Inverse Problem and Estimates for Periodic Zakharov-Shabat systems

Consider the Zakharov-Shabat (or Dirac) operator Tzs on L 2(R) L2(R) with real periodic vector potential q = (q1; q2) 2 H = L 2(T) L2(T). The spectrum of Tzs is absolutely continuous and consists of intervals separated by gaps (z n ; z + n ); n 2 Z. >From the Dirichlet eigenvalues mn; n 2 Z of the Zakharov-Shabat equation with Dirichlet boundary conditions at 0; 1, the center of the gap and the square of the gap length we construct the gap length mapping g : H ! `2 `2. Using nonlinear functional analysis in Hilbert spaces, we show that this mapping is a real analytic isomorphism. Our proof relies on new identities and estimates contained in the second part of the our paper.