Even order periodic operators on the real line

@article{Badanin2010EvenOP,
  title={Even order periodic operators on the real line},
  author={Andrey Badanin and Evgeny L. Korotyaev},
  journal={arXiv: Mathematical Physics},
  year={2010}
}
We consider $2p\ge 4$ order differential operator on the real line with a periodic coefficients. The spectrum of this operator is absolutely continuous and is a union of spectral bands separated by gaps. We define the Lyapunov function, which is analytic on a p-sheeted Riemann surface. The Lyapunov function has real or complex branch points. We prove the following results: (1) The spectrum at high energy has multiplicity two. (2) Endpoints of all gaps are periodic (or anti-periodic) eigenvalues… 

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