论文标题
$ \ mathbb {r}^d $ in Spectral barron空间上的静态schrödinger方程的规律性理论
A Regularity Theory for Static Schrödinger Equations on $\mathbb{R}^d$ in Spectral Barron Spaces
论文作者
论文摘要
光谱巴隆空间最近引起了很大的兴趣，因为它是具有无维度收敛速率的两层神经网络近似理论的自然功能空间。在本文中，我们研究了光谱巴隆空间中整个空间静态Schrödinger方程的解决方案的规律性。我们证明，如果方程的来源位于光谱barron空间$ \ mathcal {b}^s（\ mathbb {r}^d）$，并且潜在函数承认非负下限分解为正常常数，则作为一个正常常数，以及在$ \ mathcal {b}^s（\ mathbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb}^lieS lieS中，\ mathbb {r} $} $ \ mathcal {b}^{s+2}（\ Mathbb {r}^d）$。
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 $\mathcal{B}^{s+2}(\mathbb{R}^d)$.
