Inverse N-body scattering with the time-dependent Hartree-Fock approximation

TL;DR

This paper demonstrates that the time-dependent Hartree-Fock approximation allows for the unique recovery of external and pair interaction potentials in a three-dimensional inverse N-body scattering problem using high-velocity scattering states.

Contribution

It introduces a novel method to recover two potentials in quantum N-body scattering via asymptotic analysis of the scattering operator under the Hartree-Fock approximation.

Findings

The leading asymptotic term reconstructs the Fourier transform of the pair interaction.

The second asymptotic term reconstructs the X-ray transform of the external potential.

High-velocity initial states enable potential recovery.

Abstract

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.