For non-interacting Schrödinger systems the Kohn–Sham potential is unique among all Laplace form-bounded potentials, because the Hohenberg–Kohn map is one-to-one precisely when the density is positive quasi-everywhere.

Summary. The manuscript claims that for Schrödinger operators with weakly correlated ground states, the Hohenberg–Kohn theorem holds within the maximal class of form-bounded external potentials if and only if the single-particle density is positive quasi-everywhere. It further asserts that these conditions are satisfied by ground states of non-interacting Schrödinger operators with discrete ground-state energy, and therefore that the Hohenberg–Kohn theorem—and uniqueness of the Kohn–Sham potential—hold for non-interacting systems in the maximal class of Laplace form-bounded potentials. The stated key ingredient is a characterization of weakly correlated regular states whose proof relies on classical potential theory; the authors identify quasi-unique continuation of the density (rather than of the many-body wavefunction) as the continuum mechanism underlying the theorem.

Significance. If the proofs are correct, the result would be a substantial contribution to the mathematical foundations of density-functional theory: it would identify a maximal uniqueness class for external potentials in the non-interacting setting and isolate quasi-everywhere positivity of the single-particle density as the precise condition for Hohenberg–Kohn uniqueness. Uniqueness of the Kohn–Sham potential in the full Laplace form-bounded class is of clear interest to both mathematical physics and theoretical chemistry. The explicit appeal to classical potential theory and the reframing of the continuum mechanism in terms of density unique continuation are conceptually valuable, provided the supporting lemmas hold.