Inverse problem and estimates for periodic Zakharov-Shabat systems

Evgeny Korotyaev

doi:10.1515/crll.2005.2005.583.87

Abstract

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

) ⊕ L²(

). The spectrum of T_ZS is absolutely continuous and consists of intervals separated by gaps (z_n⁻ , z_n⁺ ), n ∈ ℤ. From the Dirichlet eigenvalues m_n , 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.

Inverse problem and estimates for periodic Zakharov-Shabat systems

Abstract

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