Lyapunov functions for periodic matrix-valued Jacobi operators

Abstract

We consider periodic matrix-valued Jacobi operators. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. We show that this function has (real or complex) branch points, which we call resonances. We prove that there exist two types of gaps: i) stable gaps, i.e., the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, i.e., the endpoints are resonances (real branch points). We show that some spectral data determine the spectrum (counting multiplicity) of the Jacobi operator.