A rigorous formulation of Density Functional Theory for spinless electrons in one dimension

• URL: http://arxiv.org/abs/2504.05501v1
• Date: Mon, 07 Apr 2025 20:54:47 GMT
• Title: A rigorous formulation of Density Functional Theory for spinless electrons in one dimension
• Authors: Thiago Carvalho Corso,
• Abstract summary: We consider Schr"odinger operators of the form $H_N(v,w) = -Delta + sum_ineq jN w(x_i,x_j) + sum_j=1N v(x_i)$ acting on $wedgeN mathrmL2, where the external and interaction potentials $v$ and $w$ belong to a suitable class of distributions.<n>We prove a Hohenberg-Kohn theorem that applies to the class of distributional potentials studied here
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: In this paper, we present a completely rigorous formulation of Kohn-Sham density functional theory for spinless electrons living in one dimensional space. More precisely, we consider Schr\"odinger operators of the form $H_N(v,w) = -\Delta + \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ acting on $\wedge^N \mathrm{L}^2([0,1])$, 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.

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