The Factorization Method to Solve a Class of Inverse Potential Scattering Problems for Schrödinger Equations

Abstract

This paper is concerned with the inverse scattering problems for Schrödinger equations with compactly supported potentials. For purpose of reconstructing the support of the potential, we derive a factorization of the scattering amplitude operator $A$ and prove that the ranges of $(A^∗A) ^{1/4}$ and $G$ which maps more general incident fields than plane waves into the scattering amplitude coincide. As an application we characterize the support of the potential using only the spectral data of the operator $A$.