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

In this paper, we show that the ground-state density of any non-interacting Schr & ouml;dinger 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 et 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 & ouml;dinger 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 (KS) 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 KS scheme in this simplified setting.