Flat Connections and Scattering Theory on the Line

Abstract

The scattering theory of $n \times n$ first-order systems on the line is formulated in terms of a flat connection on a vector bundle over $R \times P_1 (C)$. The relation of the scattering data to a set of transition matrices is discussed. The scattering transform is obtained as a sectionally holomorphic gauge transformation. The winding number constraints of Bar-Yaacov [“Analytic properties of scattering and inverse scattering for first order systems,” Ph.D. thesis, Yale University, New Haven, CT, 1985] and Beals and Coifman [Comm. Pure Appl. Math., 37 (1984), pp. 39–90] on the scattering data are shown to be a necessary condition for the diagram for the transition matrices to commute. The transition matrices are reconstructed from scattering data with multiple poles by solving a sequence of triangular factorization and Riemann–Hilbert problems. The inverse scattering problem is formulated as a system of singular integral equations and reduced to a Fredholm system by Plemelj’s method for a special class....