A quantitative Hohenberg-Kohn theorem and the unexpected regularity of density functional theory in one spatial dimension

TL;DR

This paper rigorously analyzes the regularity of the density-to-potential map in one-dimensional density functional theory, proving Lipschitz continuity and real analyticity, and extending the theoretical framework to complex densities and non-self-adjoint operators.

Contribution

It provides a quantitative, Lipschitz continuous, and real analytic characterization of the density-to-potential map, extending DFT to complex densities and non-self-adjoint operators.

Findings

Proves Lipschitz continuity of the density-to-potential map.

Shows the map is real analytic with respect to density and interaction strength.

Establishes the existence of an exchange-only part of the exchange-correlation potential.

Abstract

In this paper we investigate the (Kohn-Sham) density-to-potential map in the case of spinless fermions in one spatial dimension, whose existence has been rigorously established by the first author in [arXiv:2504.05501 (2025)]. Here, we focus on the regularity of this map as a function of the density and the coupling constant in front of the interaction term. More precisely, we first prove a quantitative version of the Hohenberg-Kohn theorem, thereby showing that this map is Lipschitz continuous with respect to the natural Sobolev norms in the space of densities and potentials. In particular, this implies that the inverse (Kohn-Sham) problem is not only well-posed but also Lipschitz stable. Using this result, we then show that the density-to-potential map is in fact real analytic with respect to both the density and the interaction strength. As a consequence, we obtain a holomorphic extension of the universal constrained-search functional to a suitable subset of complex-valued densities. This partially extends the DFT framework to non-self-adjoint Schr\"odinger operators. As further applications of these results, we also establish the existence of an exchange-only part of the exchange-correlation potential, and justify the G\"orling-Levy perturbation expansion for the correlation energy.