The regularity of electronic wave functions in Barron spaces

TL;DR

This paper proves that solutions to the electronic Schrödinger equation for eigenvalues below the essential spectrum are contained in spectral Barron spaces, with the hydrogen ground state exemplifying the optimality of this result.

Contribution

It establishes the regularity of electronic wave functions in spectral Barron spaces for eigenvalues below the essential spectrum, highlighting a fundamental property of these solutions.

Findings

Solutions lie in spectral Barron spaces for s<1

Hydrogen ground state exemplifies the optimality of the regularity result

Provides a mathematical characterization of electronic wave functions' regularity

Abstract

The electronic Schr\"odinger 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.