A Regularity Theory for Static Schrödinger Equations on $\mathbb{R}^d$ in Spectral Barron Spaces (2201.10072v2)

Published 25 Jan 2022 in math.AP and cs.LG

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 $\mathcal{B}^{{s+2}(\mathbb{R}^d)$.}