The regularity of electronic wave functions in Barron spaces

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.