A Regularity Theory for Static Schr\"odinger Equations on $\mathbb{R}^d$ in Spectral Barron Spaces

TL;DR

This paper establishes regularity results for solutions to static Schr"odinger equations in spectral Barron spaces, showing that solutions gain regularity under certain conditions on the source and potential functions.

Contribution

It provides the first regularity theory for static Schr"odinger equations within spectral Barron spaces, linking source and potential regularity to solution smoothness.

Findings

Solutions in spectral Barron spaces gain two derivatives in regularity.

Regularity of solutions depends on the spectral Barron space membership of source and potential.

The potential function's structure influences the solution's regularity.

Abstract

Spectral Barron spaces have received considerable interest recently as it is the natural function space for approximation theory of two-layer neural networks with a dimension-free convergence rate. In this paper we study the regularity of solutions to the whole-space static Schr\"odinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space $\mathcal{B}^s(\mathbb{R}^d)$ and the potential function admitting a non-negative lower bound decomposes as a positive constant plus a function in $\mathcal{B}^s(\mathbb{R}^d)$, then the solution lies in the spectral Barron space $\mathcal{B}^{s+2}(\mathbb{R}^d)$.