Spectral estimates for a periodic fourth-order operator

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.