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