Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Abstract

A direct and inverse scattering theory on the full line is developed for a class of firstorder selfadjoint 2n • 2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.

Nonautonomous Exponential Dichotomy

“…The following proposition can be proved as its counterpart in [6,17,11]. Contrary to the practice in [6,17,11] …”

“…Using the identities 6) we easily see that the inverses Ψ(λ , x) −1 and Φ(λ , x) −1 of the Jost matrices are the unique solutions of the dual matrix Zakharov-Shabat system (2.3) satisfyinǧ…”

“…Throughout we drop the notational system of [6,11,17]. Instead, we compromise between using Jost solutions as column vectors (as in [2,3,15,18]) and Jost solutions as square matrices (as in [6,11,17]), since either notation is advantageous in different circumstances.…”

“…Instead, we compromise between using Jost solutions as column vectors (as in [2,3,15,18]) and Jost solutions as square matrices (as in [6,11,17]), since either notation is advantageous in different circumstances. We distinguish between different related entities by using overlines for those functions analytic on the upper half complex plane.…”

Wave operators for the matrix Zakharov–Shabat system

“…13 Many mathematicians, physicists, and engineers have contributed to the development of a comprehensive theory of matrix Zakharov-Shabat systems on the line ͑e.g., Refs. 2,5,8,21,23, and 31͒. For a comprehensive theory of the closely related canonical systems on finite intervals or on the half-line, we refer to Ref.…”