A maximal Hohenberg-Kohn theorem for non-interacting systems via potential theory
arXiv:2607.12852
The paper proves that for non‑interacting Schrödinger operators, the Hohenberg‑Kohn theorem holds for the widest class of form‑bounded external potentials, provided the ground‑state density is positive quasi‑everywhere, using tools from classical potential theory.
Abstract
We show that for Schrödinger operators in a connected domain, the Hohenberg-Kohn theorem holds within the class of Laplace form-bounded external potentials if and only if the single-particle density is strictly positive quasi-everywhere. Furthermore, we show that this condition is satisfied for non-interacting Schrödinger operators whenever a ground-state exists. Consequently, we establish the Hohenberg-Kohn theorem for non-interacting systems, and thereby the uniqueness of the Kohn-Sham potential, within the maximal class of Laplace form-bounded distributions. The main ingredient to establish these results is a characterization of regular states, whose proof relies on tools from classical potential theory. Moreover, this characterization 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.
Removed the finite rank restriction on the characterization of regular states, reformulated the main results, some definitions, and proofs, and fixed some typos