固定能量下平均散射解和逆散射的完备性
Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy
摘要：
证明了具有短程电磁势$(V,A)$的薛定谔方程的平均散射解在$L^2(K)$中薛定谔方程的所有解的集合中是稠密的，其中$(V,A)$是$\ere^n,n\geq 2,$中具有光滑边界的连通有界开集.我们利用这个结果证明，如果$\ere^n, n\geq 3,$中的两个短程电磁势$(V_1,A_1)$和$(V_2,A_2)$在固定的正能量下具有相同的散射矩阵，并且如果电势$V_j$和磁场$ F_j:={\rm curl} A_j, j=1,2,$在某个球外重合，则它们必然处处重合。在以前的论文韦德和Yafaev的情况下，电势和磁场的渐近总和齐次项在无穷远进行了研究。证明了在正能量一定的情况下，所有这些项都可以由散射振幅正向的奇点唯一重构。本文的新的唯一性结果和Weder和Yafaev的结果的组合意味着，在一个固定的正能量的散射矩阵唯一地确定电势和磁场，是在无穷远的齐次项的有限和，或更一般地说，是齐次项的渐近和，实际上分别收敛到电势和磁场。
英文摘要：
We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials $(V,A)$ where $V(x)=O(|x|^{-ρ}), A(x)= O(|x|^{-ρ}), |x| \to \infty, ρ>1,$ are dense in the set of all solutions to the Schrödinger 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.
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