v-representability and Hohenberg-Kohn theorem for non-interacting Schrödinger operators with distributional potentials in the one-dimensional torus

• URL: http://arxiv.org/abs/2501.13513v2
• Date: Mon, 27 Jan 2025 11:51:02 GMT
• Title: v-representability and Hohenberg-Kohn theorem for non-interacting Schrödinger operators with distributional potentials in the one-dimensional torus
• Authors: Thiago Carvalho Corso,
• Abstract summary: We show that the ground-state density of any non-interacting Schr"odinger operator on the one-dimensional torus with potentials in a certain class of distributions is strictly positive. In particular, the density-to-potential Kohn-Sham map is single-valued, and the non-interacting functional is differentiable at every point in this space of $v$-representable densities.
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• Abstract: In this paper, we show that the ground-state density of any non-interacting Schr\"odinger 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 Schr\"odinger 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.

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