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

arXiv:2609.00872v1 Announce Type: cross 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,eta. 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+eta<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 0leqalpha<1, [ left(sum_{Iinmathcal I_sigma}prod_{iin I}langlexi_irangle^alpharight)widehat{psi_sigma}in L^1(R^{3N}). ] For a fully spin-polarized state, mathcal I_sigma={{1,ldots,N}}.

Source: arXiv cs.LG | 2026-09-02