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

TL;DR

This paper proves that for certain non-interacting Schrödinger operators with distributional potentials on a 1D torus, the ground-state density uniquely determines the potential, supporting the mathematical foundation of Kohn-Sham density functional theory.

Contribution

It establishes v-representability and the validity of the Hohenberg-Kohn theorem for non-interacting Schrödinger operators with distributional potentials on the 1D torus.

Findings

Ground-state density is strictly positive for these operators.

The density-to-potential map is single-valued and differentiable.

Complete characterization of non-interacting v-representable densities.

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.