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 $0\leq\alpha<1$, \[ \left(\sum_{I\in\mathcal I_\sigma}\prod_{i\in I}\langle\xi_i\rangle^\alpha\right)\widehat{\psi_\sigma}\in L^1(\mathbb{R}^{3N}). \] For a fully spin-polarized state, $\mathcal I_\sigma=\{\{1,\ldots,N\}\}$.