Cut-off Jastrow Factors and Spectral Barron Regularity of Coulombic Electronic Wave Functions

We study the spectral Barron regularity of Coulombic electronic eigenfunctions after extraction of a cut-off Jastrow factor. Let $H=-\mathrm{\Delta }+V$ be an $N$-electron Coulomb Hamiltonian with clamped nuclei, and let $\psi$ be an eigenfunction associated with a discrete eigenvalue below the bottom of the essential spectrum. For the cut-off Jastrow factor $F_{\mathrm{c}\mathrm{u}\mathrm{t}}$ of Fournais–Hoffmann-Ostenhof–Hoffmann-Ostenhof–Sørensen, we set $$\phi =e^{-F_{\mathrm{c}\mathrm{u}\mathrm{t}}}\psi \mathrm{.}$$ Whereas the original wave function satisfies the sharp global threshold $\psi \in {\mathcal{B}}_{\mathrm{s}\mathrm{p}}^s({\mathbb{R}}^{3N})$ for every $0\le s<1$, we prove that the Jastrow quotient gains one full order: $$\phi \in {\mathcal{B}}_{\mathrm{s}\mathrm{p}}^s({\mathbb{R}}^{3N})\text{for every }0\le s<2.$$ The endpoint $s=2$ is shown to be natural through an explicit hydrogen-like eigenfunction. The many-body proof is a global Fourier-side resolvent argument. After conjugation by the cut-off Jastrow factor, the Coulomb singularities are converted into localized angular coefficient blocks with admissible Fourier-control measures. Low frequencies are controlled by the a priori $H^1$-bound, while high frequencies are recovered by a Neumann fixed-point argument using the resolvent multiplier and annular estimates for the coefficient measures.