A distributional Gelfand–Levitan–Marchenko equation for the Helmholtz scattering problem on the line

We study an inverse scattering problem for the Helmholtz equation on whole line. The goal of this paper is to obtain a Gelfand–Levitan–Marchenko (GLM)-type Jost solution that corresponds 1D differential operator. assume simplicity refraction index compact support. Using asymptotic behavior solutions with respect wave-number, we derive generalized Povzner–Levitan representation in space tempered distributions. Then, apply Fourier transform relation describes and GLM equation. Finally, discuss possible application new medium problem.