A maximal Hohenberg-Kohn theorem for non-interacting systems via potential theory

In this paper, we show that for Schr\"odinger 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. Furthermore, we show that these conditions are satisfied for the ground state of non-interacting Schr\"odinger operators with a discrete ground state energy. Consequently, we establish the Hohenberg-Kohn theorem for non-interacting systems, and therefore the uniqueness of the Kohn-Sham potential, within the maximal class of Laplace form-bounded potentials. The key ingredient to establish these results is a characterization of weakly correlated regular states, whose proof relies on classical potential theory. Moreover, our proof reveals that, in the continuum setting, the fundamental mechanism underlying the Hohenberg-Kohn theorem is the (quasi)-unique continuation of the density rather than of the many-body wavefunction.