Reconstruction of the optical potential in the inverse quantum   scattering. Application to the relativistic inelastic NN scattering
  The numerical algorithm of the inverse quantum scattering is developed. This algorithm is based on the Marchenko theory, and includes three steps. The first one is the algebraic Pade approximation of the unitary S-matrix, what is realized by solving a system of linear equations. Second step is the exact solution of the Marchenko equation. The used approximant reduces it to another system of linear equations. At this step we get the real-valued potential. It is shown numerically that the developed algorithm is able not only to generate the given S-matrix dependence, but converges to the initial potential. At third step we construct the optical complex-valued potential which gives the needed S-matrix. It is shown that the modern phase shift analysis data allow to construct the nucleon-nucleon optical potentials of two kinds. These potentials describe the deuteron properties and the phase shift analysis data up to 3 GeV and they have different behavior at short distances. One is a repulsive core potential and another is a Moscow attractive potential with forbidden states. 
N.A. Khokhlov, V.A. Knyr