Inverse Scattering Problem for the Impulsive Schr¨odinger Equation with a Polynomial Spectral Dependence in the Potential

In the present work, under some differentiability conditions on the potential functions , we first reduce the inverse scattering problem (ISP) for the polynomial pencil of the Schr¨odinger equation to the corresponding ISP for the generalized matrix Schr¨odinger equation. Then ISP will be solved in analogy of the Marchenko method. We aim to establish an effective algorithm for uniquely reconstructing of the potential functions of the equation in that case when there is no a discrete spectrum.