5. Inverse Scattering

In this chapter we analyze the inverse scattering problem of recovery of the corresponding input data set D in the Faddeev class from a scattering data set S in the Marchenko class. We discuss the nonuniqueness arising in the inverse scattering problem if the scattering matrix is defined one way with the Dirichlet boundary condition and in a different way with a non-Dirichlet boundary condition, as usually done in the standard literature. We present the solution to the inverse scattering problem by analyzing the solvability of the Marchenko integral equation and the derivative Marchenko integral equation. We give a proof of the main characterization result presented in Theorem 2.​6.​1. By establishing the equivalents among various characterization properties, we provide the proofs for various alternate characterization results presented in Sect. 2.​7. We consider the inverse scattering problem from a given scattering matrix without having the bound-state information. From the given scattering matrix alone we show how to construct a scattering data set belonging to the Marchenko class so that the constructed scattering data set can be used as input into a properly posed inverse scattering problem. We present a proof of the two equivalent characterization results stated in Theorems 2.​7.​9 and 2.​7.​10, where the characterization involves the use of Levinson’s theorem. Next, we introduce and present a proof of Parseval’s equality expressing the completeness of the set consisting of the physical solution and the bound-state solutions in the study of the matrix Schrödinger operator on the half line with the general self-adjoint boundary condition. We present the generalized Fourier map and establish its properties. We then prove the characterization result stated in Theorem 2.​8.​1 involving the use of the generalized Fourier map. Moreover, we consider the characterization of the scattering data when the potentials are restricted to the class \(L^1_p(\mathbf R^+)\) for p > 1 instead of only p = 1 in the Faddeev class. Finally, we formulate the characterization of the scattering data in the special case of the purely Dirichlet boundary condition, which allows us to make a comparison and contrast with the characterization result of Agranovich and Marchenko.