Estimates, asymptotics and trace formulas for periodic vector NLS equations, II

We consider a first order operator with a smooth periodic 3x3 matrix potential on the real line. It is the Lax operator for the periodic vector NLS equation. Its spectrum covers the real line and it is union of the spectral bands of multiplicity 3, separated by intervals (gaps) of multiplicity 1. We prove and describe the following: \\ $\cdot$ The geometry of the Riemann surface and its branch points. \\ $\cdot$ The asymptotics of branch points are determined and they are real at high energy. \\ $\cdot$ Trace formulas for integral of motions, including the Hamiltonian of the NLS equation. \\ $\cdot$ Estimates of the Hamiltonian in terms of gap lengths. The proof is based on the analysis of averaged quasi-momentum as a conformal mapping of the upper half plane on the domain on the upper half plane and on the asymptotics of the monodromy matrix and multipliers at high energy.