Abstract

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.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.