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

Abstract:In this paper, we show 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. Furthermore, we show that these conditions are satisfied for the ground state of non-interacting Schrödinger 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.

Fixed some typos, simplified some remarks, and slightly modified the proof of Lemma 5.1 to include the case of form-bounded potentials with arbitrary form bound

Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Quantum Physics (quant-ph)

arXiv:2607.12852 [math-ph]

(or arXiv:2607.12852v2 [math-ph] for this version)