v-representability and Hohenberg-Kohn theorem for non-interacting Schrödinger operators with distributional potentials in the one-dimensional torus
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| Format: | Preprint |
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2025
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| _version_ | 1866908276481851392 |
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| author | Corso, Thiago Carvalho |
| author_facet | Corso, Thiago Carvalho |
| contents | In this paper, we show that the ground-state density of any non-interacting Schrödinger operator on the one-dimensional torus with potentials in a certain class of distributions is strictly positive. This result together with recent results from [Sutter el al (2024), J. Phys. A: Math. Theor. 57 475202] provides a complete characterization of the set of non-interacting v-representable densities on the torus. Moreover, we prove that, for said class of non-interacting Schrödinger operators with distributional potentials, the Hohenberg-Kohn theorem holds, i.e., the external potential is uniquely determined by the ground-state density. In particular, the density-to-potential Kohn-Sham map is single-valued, and the non-interacting Lieb functional is differentiable at every point in this space of $v$-representable densities. These results contribute to establishing a solid mathematical foundation for the Kohn-Sham scheme in this simplified setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_13513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | v-representability and Hohenberg-Kohn theorem for non-interacting Schrödinger operators with distributional potentials in the one-dimensional torus Corso, Thiago Carvalho Mathematical Physics Analysis of PDEs Quantum Physics Primary: 81Q10, secondary: 81V74, 34L40 In this paper, we show that the ground-state density of any non-interacting Schrödinger operator on the one-dimensional torus with potentials in a certain class of distributions is strictly positive. This result together with recent results from [Sutter el al (2024), J. Phys. A: Math. Theor. 57 475202] provides a complete characterization of the set of non-interacting v-representable densities on the torus. Moreover, we prove that, for said class of non-interacting Schrödinger operators with distributional potentials, the Hohenberg-Kohn theorem holds, i.e., the external potential is uniquely determined by the ground-state density. In particular, the density-to-potential Kohn-Sham map is single-valued, and the non-interacting Lieb functional is differentiable at every point in this space of $v$-representable densities. These results contribute to establishing a solid mathematical foundation for the Kohn-Sham scheme in this simplified setting. |
| title | v-representability and Hohenberg-Kohn theorem for non-interacting Schrödinger operators with distributional potentials in the one-dimensional torus |
| topic | Mathematical Physics Analysis of PDEs Quantum Physics Primary: 81Q10, secondary: 81V74, 34L40 |
| url | https://arxiv.org/abs/2501.13513 |