We consider an inverse $N$-body scattering problem of determining two potentials—an external potential acting on all particles and a pair interaction potential—from the scattering particles. This paper finds that the time-dependent Hartree-Fock approximation for a three-dimensional inverse $N$-body scattering in quantum mechanics enables us to recover the two potentials from the scattering states with high-velocity initial states. The main ingredient of mathematical analysis in this paper is based on the asymptotic analysis of the scattering operator defined in terms of a scattering solution to the Hartree-Fock equation at high energies. We show that the leading part of the asymptotic expansion of the scattering operator uniquely reconstructs the Fourier transform of the pair interaction, and the second term of the expansion uniquely reconstructs the $X$-ray transform of the external potential.

中文翻译：

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与时间相关的hartree-fock近似的$ N $体散射逆

我们考虑一个反向的N体散射问题，该问题确定了散射粒子的两个电势-作用在所有粒子上的外部电势和一对相互作用电势。本文发现，量子力学中三维逆$ N $-体散射的时变Hartree-Fock近似使我们能够从具有高速初始状态的散射状态中恢复两个电势。本文中数学分析的主要成分是基于对散射算子的渐近分析，该算子是根据高能Hartree-Fock方程的散射解来定义的。我们证明了散射算子渐近展开的前导部分唯一地重构了对相互作用的傅立叶变换，