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

翻译：光谱 Barron 空间最近引起了相当大的兴趣, 因为它是两层神经网络的近似理论的自然功能空间, 且无维趋同率 。 在本文中, 我们研究了光谱 Barron 空间中整个空间静态 Schr\" 量子方程式解决方案的规律性 。 我们证明, 如果光谱 Barron 空间是光谱 $\ mathcal{B\\\\\\\\\\\ b\\\\\\\\\\\\\ b\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\ \\\\\\ \\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\