A Regularity Theory for Static Schrödinger Equations on \(\boldsymbol{\mathbb{R}^d}\) in Spectral Barron Spaces
A Regularity Theory for Static Schrödinger Equations on \(\boldsymbol{\mathbb{R}^d}\) in Spectral Barron Spaces
复制标题
谱巴伦空间中(oldsymbol{mathbb{R}^d})静态薛定谔方程的正则理论
DOI：
10.1137/22m1478719
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发表时间：
2023
影响因子：
2
通讯作者：
Zhou, Shengxuan
中科院分区：
文献类型：
--
作者：
Chen, Ziang;Lu, Jianfeng;Lu, Yulong;Zhou, Shengxuan
谱巴伦空间是研究具有无量纲收敛速度的双层神经网络逼近理论的自然函数空间，近年来受到了广泛的关注。本文研究谱巴伦空间中全空间静态薛定谔方程解的正则性。证明了若方程的源在谱巴伦空间中，且具有非负下界的势函数分解为一个正常数加上一个函数，则方程的解在谱巴伦空间中.
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 spaceand the potential function admitting a nonnegative lower bound decomposes as a positive constant plus a function in, then the solution lies in the spectral Barron space.