2×2-Matrix Riemann-Hilbert Transform and Its Connection to the Continuous Scattering Data

Abstract

A basic 2×2-matrix Riemann-Hilbert problem is studied and its complete connection to the associated eigenvalue problem is made clear. A potential formula independent on the spectral parameter λ is derived by considering a transformation contributed from the continuous data. As a by-product we get a mapping from the variation of scattering data to the one of potentials.