The inverse scattering problem at fixed energy based on the Marchenko   equation for an auxiliary Sturm-Liouville operator
  A new approach is proposed to the solution of the quantum mechanical inverse scattering problem at fixed energy. The method relates the fixed energy phase shifts to those arising in an auxiliary Sturm-Liouville problem via the interpolation theory of the Weyl-Titchmarsh m-function. Then a Marchenko equation is solved to obtain the potential. 
Tamas Palmai, Barnabas Apagyi
math-ph math.MP quant-ph