On an Inverse Scattering Problem for the Quadratic Pencil of the Sturm-Liouville Operator With a Piece-Wise Constant Coefficient

In this work, direct and inverse scattering problems on the real axis for the quadratic pencil of the Sturm-Liouville operator with piece-wise constant coefficient are studied. The new integral representations for solutions are given, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proven, the necessary and sufficient conditions for recovering of the potentials are examined.