Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 457

Inverse Problem and Estimates for Periodic Zakharov-Shabat systems

Evgeni Korotyaev

Consider the Zakharov-Shabat (or Dirac) operator $T_{zs}$ on $L^2(R )os L^2(R )$ with real periodic vector potential $q=(q_1,q_2)in H=L^2(T )os L^2(T )$. The spectrum of $T_{zs}$ is absolutely continuous and consists of intervals separated by gaps $(z_n^-,z_n^+), ninZ$. >From the Dirichlet eigenvalues $m_n, n in Z$ 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 $g: H o ell^2osell^2$. 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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