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arXiv · 2501.13513

v-representability and Hohenberg-Kohn theorem for non-interacting Schr\"odinger operators with distributional potentials in the one-dimensional torus

Thiago Carvalho Corso
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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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Thiago Carvalho Corso. 2025-01-23. v-representability and Hohenberg-Kohn theorem for non-interacting Schr\"odinger operators with distributional potentials in the one-dimensional torus. https://doi.org/10.1088/1751-8121%2Fadc04c

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