Some Results on Inverse Scattering

TL;DR

This paper reviews various results in inverse scattering theory, covering topics from property C to material design, highlighting advances in stable inversion, incomplete data, and resonance analysis.

Contribution

The paper compiles and discusses the author's significant findings across multiple aspects of inverse scattering, including new methods and theoretical insights.

Findings

Stable inversion of fixed-energy 3D scattering data achieved

Inverse scattering with incomplete data addressed

Resonance properties and their perturbations analyzed

Abstract

A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: 1) Property $C$ and applications, 2) Stable inversion of fixed-energy 3D scattering data and its error estimate, 3) Inverse scattering with ''incomplete`` data, 4) Inverse scattering for inhomogeneous Schr\"odinger equation, 5) Krein's inverse scattering method, 6) Invertibility of the steps in Gel'fand-Levitan, Marchenko, and Krein inversion methods, 7) The Newton-Sabatier and Cox-Thompson procedures are not inversion methods, 8) Resonances: existence, location, perturbation theory, 9) Born inversion as an ill-posed problem, 10) Inverse obstacle scattering with fixed-frequency data, 11) Inverse scattering with data at a fixed energy and a fixed incident direction, 12) Creating materials with a desired refraction coefficient and wave-focusing properties.