The regularity of electronic wave functions in Barron spaces
Harry Yserentant
TL;DR
The paper investigates the regularity of electronic wave functions for the electronic Schrödinger equation with Coulomb interactions, focusing on bound states below the ionization threshold. It develops a momentum-space framework and $L_1$-based Barron-space analysis, proving that all bound-state eigenfunctions lie in the spectral Barron spaces $\mathcal{B}^s(\mathbb{R}^{3N})$ for every $s<1$, with the angularly singular hydrogen ground state demonstrating the sharpness of the $s<1$ bound. The method combines Fourier-transform representations of Coulomb interactions, operator bounds, and a contractive fixed-point argument (via a high-frequency cutoff) to establish the desired regularity and quantitative decay in the Barron norm, which scales like $\sim 1/(1-s)$ as $s \to 1$. This result provides a rigorous link between quantum-mechanical regularity and Barron-space representations, supporting high-dimensional numerical approaches (e.g., neural-network-based PDE solvers) by clarifying the attainable regularity of electronic eigenfunctions.
Abstract
The electronic Schrödinger equation describes the motion of $N$ electrons under Coulomb interaction forces in a field of clamped nuclei. It is proved that its solutions for eigenvalues below the essential spectrum lie in the spectral Barron spaces $\mathcal{B}^s(\mathbb{R}^{3N})$ for $s<1$. The example of the hydrogen ground state shows that this result cannot be improved.