Inverse problem and estimates for periodic Zakharov-Shabat systems

Consider the Zakharov-Shabat (or Dirac) operator T zs on L 2 (R) \\Phi L 2 (R) with real periodic vector potential q = (q 1 ; q 2 ) 2 H = L 2 (T) \\Phi L 2 (T). The spectrum of T zs is absolutely continuous and consists of intervals separated by gaps (z \\Gamma n ; z + n ); n 2 Z. ?From the Dirichlet eigenvalues m n ; 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 \\Phi` 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.

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