On increasing stability in the two dimensional inverse source scattering problem with many frequencies

TL;DR

This paper investigates how increasing the frequency range improves the stability and uniqueness in reconstructing a source term in a 2D inverse scattering problem governed by the Helmholtz equation, using advanced mathematical techniques.

Contribution

It provides new sharp stability and uniqueness estimates for the inverse source problem with larger frequency intervals, enhancing understanding of the problem's stability properties.

Findings

Established sharper stability bounds for larger frequency ranges.

Proved uniqueness of the inverse source solution under new conditions.

Extended the analytic continuation bounds for scattering solutions.

Abstract

In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain $\Omega$ with sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and exact observability for wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate with larger wave numbers interval.