Uniqueness criteria for inverse scattering problem in terms of transmission matrix in boundary condition for a first order system of ordinary differential equations

The inverse scattering problem (ISP) involves the recovery of the matrix coefficient of a first-order system on the half-line from its scattering matrix. Specifically, when the matrix coefficient exhibits a triangular structure, the system possesses a Volterra-type integral transformation operator at infinity. This transformation operator facilitates the determination of the scattering matrix on the half-line through matrix Riemann-Hilbert factorization. Solving the ISP on the half-line entails reducing it to an ISP on the whole line for the considered system. This reduction involves extending the coefficients to the whole line by zero. The uniqueness criteria in terms of transmission matrix in boundary condition for the ISP is also established.