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

Thiago Carvalho Corso

TL;DR

This work addresses v-representability and the Hohenberg-Kohn framework for non-interacting Schrödinger operators on the 1D torus with distributional potentials in $H^{-1}(oldsymbol{\mathbb{T}})$. It establishes the strict positivity of ground-state densities, provides a complete non-interacting $\mathcal V$-representability characterization via $\mathcal D_N$, and proves a non-interacting HK theorem that guarantees a unique density-to-potential map up to constants, with an explicit smooth Kohn–Sham map for $N=1$ given by $v^{KS}_1(\rho)=\frac{\Delta\sqrt{\rho}}{\sqrt{\rho}}$. The results rely on a Perron–Frobenius–type analysis, Courant nodal theory, and a spectral-dual-space approach to handle distributional potentials, illustrating a rigorous foundation for KS-DFT in this simplified setting. However, HK does not extend to excited states, highlighting intrinsic limitations when distributional potentials (e.g., Dirac deltas) are allowed.

Abstract

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.

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

TL;DR

This work addresses v-representability and the Hohenberg-Kohn framework for non-interacting Schrödinger operators on the 1D torus with distributional potentials in . It establishes the strict positivity of ground-state densities, provides a complete non-interacting -representability characterization via , and proves a non-interacting HK theorem that guarantees a unique density-to-potential map up to constants, with an explicit smooth Kohn–Sham map for given by . The results rely on a Perron–Frobenius–type analysis, Courant nodal theory, and a spectral-dual-space approach to handle distributional potentials, illustrating a rigorous foundation for KS-DFT in this simplified setting. However, HK does not extend to excited states, highlighting intrinsic limitations when distributional potentials (e.g., Dirac deltas) are allowed.

Abstract

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 -representable densities. These results contribute to establishing a solid mathematical foundation for the Kohn-Sham scheme in this simplified setting.
Paper Structure (12 sections, 14 theorems, 88 equations)

This paper contains 12 sections, 14 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation
  4. Main results
  5. Outline of the paper
  6. Mathematical background
  7. Sobolev and dual spaces on the torus
  8. Quadratic form definition of Hamiltonian
  9. Single-particle density matrix and natural orbitals
  10. Non-degeneracy of the ground-state
  11. The non-interacting Hohenberg-Kohn theorem for distributional potentials
  12. Concluding remarks

Key Result

Theorem 2.1

Let $v \in \mathcal{V}$ and $h(v) = -\Delta +v$ be the single-particle operator defined in eq:quadraticform, then the ground-state of $h(v)$ is non-degenerate, and the unique (up to a global phase) normalized ground-state wave-function $\varphi_v \in \mathrm{H}^1(\mathbb{T})$ is strictly positive ev

Theorems & Definitions (32)

  • Theorem 2.1: Non-degenerate single-particle ground-state
  • Remark 2.2: Existence of ground-state
  • Corollary 2.3: Necessary conditions for non-interacting $\mathcal{V}$-representabilitiy
  • Theorem 2.4: Characteriztion of $\mathcal{V}$-representable densities
  • Theorem 2.5: Hohenberg-Kohn theorem
  • Remark 2.6
  • Theorem 2.7: Pure-state $\mathcal{V}$-representability for $N=1$
  • Theorem 2.8: No Hohenberg-Kohn for excited states
  • Definition 3.1: Sobolev spaces
  • Lemma 3.2: Algebra property of $\mathrm{H}^1(\mathbb{T})$
  • ...and 22 more