Direct and Inverse Asymptotic Scattering Problems for Dirac–Krein Systems

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).