In search of the Hohenberg-Kohn theorem

TL;DR

This paper rigorously investigates the conditions under which the Hohenberg-Kohn theorem guarantees the uniqueness of external potentials for a quantum many-body system, exploring trade-offs between potential and density conditions.

Contribution

It provides a detailed analysis of sufficient conditions for potential-density uniqueness in density functional theory within a rigorous mathematical framework.

Findings

Conditions range from positivity of density to integrability of potentials.

Localizability of the theorem under less strict conditions is examined.

Trade-offs between potential regularity and density positivity are characterized.

Abstract

The Hohenberg-Kohn theorem, a cornerstone of electronic density functional theory, concerns uniqueness of external potentials yielding given ground densities of an ${\mathcal N}$-body system. The problem is rigorously explored in a universe of three-dimensional Kato-class potentials, with emphasis on trade-offs between conditions on the density and conditions on the potential sufficient to ensure uniqueness. Sufficient conditions range from none on potentials coupled with everywhere strict positivity of the density, to none on the density coupled with something a little weaker than local $3{\mathcal N}/2$-power integrability of the potential on a connected full-measure set. A second theme is localizability, that is, the possibility of uniqueness over subsets of ${\mathbb R}^3$ under less stringent conditions.