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A quantitative Hohenberg-Kohn theorem and the unexpected regularity of density functional theory in one spatial dimension

Thiago Carvalho Corso, Andre Laestadius

TL;DR

This work analyzes the density-to-potential map for spinless fermions in one dimension, proving a quantitative Hohenberg–Kohn theorem that yields a Lipschitz-stable inverse KS problem and establishing real-analytic dependence on both the ground-state density and the interaction strength.By leveraging an implicit-function framework and a detailed operator-theoretic setup in Sobolev spaces, the authors extend DFT to complex-valued densities and non-self-adjoint Hamiltonians via a holomorphic extension of the Levy–Lieb functional, and they derive rigorous results for the exchange–correlation functional, including an exchange-only potential and an absolutely convergent Görling–Levy perturbation series.The results illuminate why the one-dimensional setting exhibits unexpected regularity in DFT and open avenues for TDDFT extensions, while also clarifying how the choice of functional-analytic topology affects inverse-problem stability.

Abstract

In this paper we investigate the (Kohn-Sham) density-to-potential map in the case of spinless fermions in one spatial dimension, whose existence has been rigorously established by the first author in [arXiv:2504.05501 (2025)]. Here, we focus on the regularity of this map as a function of the density and the coupling constant in front of the interaction term. More precisely, we first prove a quantitative version of the Hohenberg-Kohn theorem, thereby showing that this map is Lipschitz continuous with respect to the natural Sobolev norms in the space of densities and potentials. In particular, this implies that the inverse (Kohn-Sham) problem is not only well-posed but also Lipschitz stable. Using this result, we then show that the density-to-potential map is in fact real analytic with respect to both the density and the interaction strength. As a consequence, we obtain a holomorphic extension of the universal constrained-search functional to a suitable subset of complex-valued densities. This partially extends the DFT framework to non-self-adjoint Schrödinger operators. As further applications of these results, we also establish the existence of an exchange-only part of the exchange-correlation potential, and justify the Görling-Levy perturbation expansion for the correlation energy.

A quantitative Hohenberg-Kohn theorem and the unexpected regularity of density functional theory in one spatial dimension

TL;DR

This work analyzes the density-to-potential map for spinless fermions in one dimension, proving a quantitative Hohenberg–Kohn theorem that yields a Lipschitz-stable inverse KS problem and establishing real-analytic dependence on both the ground-state density and the interaction strength.By leveraging an implicit-function framework and a detailed operator-theoretic setup in Sobolev spaces, the authors extend DFT to complex-valued densities and non-self-adjoint Hamiltonians via a holomorphic extension of the Levy–Lieb functional, and they derive rigorous results for the exchange–correlation functional, including an exchange-only potential and an absolutely convergent Görling–Levy perturbation series.The results illuminate why the one-dimensional setting exhibits unexpected regularity in DFT and open avenues for TDDFT extensions, while also clarifying how the choice of functional-analytic topology affects inverse-problem stability.

Abstract

In this paper we investigate the (Kohn-Sham) density-to-potential map in the case of spinless fermions in one spatial dimension, whose existence has been rigorously established by the first author in [arXiv:2504.05501 (2025)]. Here, we focus on the regularity of this map as a function of the density and the coupling constant in front of the interaction term. More precisely, we first prove a quantitative version of the Hohenberg-Kohn theorem, thereby showing that this map is Lipschitz continuous with respect to the natural Sobolev norms in the space of densities and potentials. In particular, this implies that the inverse (Kohn-Sham) problem is not only well-posed but also Lipschitz stable. Using this result, we then show that the density-to-potential map is in fact real analytic with respect to both the density and the interaction strength. As a consequence, we obtain a holomorphic extension of the universal constrained-search functional to a suitable subset of complex-valued densities. This partially extends the DFT framework to non-self-adjoint Schrödinger operators. As further applications of these results, we also establish the existence of an exchange-only part of the exchange-correlation potential, and justify the Görling-Levy perturbation expansion for the correlation energy.

Paper Structure

This paper contains 19 sections, 20 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main contributions
  4. Outline of the paper:
  5. Main results
  6. Applications
  7. Proof strategy
  8. Connection with a previous result on the instability of the inverse problem
  9. Mathematical preliminaries
  10. Operators and resolvents on the Sobolev scales
  11. Trace class operators in the Sobolev scale
  12. The density map
  13. A quantitative version of the Hohenberg--Kohn theorem
  14. Analyticity of density-to-potential map
  15. Proof of corollaries
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $w\in \mathcal{W}(I)$, then for any $\rho \in \mathcal{D}_N$ there exists a constant $C= C(\rho;w)>0$ and a neighborhood $U\subset \mathcal{D}_N$ of $\rho$, such that

Theorems & Definitions (49)

  • Theorem 2.1: Quantitative Hohenberg--Kohn Theorem
  • Theorem 2.2: Analytic density-to-potential map
  • Remark 2.3: Tangent space of $\mathcal{D}_N$
  • Remark 2.4: Analytic maps
  • Remark 2.5: Periodic case
  • Remark 2.6: Kohn--Sham kinetic energy
  • Corollary 2.7: Analytic exchange--correlation
  • Corollary 2.8: Görling--Levy perturbation expansion
  • Remark 2.9: Splitting of AC potential
  • Theorem 2.10: Holomorphic extension of DFT
  • ...and 39 more