长程电磁场中多维相对论性牛顿方程的高能逆散射
On inverse scattering at high energies for the multidimensional relativistic Newton equation in a long range electromagnetic field
摘要：
我们定义了相对论性牛顿方程在电场$-\nabla V\in C^1(\R^n,\R^n)$、$n\ge 2$和磁场$B\in C^1(\R^n,A_n(\R))$中的散射数据，对于某些$α\in (0,1]$，这些磁场在无穷远衰减，就像$r^{-α-1}$，其中$A_n(\R)$是$n\times n$反对称矩阵的空间。我们提供的散射解决方案和散射数据上的估计，我们证明，特别是在高能量的散射数据唯一确定的短程部分的$(\nabla V,B)$的知识的长期尾巴的$(\nabla V,B)$。在散射数据的固定能量的玻恩近似也被认为是。然后，我们改变了散射数据的定义，以研究其在其他渐近制度的行为。这项工作推广了[Jollivet，2007]，其中考虑了短程电磁场。
英文摘要：
We define scattering data for the relativistic Newton equation in an electric field $-\nabla V\in C^1(\R^n,\R^n)$, $n\ge 2$, and in a magnetic field $B\in C^1(\R^n,A_n(\R))$ that decay at infinity like $r^{-α-1}$ for some $α\in (0,1]$, where $A_n(\R)$ is the space of $n\times n$ antisymmetric matrices. We provide estimates on the scattering solutions and on the scattering data and we prove, in particular, that the scattering data at high energies uniquely determine the short range part of $(\nabla V,B)$ up to the knowledge of the long range tail of $(\nabla V,B)$. The Born approximation at fixed energy of the scattering data is also considered. We then change the definition of the scattering data to study their behavior in other asymptotic regimes. This work generalizes [Jollivet, 2007] where a short range electromagnetic field was considered.
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