This paper proves a strong version of the Hohenberg–Kohn theorem for a wide class of non‑interacting quantum systems. In plain terms, the authors show that under broad mathematical conditions the one‑particle density of a ground state uniquely determines the external potential, and they identify exactly when this uniqueness holds.

The Hohenberg–Kohn theorem is a basic result behind density functional theory. It says that the ground‑state electron density determines the external potential (up to an additive constant), and so in principle all ground‑state properties follow from the density. The Kohn–Sham potential is the effective potential used in practical versions of the theory for non‑interacting reference systems. The paper focuses on when that potential is uniquely determined by the density.

Technically, the authors work with Schrödinger operators and a broad class of external potentials called form‑bounded potentials. A potential being form‑bounded means it is allowed to be singular or grow, but only so far that certain energy estimates still hold. They show that the Hohenberg–Kohn statement is true within the maximal such class if and only if the single‑particle density is positive quasi‑everywhere. “Quasi‑everywhere” is a technical notion from potential theory meaning the density is positive except possibly on a very small exceptional set. They also prove these positivity conditions hold for ground states of non‑interacting Schrödinger operators when the lowest energy level is discrete.

The proof uses tools from classical potential theory. A key intermediate result is a characterization of what the authors call weakly correlated regular states. That characterization lets them connect the behavior of the density to the uniqueness of the potential. One conceptual takeaway is that, in the continuum models they study, it is the (quasi‑)unique continuation of the density — the way the density cannot vanish on a large set — that underlies the Hohenberg–Kohn result, rather than any continuation property of the full many‑body wavefunction.