A rigorous formulation of density functional theory for spinless fermions in one dimension

Abstract In this paper, we present a completely rigorous formulation of Kohn–Sham density functional theory for spinless fermions living in one-dimensional space. More precisely, we consider Schrödinger operators of the form aligned HN (v, w) = - + ₈ ₉N w (xᵢ, xⱼ) + ₉=₁N v (xᵢ) acting on N L² (0, 1), aligned H N (v, w) = - Δ + ∑ i ≠ j N w (x i, x j) + ∑ j = 1 N v (x i) acting on ⋀ N L 2 (0, 1), where the external and interaction potentials v and w belong to a suitable class of distributions. In this setting, we obtain a complete characterization of the set of pure-state v -representable densities on the interval. Then, we prove a Hohenberg–Kohn theorem that applies to the class of distributional potentials studied here. Lastly, we establish the differentiability of the exchange-correlation functional and therefore the existence of a unique exchange-correlation potential. We then combine these results to provide a rigorous formulation of the Kohn–Sham scheme. In particular, these results show that the Kohn–Sham scheme is rigorously exact in this setting.