# Time-dependent methods in inverse scattering problems for the Hartree-Fock equation

## TL;DR

This paper develops a new method for reconstructing two-body interactions in quantum many-body systems described by the Hartree-Fock equation, using high-velocity limits of scattering operators and advanced mathematical tools.

## Contribution

It establishes a uniqueness theorem and introduces a novel reconstruction procedure for short-range two-body interactions in the Hartree-Fock framework.

## Key findings

- Proves a uniqueness theorem for inverse scattering in Hartree-Fock systems.
- Develops a reconstruction method based on high-velocity limits of scattering operators.
- Shows the equivalence of high-velocity and small-amplitude limits of the scattering operator.

## Abstract

The inverse scattering theory for many-body systems in quantum mechanics is an important and difficult issue not only in physics---atomic physics, molecular physics and nuclear physics---but also mathematics. The major purpose in this paper is to establish a reconstruction procedure of two-body interactions from scattering solutions for a Hartree-Fock equation. More precisely, this paper gives a uniqueness theorem and proposes a new reconstruction procedure of the short-range and two-body interactions from a high-velocity limit of the scattering operator for the Hartree-Fock equation. Moreover, it will be found that the high-velocity limit of the scattering operator is equal to a small-amplitude limit of it. The main ingredients of mathematical analysis in this paper are based on the theory of integral equations of the first kind and a Strichartz type estimates on a solution to the free Schr\"{o}dinger equation.