## Abstract

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 nonnegative 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}).

Original language | English (US) |
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Pages (from-to) | 557-570 |

Number of pages | 14 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 55 |

Issue number | 1 |

DOIs | |

State | Published - 2023 |

Externally published | Yes |

### Bibliographical note

Funding Information:*Received by the editors February 17, 2022; accepted for publication (in revised form) November 14, 2022; published electronically February 23, 2023. https://doi.org/10.1137/22M1478719 Funding: The work of the first and second authors was supported in part by National Science Foundation grant DMS-2012286. The work of the third author was supported by National Science Foundation grant DMS-2107934. \dagger Department of Mathematics, Duke University, Durham, NC 27708 USA (ziang@math.duke.edu). \ddagger Departments of Mathematics, Physics, and Chemistry, Duke University, Durham, NC 27708 USA (jianfeng@math.duke.edu). \S Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003 USA (yulonglu@umass.edu). \P Beijing International Center for Mathematical Research, Peking University, Beijing 100871, People`s Republic of China (zhoushx19@pku.edu.cn).

Publisher Copyright:

© 2023 Society for Industrial and Applied Mathematics.

## Keywords

- Barron space
- neural networks
- regularity theory
- Schrödinger equation

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