Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions

• URL: http://arxiv.org/abs/2609.00872v1
• Date: Tue, 01 Sep 2026 08:05:47 GMT
• Title: Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions
• Abstract summary: We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians.<n>For fixed-spin components with two occupied spin blocks, it reduces to $s++1$.<n>For a fully spin-polarized state, $mathcal I_=1,ldots,N$.
• Score: 5.361030050051251
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\mathcal I_σ$ denotes the family of occupied same-spin blocks determined by $σ$, then every fixed-spin spatial component $ψ_σ$ satisfies, for every $0\leqα<1$, \[ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_σ=\{\{1,\ldots,N\}\}$.

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