# Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions

arXiv: https://arxiv.org/abs/2608.22252 · HTML: https://mlsanity.com/paper/2608.22252/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 12 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Built on by (most cited first, 1 of 1)

- [Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions](https://mlsanity.com/paper/2609.00872/) (2026)

## Abstract (arXiv)

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[Definition~1.4]FournaisEtAl2005, for a Coulombic eigenfunction psi we define the successive quotients by [ φ=e^-F2,cutpsi quadandquad φ3=e^-F3,cutφ=e^-(F2,cut+F3,cut)psi. ] Then [ φ,φ3∈Bs(ℝ3N) qquadfor 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. We also determine the exact endpoint growth. Writing varepsilon=2-s, we prove that, for either u=φ or u=φ3, there is a computable constant M independent of varepsilon such that [ left|uright|_B2-varepsilon≤fracMvarepsilon2left|uright|B1. ] For the unperturbed two-electron atom we prove, with a constant independent of varepsilon, [ left|left|φ3right|_B2-varepsilon-frac32π Zlvertφ3(0,0)rvertvarepsilon2right|≤fracCvarepsilon. ] Hence the quadratic rate in the upper bound is sharp whenever lvertφ3(0,0)rvertneq0, as is the case for the ground state.