A rigorous formulation of Density Functional Theory for spinless fermions in one dimension
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910063963144192 |
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| author | Corso, Thiago Carvalho |
| author_facet | Corso, Thiago Carvalho |
| contents | 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 $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_05501 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A rigorous formulation of Density Functional Theory for spinless fermions in one dimension Corso, Thiago Carvalho Mathematical Physics Analysis of PDEs Spectral Theory Quantum Physics Primary: 35J10 Secondary:81Q05, 81V74, 46N50 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 $H_N(v,w) = -Δ+ \sum_{i\neq j}^N w(x_i,x_j) + \sum_{j=1}^N v(x_i)$ 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. |
| title | A rigorous formulation of Density Functional Theory for spinless fermions in one dimension |
| topic | Mathematical Physics Analysis of PDEs Spectral Theory Quantum Physics Primary: 35J10 Secondary:81Q05, 81V74, 46N50 |
| url | https://arxiv.org/abs/2504.05501 |