Geodesic
Direct and Inverse Asymptotic Scattering Problems for Dirac--Krein Systems
Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 3, pp. 17-33

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The asymptotic scattering matrix $s_{\varepsilon}(\lambda)$ for a Dirac–Krein system with signature matrix $J=\operatorname{diag}\{I_p,-I_p\}$, integrable potential, and the boundary condition $u_1(0,\lambda)=u_2(0,\lambda)\varepsilon(\lambda)$ with a coefficient $\varepsilon(\lambda)$ that belongs to the Schur class of holomorphic contractive $p\times p$ matrix-valued functions in the open upper half-plane is defined. The inverse asymptotic scattering problem for a given $s_{\varepsilon}$ is analyzed by Krein's method. Earlier studies by Krein and others focused on the case in which $\varepsilon=I_p$ (or a constant unitary matrix).
Keywords: inverse problem, asymptotic scattering matrix, matrix-valued function, Hilbert space, linear bounded operator, Nehari problem, Schur problem, Hankel operator, Toeplitz operator, Wiener class. 
@article{FAA_2007_41_3_a1,
     author = {D. Z. Arov and H. Dym},
     title = {Direct and {Inverse} {Asymptotic} {Scattering} {Problems} for {Dirac--Krein} {Systems}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {17--33},
     publisher = {mathdoc},
     volume = {41},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2007_41_3_a1/}
}
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D. Z. Arov; H. Dym. Direct and Inverse Asymptotic Scattering Problems for Dirac--Krein Systems. Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 3, pp. 17-33. http://geodesic.mathdoc.fr/item/FAA_2007_41_3_a1/