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 \(H=-\Delta+V\) 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 \(F_{\rm cut}\) of Fournais--Hoffmann-Ostenhof--Hoffmann-Ostenhof--Sørensen, we set \[ \phi=e^{-F_{\rm cut}}\psi . \] Whereas the original wave function satisfies the sharp global threshold \(\psi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N})\) for every \(0\leq s<1\), we prove that the Jastrow quotient gains one full order: \[ \phi\in \mathcal B_{\rm sp}^s(\mathbb R^{3N}) \qquad \text{for every } 0\le s<2 . \] 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.