The Lyapunov function for Schrödinger operators with a periodic 2×2 matrix potential

We consider the Schrödinger operator on the real line with a 2×2 matrix-valued 1-periodic potential. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define a Lyapunov function which is analytic on a two-sheeted Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove the existence of real as well as non-real resonances for specific potentials. We determine the asymptotics of the periodic and the anti-periodic spectrum and of the resonances at high energy. We show that there exist two type of gaps: (1) stable gaps, where the endpoints are the periodic and the anti-periodic eigenvalues, (2) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We also show that periodic and anti-periodic spectrum together determine the spectrum of the matrix Hill operator.  2005 Elsevier Inc. All rights reserved.