Even Order Periodic Operators on the Real Line

We consider a 2 p ≥4 order differential operator on the real line with periodic coefficients. The spectrum of this operator is absolutely continuous and consists of 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 2. (2) The endpoints of all gaps are either periodic (or anti-periodic) eigenvalues or the real branch points. (3) The spectrum of the operator has an infinite number of open gaps and there is only a finite number of nonreal branch points in the generic case. (4) The high-energy asymptotics of the periodic and anti-periodic eigenvalues and of the branch points are determined.