Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions
Abstract
We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier L1 norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set I of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order s and the coordinate orders α,β. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to s+α+β<1; in the fully spin-polarized class it reduces to s+α<1. In particular, if ℐσ denotes the family of occupied same-spin blocks determined by σ, then every fixed-spin spatial component ψσ satisfies, for every 0≤α<1, (∑I∈ℐσ∏i∈ I⟨ξi⟩α)ψσ̂∈ L1(ℝ3N). For a fully spin-polarized state, ℐσ={{1,…,N}}.
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BibTeX
@misc{ming2026sharp,
title = {{Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions}},
author = {Pingbing Ming and Hao Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00872},
eprint = {2609.00872},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00872},
}
RIS
TY - UNPB
TI - Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions
AU - Pingbing Ming
AU - Hao Yu
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00872
UR - https://arxiv.org/abs/2609.00872
AB - We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians. The mixed norm is a Fourier L^1 norm with one isotropic weight and coordinate-product weights, and therefore detects regularity invisible to the isotropic Barron scale. For a nonempty set I of electron indices on which the wave function is antisymmetric, we derive an explicit admissible region for the isotropic order s and the coordinate orders α,β. This region is optimal as a uniform statement over the class of clamped-nuclei Coulomb Hamiltonians. For fixed-spin components with two occupied spin blocks, it reduces to s+α+β<1; in the fully spin-polarized class it reduces to s+α<1. In particular, if ℐ_σ denotes the family of occupied same-spin blocks determined by σ, then every fixed-spin spatial component ψ_σ satisfies, for every 0≤α<1, (∑_I∈ℐ_σ∏_i∈ I⟨ξ_i⟩^α)ψ_σ̂∈ L^1(ℝ^3N). For a fully spin-polarized state, ℐ_σ={{1,…,N}}.
ER -