math.APAug 23, 2026

Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions

Authors: Pingbing Ming, Hao Yu

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Abstract

We establish sharp Barron regularity for Coulombic many-electron wave functions after extraction of the universal cut-off Jastrow factors. Following the factorization of Fournais et al.\cite[Definition1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction ψψ we define the successive quotients by

φ=e−F2,cutψandφ3=e−F3,cutφ=e−(F2,cut+F3,cut)ψ.φ=e^{-F_{2,\mathrm{cut}}}ψ\quad\text{and}\quad φ_3=e^{-F_{3,\mathrm{cut}}}φ=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}ψ.

Then

φ,φ3∈Bs(R3N)for every s<2.φ,φ_3\in\mathcal{B}^s(\mathbb{R}^{3N}) \qquad\text{for every }s<2.

This range is optimal among universal factorizations. No factor depending only on the particle number and the nuclear data, but not on the eigenfunction or its eigenvalue, can make every corresponding quotient belong to B2\mathcal{B}^2. We also determine the exact endpoint growth. Writing ε=2−s\varepsilon=2-s, we prove that, for either u=φu=φ or u=φ3u=φ_3, there is a computable constant MM independent of ε\varepsilon such that

∥u∥B2−ε≤Mε2∥u∥B1.\left\|u\right\|_{\mathcal{B}^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\left\|u\right\|_{\mathcal{B}^1}.

For the unperturbed two-electron atom we prove, with a constant independent of ε\varepsilon,

∣∥φ3∥B2−ε−32πZ∣φ3(0,0)∣ε2∣≤Cε.\left|\left\|φ_3\right\|_{\mathcal{B}^{2-\varepsilon}}-\frac{32πZ\lvertφ_3(0,0)\rvert}{\varepsilon^2}\right|\leq\frac{C}{\varepsilon}.

Hence the quadratic rate in the upper bound is sharp whenever ∣φ3(0,0)∣≠0\lvertφ_3(0,0)\rvert\neq0, as is the case for the ground state.

Figures & tables

Figure 1. Frequency-region decomposition used in Steps 2 and 4 of the proof of Proposition 6.1 . Panel (a) shows the two nested cones and the high–high, one-radius, and bounded parts of their complement. The threshold 88 only separates the one-radius strips from the bounded core; it is not part of the definition of D4±D_{4}^{\pm} . In Panel (b), if Sj,k⊂Ω4c\mathcal{S}_{j,k}\subset\Omega_{4}^{c} is nonempty with j,k≥2j,k\geq 2 , then ∣j−k∣≤2|j-k|\leq 2 . Thus, for each n=max⁡{j,k}n=\max\{j,k\} , at most five shell pairs occur, and only one dyadic index is independently unbounded. Boundary sets may be assigned to either adjacent region.
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