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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 language | English |
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Pages (from-to) | 703-736 |
Number of pages | 34 |
Journal | St. Petersburg Mathematical Journal |
Volume | 22 |
Issue number | 5 |
DOIs | |
State | Published - 1 Dec 2011 |
ID: 5288278