  The inverse scattering theory for many-body systems in quantum mechanics is
an important and difficult issue not only in physics---atomic physics,
molecular physics and nuclear physics---but also mathematics. The major purpose
in this paper is to establish a reconstruction procedure of two-body
interactions from scattering solutions for a Hartree-Fock equation. More
precisely, this paper gives a uniqueness theorem and proposes a new
reconstruction procedure of the short-range and two-body interactions from a
high-velocity limit of the scattering operator for the Hartree-Fock equation.
Moreover, it will be found that the high-velocity limit of the scattering
operator is equal to a small-amplitude limit of it. The main ingredients of
mathematical analysis in this paper are based on the theory of integral
equations of the first kind and a Strichartz type estimates on a solution to
the free Schr\"{o}dinger equation.
