A Regularity Theory for Static Schrödinger Equations on \(\boldsymbol{\mathbb{R}}\)d in Spectral Barron Spaces

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 Schrödinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space and the potential function admitting a nonnegative lower bound decomposes as a positive constant plus a function in , then the solution lies in the spectral Barron space .