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Asymptotics of the Kantorovich Potential for the Optimal Transport with Coulomb Cost

Rodrigue Lelotte

TL;DR

This work rigorously establishes the asymptotics of the Kantorovich potential for multimarginal optimal transport with Coulomb and generalized Riesz costs on unbounded, connected supports. It proves that the Kantorovich potential satisfies $u(r) = \frac{N-1}{|r|^s} + C_u + o\left(\frac{1}{|r|^s}\right)$ with $C_u<0$ for all $s>0$ and $d\ge1$, and links the limit to a dual charge representation when $s=d-2$. The paper also proves a dissociation-at-infinity result: only one particle can escape to infinity, and shows that the dual charge mass equals $N-1$ under mild geometric conditions. Furthermore, it derives HVZ-type dissociation and binding inequalities for associated energies, concluding that the unbounded external Kantorovich potential can bind at most $N-1$ electrons and that the energy minimisers exhibit a rich structure tied to the support of minimisers. These results provide a rigorous mathematical justification of asymptotics predicted by chemistry and have implications for density functional theory and strongly correlated systems.

Abstract

We prove a conjecture regarding the asymptotic behavior at infinity of the Kantorovich potential for the Multimarginal Optimal Transport with Coulomb and Riesz costs.

Asymptotics of the Kantorovich Potential for the Optimal Transport with Coulomb Cost

TL;DR

This work rigorously establishes the asymptotics of the Kantorovich potential for multimarginal optimal transport with Coulomb and generalized Riesz costs on unbounded, connected supports. It proves that the Kantorovich potential satisfies with for all and , and links the limit to a dual charge representation when . The paper also proves a dissociation-at-infinity result: only one particle can escape to infinity, and shows that the dual charge mass equals under mild geometric conditions. Furthermore, it derives HVZ-type dissociation and binding inequalities for associated energies, concluding that the unbounded external Kantorovich potential can bind at most electrons and that the energy minimisers exhibit a rich structure tied to the support of minimisers. These results provide a rigorous mathematical justification of asymptotics predicted by chemistry and have implications for density functional theory and strongly correlated systems.

Abstract

We prove a conjecture regarding the asymptotic behavior at infinity of the Kantorovich potential for the Multimarginal Optimal Transport with Coulomb and Riesz costs.
Paper Structure (6 sections, 6 theorems, 59 equations)

This paper contains 6 sections, 6 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Dissociation at infinity
  3. Proof of the main results
  4. Proof of \ref{['asym_k']}
  5. Proof of \ref{['dual_charge_mass']}
  6. Dissociation and binding inequalities

Key Result

Theorem 1

Let $\Omega \subset \mathbb R^d$ be an unbounded and connected open set, and let $\rho \in L^1(\Omega, \mathbb R_+)$ with $\int_{\mathbb R^d} \rho = N$ be such that $\rho> 0$ almost everywhere on $\Omega$. Let $u$ be the Kantorovich potential for the problem SCE_d_original. Then for a constant $C_u$ which is negative.

Theorems & Definitions (14)

  • Theorem 1: Asymptotics of the Kantorovich potential
  • Remark 1
  • Remark 2
  • Corollary 2: Total mass of the dual charge
  • Theorem 3: Dissociation at infinity
  • proof : Proof of \ref{['main_thm']}
  • Remark 3
  • Lemma 4: lelotteExternalDualCharge2024
  • proof : Proof of Lemma \ref{['lem:C_u']}
  • Remark 4
  • ...and 4 more