Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy
Title:
Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy
Author:
Weder, Ricardo
Appeared in:
Communications in partial differential equations
Paging:
Volume 32 (2007) nr. 5 pages 675-691
Year:
2007-05
Contents:
We prove that the averaged scattering solutions to the Schrodinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|-ρ), A(x) = O(|x|-ρ), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrodinger equation that are in L2(K) where K is any connected bounded open set in n,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V1, A1) and (V2, A2) in n, n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials Vj and the magnetic fields Fj: = curl Aj, 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.
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