Numerical results for Saito’s uniqueness theorem in inverse scattering theory

Abstract

We consider an inverse scattering problem for the Schrödinger operator in two dimensions. The aim of this work is to discuss some first numerical results on Saito’s formula. Saito’s formula is an explicit integral formula, which at the high-frequency limit gives a uniqueness result for the inverse scattering problem. The numeric approach is quite straight-forward: we take a large enough fixed wave number and evaluate the integrals in Saito’s formula numerically. The potential function can then be recovered from the blurry measurements by using the fast Fourier transform and a high-pass filter. We also discuss in detail how the synthetic data is generated via a matrix-based approach. Several numerical examples are shown to demonstrate the results.