What the paper proves in plain language: the authors show that, for a wide class of time-varying one-particle densities, there is a unique external potential (up to adding a time-dependent constant) that makes the full many-electron Schrödinger equation produce that density. In other words, if you specify how the electron density should evolve in time, they prove you can recover the physical force (the external potential) that generates it — at least on a periodic box (a mathematical torus) and under certain regularity assumptions.

What the researchers did: they solved an “inverse problem” for the time-dependent many-body Schrödinger equation. Starting from a given density and a compatible initial wavefunction, they construct an external potential so that the N‑particle wavefunction produced by the Schrödinger evolution has the prescribed one-particle density. The proof handles realistic Coulomb interactions (the usual 1/|x| repulsion between electrons) and allows for extended nuclei (smooth charge distributions) moving in time. The potential they find is unique except for an arbitrary function of time, which has no effect on the density.

How the argument works at a high level: a standard route in this area writes the potential in terms of the density and the many-body wavefunction via a relation known in physics as the force-balance or van Leeuwen equation. That relation typically involves many spatial derivatives and causes a technical “loss of derivatives” that blocks straightforward solution methods. The authors rewrite the equation to reduce the derivative loss to only one derivative and then prove existence by working in a class of functions that are analytic in space in a suitable sense. A key technical idea is to require analyticity only in the center-of-mass coordinate rather than in all particle coordinates; this avoids problems caused by the cusps that the Coulomb interaction creates when two electrons meet.