Sharp Barron Regularity Results for Coulombic Many-Electron Wave Functions

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.~\cite[Definition~1.4]{FournaisEtAl2005}, for a Coulombic eigenfunction $\psi$ we define the successive quotients by \[ ϕ=e^{-F_{2,\mathrm{cut}}}ψ\quad\text{and}\quad ϕ_3=e^{-F_{3,\mathrm{cut}}}ϕ=e^{-(F_{2,\mathrm{cut}}+F_{3,\mathrm{cut}})}ψ. \] Then \[ ϕ,ϕ_3\in\mathcal{B}^s(\mathbb{R}^{3N}) \qquad\text{for every }s<2. \] 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 ${\mathcal{B}}^2$. We also determine the exact endpoint growth. Writing $\epsilon =2-s$, we prove that, for either $u=\varphi$ or $u={\varphi }_3$, there is a computable constant $M$ independent of $\epsilon$ such that \[ \left\|u\right\|_{\mathcal{B}^{2-\varepsilon}}\leq\frac{M}{\varepsilon^2}\left\|u\right\|_{\mathcal{B}^1}. \] For the unperturbed two-electron atom we prove, with a constant independent of $\epsilon$, \[ \left|\left\|ϕ_3\right\|_{\mathcal{B}^{2-\varepsilon}}-\frac{32πZ\lvertϕ_3(0,0)\rvert}{\varepsilon^2}\right|\leq\frac{C}{\varepsilon}. \] Hence the quadratic rate in the upper bound is sharp whenever $|{\varphi }_3(0,0)|\ne 0$, as is the case for the ground state.