A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition
Abstract
We propose spectral and spectral-element methods for accurately computing many eigenvalues of the Schrödinger operator with Robin boundary condition on simple and complex geometries, respectively. Due to their spectral-type accuracy in approximating those eigenfunctions corresponding to high-index eigenvalues, which are usually highly oscillatory, the proposed approaches have excellent resolution in computing thousands of eigenvalues accurately and efficiently with the resolution property that the number of eigenvalues with reasonable accuracy is proportional to the number of degrees of freedom. Using a standard HPC node with 128 GB of memory, we can obtain numerically more than 5,000 reliable eigenvalues with relative errors below 10−8 for different complex domains in two dimensions (2D). Based on these computed eigenvalues, we first confirm some theoretical results on the Robin-to-Neumann (RtN) gaps of the Laplacian operator in 2D, which were recently studied by Rudnick \textit{et al.} [\textit{Comm. Math. Phys.} 388 (2021)]. Then we systematically study the RtN gaps of the Schrödinger operator and their convergence rates. Based on our extensive numerical results, we formulate a unified conjecture on the cumulative averages of the RtN gaps of the Schrödinger operator.
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BibTeX
@misc{bao2026spectralelement,
title = {{A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition}},
author = {Weizhu Bao and Fukeng Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12369},
eprint = {2609.12369},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12369},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition
AU - Weizhu Bao
AU - Fukeng Huang
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12369
UR - https://arxiv.org/abs/2609.12369
AB - We propose spectral and spectral-element methods for accurately computing many eigenvalues of the Schrödinger operator with Robin boundary condition on simple and complex geometries, respectively. Due to their spectral-type accuracy in approximating those eigenfunctions corresponding to high-index eigenvalues, which are usually highly oscillatory, the proposed approaches have excellent resolution in computing thousands of eigenvalues accurately and efficiently with the resolution property that the number of eigenvalues with reasonable accuracy is proportional to the number of degrees of freedom. Using a standard HPC node with 128 GB of memory, we can obtain numerically more than 5,000 reliable eigenvalues with relative errors below 10^−8 for different complex domains in two dimensions (2D). Based on these computed eigenvalues, we first confirm some theoretical results on the Robin-to-Neumann (RtN) gaps of the Laplacian operator in 2D, which were recently studied by Rudnick \textit{et al.} [\textit{Comm. Math. Phys.} 388 (2021)]. Then we systematically study the RtN gaps of the Schrödinger operator and their convergence rates. Based on our extensive numerical results, we formulate a unified conjecture on the cumulative averages of the RtN gaps of the Schrödinger operator.
ER -