Direct and Inverse Scattering Problems for Sturm-Liouville Operator with Discontinuous Coefficient Under Discontinuity Conditions

This paper deals with the study of the direct and inverse problems of scattering theory for Sturm-Liouville operator with piecewise continuous coefficient under the discontinuity conditions at the interior point of the positive semi-axis. The scattering properties are examined and the eigenfunction expansion is obtained. The fundamental equation or Marchenko-type equation of the inverse scattering problem is derived and an algorithm of the reconstruction of the potential according to scattering data of this problem is given. Moreover, the continuity of scattering function of this problem is examined.