A REGULARITY THEORY FOR STATIC SCHRÖDINGER EQUATIONS ON R ^d IN SPECTRAL BARRON SPACES

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 B ^s(R ^d) and the potential function admitting a nonnegative lower bound decomposes as a positive constant plus a function in B ^s(R ^d), then the solution lies in the spectral Barron space B ^s+2(R ^d).