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.
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