Inverse Scattering on the Line—an Overview

This chapter describes the inverse scattering on the line. Historically interest in inverse scattering theory began with the frustration of theoretical physicists in trying to construct a rigorous quantum field theory. This “existential” theory took the name S-matrix theory. Using a technique which the chapter refers to as Darboux transformations, or Backlund transformations, by example that h was not determined by S(h). Thus S-matrix theory was indeed weaker than the Hamiltonian theory. Four major technical problems arise, the first in the direct scattering theory, the remaining three in the inverse theory.