Polytéchnique, France (April–July, 2010). The authors are grateful to these institutions for their hospitality.
Article copyright: © Copyright 2011 American Mathematical Society