Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy

TL;DR

This paper proves the density of averaged scattering solutions for Schrödinger equations with short-range electromagnetic potentials and demonstrates that the scattering matrix at a fixed energy uniquely determines these potentials and fields.

Contribution

It establishes the density of averaged solutions and proves uniqueness of electromagnetic potentials from fixed-energy scattering data, extending previous asymptotic results.

Findings

Averaged scattering solutions are dense in $L^2(K)$.

Scattering matrix at fixed energy uniquely determines potentials.

Results extend to potentials with asymptotic sums of homogeneous terms.

Abstract

We prove that the averaged scattering solutions to the Schr\"odinger equation with short-range electromagnetic potentials $(V,A)$ where $V(x)=O(|x|^{-\rho}), A(x)= O(|x|^{-\rho}), |x| \to \infty, \rho >1,$ are dense in the set of all solutions to the Schr\"odinger equation that are in $L^2(K)$ where $K$ is any connected bounded open set in $\ere^n,n\geq 2,$ with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials $(V_1,A_1)$ and $(V_2,A_2)$ in $\ere^n, n\geq 3,$ have the same scattering matrix at a fixed positive energy and if the electric potentials $V_j$ and the magnetic fields $ F_j:={\rm curl} A_j, j=1,2,$ coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field.