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 $L^1$ 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 $\alpha ,\beta$. 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 $s+\alpha +\beta <1$; in the fully spin-polarized class it reduces to $s+\alpha <1$. In particular, if ${\mathcal{I}}_{\sigma }$ denotes the family of occupied same-spin blocks determined by $\sigma$, then every fixed-spin spatial component ${\psi }_{\sigma }$ satisfies, for every $0\le \alpha <1$, \[ \left(\sum_{I\in\mathcal I_σ}\prod_{i\in I}\langleξ_i\rangle^α\right)\widehat{ψ_σ}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, ${\mathcal{I}}_{\sigma }=\{\{1,\dots ,N\}\}$.

arXiv comment22 pages; no figures