Spectral theory for periodic vector NLS equations

We consider a first order operator with a periodic 3x3 matrix potential on the real line. This operator appears in the problem of the periodic vector NLS equation. The spectrum of the operator covers the real line, it is union of the spectral bands of multiplicity 3, separated by spectral intervals of multiplicity 1. The main results of this work are the following: The Lyapunov function on the corresponding 2 or 3-sheeted Riemann surface is described. Necessary and sufficient conditions are given when the Riemann surface is 2-sheeted. The asymptotics of 2-periodic eigenvalues are determined. One constructs an entire function, which is positive on the spectrum of multiplicity 3 and is negative on its gaps. The Borg type results about inverse problems are solved.