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Bound on the excess charge for bosonic systems interacting through Coulomb potentials

Rafael D. Benguria, Juan Manuel González-Brantes

TL;DR

The paper proves an improved upper bound on the excess charge $N-Z$ for non-relativistic bosonic atoms with Coulomb interactions, showing $N < 1.4811 Z + 3.1516 Z^{1/3}$ for $Z \ge 12$, refining Lieb's 1984 bound in the bosonic case. The authors adapt Nam's strategy within the Hartree framework, leveraging Virial Theorem relations for the one-particle functional, the Coulomb Uncertainty Principle, and density-based inequalities (Lieb–Oxford, Hoffmann-Ostenhof, GNS) to connect many-body energy to single-particle quantities. The proof proceeds by deriving a lower bound on the $N$-particle energy, bounding the Coulomb and kinetic terms via density functionals, and employing eigenvalue estimates to control the particle number relative to $Z$, culminating in a precise $N/Z$ bound and a computable error term. The result tightens the ionization bound for bosonic atoms and provides a robust, computable threshold applicable to Coulomb-interacting many-body systems.

Abstract

We prove an upper bound on the excess charge for non-relativistic atomic systems, independent of the particle statistics. This result improves Lieb's bound of 1984 for any $Z \ge 12$.

Bound on the excess charge for bosonic systems interacting through Coulomb potentials

TL;DR

The paper proves an improved upper bound on the excess charge for non-relativistic bosonic atoms with Coulomb interactions, showing for , refining Lieb's 1984 bound in the bosonic case. The authors adapt Nam's strategy within the Hartree framework, leveraging Virial Theorem relations for the one-particle functional, the Coulomb Uncertainty Principle, and density-based inequalities (Lieb–Oxford, Hoffmann-Ostenhof, GNS) to connect many-body energy to single-particle quantities. The proof proceeds by deriving a lower bound on the -particle energy, bounding the Coulomb and kinetic terms via density functionals, and employing eigenvalue estimates to control the particle number relative to , culminating in a precise bound and a computable error term. The result tightens the ionization bound for bosonic atoms and provides a robust, computable threshold applicable to Coulomb-interacting many-body systems.

Abstract

We prove an upper bound on the excess charge for non-relativistic atomic systems, independent of the particle statistics. This result improves Lieb's bound of 1984 for any .

Paper Structure

This paper contains 2 sections, 4 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:excess_charge']}.

Key Result

Theorem 1.1

Let us consider the $N$-particle Hamiltonian acting on the symmetric tensor product of $N$ copies of $L^{2}\qty(\mathds{R}^{3})$ (Bosonic Atom). Then, the maximum number of particles, $N$, for which the Hamiltonian has a bound state satisfies for $Z \geq 12$.

Theorems & Definitions (7)

  • Theorem 1.1: Bound on the excess charge for bosonic atoms
  • Remark 1.2
  • Corollary 1.3: Lower bound on $\mathcal{E}\qty(\psi)$
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Kinetic Energy Bound in terms of the density; Maria and Thomas Hoffmann-Ostenhof, 1977 HoffmannOstenhof1977
  • Theorem 1.7: Bound on the Coulomb Energy in terms of the density; E. H. Lieb and S. Oxford, 1981 LiebOxford1981