标题： 高能和光滑性渐近展开式对于狄拉克方程的散射振幅及其应用 显示英文标题

标题： High-energy and smoothness asymptotic expansion of the scattering amplitude for the Dirac equation and applications

摘要： 我们得到了短程电磁势下狄拉克方程散射振幅对角奇点的显式公式。 利用这个展开式，我们从散射振幅的高能极限中唯一地重构出电势和磁场。 此外，假设电势和磁场是齐次项的渐近和，我们给出了从已知某个能量$E.$的散射振幅中唯一重构这些渐近的方法。 此外，我们证明了狄拉克方程的平均散射解的集合在所有属于$L^{2}\left( \Omega\right) ,$的解的集合中是稠密的，其中$\Omega$是$\mathbb{R}^{3}$中任意一个具有光滑边界的连通有界开集，我们还表明，如果我们知道$\mathbb{R}^{3}\setminus\Omega$的电势和磁场，那么对于某个能量$E$给出的散射振幅可以唯一地确定这些电势和磁场在$\mathbb{R}^{3}$中的任何地方。 结合这一唯一性结果与电势和磁场渐近性的重构过程，我们证明了对于某些$E$，已知的散射振幅唯一地确定了一个电势和一个磁 场，它们分别是齐次项的渐近和，分别收敛于电势和磁场。 此外，我们讨论了散射矩阵核的对称性，这些对称性来自于狄拉克算子的宇称、电荷共轭和时间反演变换。 显示英文摘要

摘要： We obtain an explicit formula for the diagonal singularities of the scattering amplitude for the Dirac equation with short-range electromagnetic potentials. Using this expansion we uniquely reconstruct an electric potential and magnetic field from the high-energy limit of the scattering amplitude. Moreover, supposing that the electric potential and magnetic field are asymptotic sums of homogeneous terms we give the unique reconstruction procedure for these asymptotics from the scattering amplitude, known for some energy $E.$ Furthermore, we prove that the set of the averaged scattering solutions to the Dirac equation is dense in the set of all solutions to the Dirac equation that are in $L^{2}\left( \Omega\right) ,$ where $\Omega$ is any connected bounded open set in $\mathbb{R}^{3}$ with smooth boundary, and we show that if we know an electric potential and a magnetic field for $\mathbb{R}^{3}\setminus\Omega$, then the scattering amplitude, given for some energy $E$, uniquely determines these electric potential and magnetic field everywhere in $\mathbb{R}^{3}$. Combining this uniqueness result with the reconstruction procedure for the asymptotics of the electric potential and the magnetic field we show that the scattering amplitude, known for some $E$, uniquely determines a electric potential and a magnetic field, that are asymptotic sums of homogeneous terms, which converges to the electric potential and the magnetic field, respectively. Moreover, we discuss the symmetries of the kernel of the scattering matrix, which follow from the parity, charge-conjugation and time-reversal transformations for the Dirac operator.