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

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 mathcal Iσ denotes the family of occupied same-spin blocks determined by σ, then every fixed-spin spatial component psiσ satisfies, for every 0≤α<1, [ left(sumI∈mathcal Iσprodi∈ Ilanglexiirangleαright)widehatpsiσ∈ L1(ℝ3N). ] For a fully spin-polarized state, mathcal Iσ=1,ldots,N.