电磁场中多维Newton方程的高能逆散射
On inverse scattering at high energies for the multidimensional Newton equation in electromagnetic field
摘要：
考虑静态电磁场$$\ddot x = F(x,\dot x), F(x,\dot x)=-\nabla V(x)+B(x)\dot x, \dot x={dx\over dt}, x\in C^2(\R,\R^n),\eqno{(*)}$$中的多维（非相对论）牛顿方程，其中$V \in C^2(\R^n,\R),$ $B(x)$是$n\times n$实反对称矩阵，其元素为$B_{i,k}(x)$，$B_{i,k}\in C^1(\R^n,\R)$（且$B$满足闭合条件），且$|\pa^{j_1}_xV(x)| +|\pa^{j_2}_xB_{i,k}(x)| \le β_{|j_1|} (1+|x|)^{-(α+|j_1|)}$对于$x\in \R^n,$ $1\le|j_1|\le 2,$ $0\le|j_2|\le 1$，$|j_2|=|j_1|-1$，$i,k=1... n$和某些$α> 1$ .我们估计和渐近散射的解决方案和散射数据的方程$(*)$的情况下，小角度散射。我们表明，在高能量的速度值的散射算子的分量唯一地确定X射线变换$P\nabla V$和$PB_{i,k}$（足够丰富的一组直线）。应用X射线变换$P$的反演结果，我们得到，对于$n\ge 2$，在高能量下的散射算子的速度值分量唯一地确定了$(\nabla V,B)$。我们还考虑从我们的高能量渐近发现的散射算子的配置值组件的恢复$(\nabla V,B)$的问题。本工作的结果是通过发展[R. Novikov，1999]的$(*)$与$B\equiv 0$和[Jollivet，2005]的相对论版本的$(*)$。我们强调，有一个有趣的差异，在渐近散射的解决方案和散射数据$(*)$一方面，其相对论版本的另一方面。
英文摘要：
We consider the multidimensional (nonrelativistic) Newton equation in a static electromagnetic field $$\ddot x = F(x,\dot x), F(x,\dot x)=-\nabla V(x)+B(x)\dot x, \dot x={dx\over dt}, x\in C^2(\R,\R^n),\eqno{(*)}$$ where $V \in C^2(\R^n,\R),$ $B(x)$ is the $n\times n$ real antisymmetric matrix with elements $B_{i,k}(x)$, $B_{i,k}\in C^1(\R^n,\R)$ (and $B$ satisfies the closure condition), and $|\pa^{j_1}_xV(x)| +|\pa^{j_2}_xB_{i,k}(x)| \le β_{|j_1|} (1+|x|)^{-(α+|j_1|)}$ for $x\in \R^n,$ $1\le|j_1|\le 2,$ $0\le|j_2|\le 1$, $|j_2|=|j_1|-1$, $i,k=1... n$ and some $α> 1$. We give estimates and asymptotics for scattering solutions and scattering data for the equation $(*)$ for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms $P\nabla V$ and $PB_{i,k}$ (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform $P$ we obtain that for $n\ge 2$ the velocity valued component of the scattering operator at high energies uniquely determines $(\nabla V,B)$. We also consider the problem of recovering $(\nabla V,B)$ from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of [R. Novikov, 1999] for $(*)$ with $B\equiv 0$ and of [Jollivet, 2005] for the relativistic version of $(*)$. We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for $(*)$ on the one hand and for its relativistic version on the other.
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