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Licensed Unlicensed Requires Authentication Published by De Gruyter November 7, 2005

Inverse problem and estimates for periodic Zakharov-Shabat systems

  • Evgeny Korotyaev
From the journal


Consider the Zakharov-Shabat operator TZS on L2(ℝ) ⊕ L2(ℝ) with real periodic vector potential q = (q1, q2) ∈ H = L2(

) ⊕ L2(
). The spectrum of TZS  is absolutely continuous and consists of intervals separated by gaps (zn , zn+ ), n ∈ ℤ. From the Dirichlet eigenvalues mn n ∈ 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 → ℓ2 ⊕ ℓ2. Using nonlinear functional analysis in Hilbert spaces, we show that this mapping is a real analytic isomorphism. Our proof relies on new identities and a priori estimates contained in the second part of the paper. In order to get these estimates we obtain new results in conformal mapping theory.

Published Online: 2005-11-07
Published in Print: 2005-06-27

© Walter de Gruyter

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