Strict positivity of interacting Coulomb ground-state densities

Prove that ground-state densities of interacting Coulomb systems in three dimensions are strictly positive everywhere.

Background

Strict positivity of the initial density is required to apply the paper’s inverse theorem. The authors note that positivity is known in several settings, including the non-interacting case, sufficiently weak interactions, and one-dimensional fermionic systems.

For three-dimensional interacting Coulomb systems, strict positivity is expected but remains unproved, leaving a gap in the direct application of the inverse theorem to general interacting ground states.

References

In dimension $D=3$, it is expected that densities of interacting ground states are always strictly positive for Coulomb systems~cite{FouHofSor-07}, but no proof of this claim has yet been given.

— The inverse problem of time-dependent density functional theory on the torus  (2609.08984 - Lauritsen et al., 8 Sep 2026) in Remark 2.8, “Ground states”