A Regularity Theory for Static Schrödinger Equations on ℝ^d in Spectral Barron Spaces

01/25/2022

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by   Ziang Chen, et al.

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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 non-negative 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).