Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions

arXiv:2608.22252v1 Announce Type: cross Abstract: We establish sharp Barron regularity for Coulombic many-electron wave functions after extraction of the universal cut-off Jastrow factors. Following the factorization of Fournais et al.ite[Definition1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction psi we define the successive quotients by [ phi=e^{-F_{2,cut}}psi quadext{and}quad phi_3=e^{-F_{3,cut}}phi=e^{-(F_{2,cut}+F_{3,cut})}psi. ] Then [ phi,phi_3inB^s(R^{3N}) qquadext{for every }s<2. ] This range is optimal among universal factorizations. No factor depending only on the particle number and the nuclear data, but not on the eigenfunction or its eigenvalue, can make every corresponding quotient belong to B^2. We also determine the exact endpoint growth. Writing arepsilon=2-s, we prove that, for either u=phi or u=phi_3, there is a computable constant M independent of arepsilon such that [ left|uright|{B^{2-arepsilon}}leqfrac{M}{arepsilon^2}left|uright|{B^1}. ] For the unperturbed two-electron atom we prove, with a constant independent of arepsilon, [ left|left|phi_3right|_{B^{2-arepsilon}}-frac{32pi Zlvertphi_3(0,0)rvert}{arepsilon^2}right|leqfrac{C}{arepsilon}. ] Hence the quadratic rate in the upper bound is sharp whenever lvertphi_3(0,0)rvertneq0, as is the case for the ground state.

Source: arXiv cs.LG | 2026-08-25