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 $$\begin{aligned} H_N(v,w) = -\Delta + \sum _{i\ne j}^N w(x_i,x_j) + \sum _{j=1}^N v(x_i) \quad \hbox {acting on }\bigwedge ^N \textrm{L}^2([0,1]), \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>v</mml:mi> <mml:mo>,</mml:mo> <mml:mi>w</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>≠</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <mml:mi>w</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>N</mml:mi> </mml:munderover> <mml:mi>v</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace/> <mml:mtext>acting on</mml:mtext> <mml:mspace/> <mml:mover> <mml:mo>⋀</mml:mo> <mml:mi>N</mml:mi> </mml:mover> <mml:msup> <mml:mtext>L</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> 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.