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

$$\begin{aligned} H_N(v,w) = -\Delta + \sum _{i\ne j}^N w(x_i,x_j) + \sum _{j=1}^N v(x_i) \quad \hbox {acting on }\bigwedge ^N \textrm{L}^2([0,1]), \end{aligned}$$

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.