A Regularity Theory for Static Schrödinger Equations on $\mathbb{R}^d$ in Spectral Barron Spaces
This provides theoretical foundations for approximation theory in neural networks, addressing a domain-specific mathematical analysis problem, but it is incremental as it extends existing regularity results to spectral Barron spaces.
The paper tackles the regularity of solutions to static Schrödinger equations on ℝ^d in spectral Barron spaces, proving that if the source and potential meet specific conditions in these spaces, the solution gains two orders of regularity, lying in a higher-order Barron space.
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 $\mathcal{B}^s(\mathbb{R}^d)$ 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 $\mathcal{B}^{s+2}(\mathbb{R}^d)$.