Hohenberg–Kohn conjecture for density-potential determination

Let $X$ be the density space, let $X^*$ be its dual space of external potentials, let $\mathcal{D}^N$ denote the set of $N$-particle densities, and let ${\underline{\partial}}F[\rho]$ be the relevant subdifferential of the universal functional $F$ at $\rho$. A density is a ground-state density for a potential when it arises as a ground state of the corresponding external-potential Hamiltonian.

Hohenberg–Kohn conjecture. For every $\rho\in\mathcal{D}^N$, either

$${\underline{\partial}} F[\rho] = \{ -v + \mu \mid \mu \in \mathbb{R} \}$$

for some $v\in X^*$, or

$${\underline{\partial}} F[\rho] = \emptyset,$$

whenever there are no potentials for which $\rho$ is a ground-state density. Thus, whenever the density is $v$-representable, it determines the potential up to an additive constant.

The conjecture is the rigorous form of the Hohenberg–Kohn density-as-basic-variable principle in density functional theory. Its original proof has a non-rigorous pointwise division by the wavefunction, and it remains partially open for general potentials $v\in X^*$; unique continuation is known for, among others, suitable locally integrable potentials including Coulomb potentials.