Increasing stability estimates for the inverse potential scattering problems

TL;DR

This paper demonstrates that the stability of inverse potential scattering problems improves with increasing frequency, using a novel analytic continuation method and scattering theory in 2D and 3D.

Contribution

It introduces a new approach for increasing stability estimates in inverse scattering problems, applicable to both electric and magnetic potentials in multiple dimensions.

Findings

Stability improves with higher frequencies.

New method based on analytic continuation and scattering theory.

Applicable to both electric and magnetic potentials.

Abstract

This paper is mainly concerned with the inverse scattering problem of determining the unknown potential for the classical Schr\"odinger equation in two and three dimensions. We establish the increasing stability of the inverse scattering problem from either multi-frequency near-field data or multi-frequency far-field pattern. The stability estimate consists of the Lipschitz type data discrepancy and the logarithmic high frequency tail of the potential function, where the latter decreases as the upper bound of the frequency increases. A novel method is proposed for the proof, which is based on choosing appropriate incident plane waves and an application of the quantitative analytic continuation. A key ingredient in the analysis is employing scattering theory to obtain an analytic region and resolvent estimates in this region for the resolvent in two and three dimensions. We further apply this method to study the inverse scattering problem of determining both the magnetic potential and electric potential for the three-dimensional magnetic Schr\"odinger equation.