A REGULARITY THEORY FOR STATIC SCHRÖDINGER EQUATIONS ON R d IN SPECTRAL BARRON SPACES

Ziang Chen, Jianfeng Lu, Yulong Lu, Shengxuan Zhou

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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 languageEnglish (US)
Pages (from-to)557-570
Number of pages14
JournalSIAM Journal on Mathematical Analysis
Volume55
Issue number1
DOIs
StatePublished - 2023
Externally publishedYes

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

Fingerprint

Dive into the research topics of 'A REGULARITY THEORY FOR STATIC SCHRÖDINGER EQUATIONS ON R d IN SPECTRAL BARRON SPACES'. Together they form a unique fingerprint.

Cite this