一种计算具有Robin边界条件的Schrödinger算子众多特征值及其渐近性的谱元方法
A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition
- National University of Singapore（新加坡国立大学）
- Eastern Institute of Technology（东方理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
提出谱元法高效精确计算Robin边界条件下Schrödinger算子的数千个特征值，验证二维RtN间隙理论并系统研究其收敛性，进而提出累积平均值的统一猜想。
AI中文摘要：
我们分别针对简单和复杂几何域上具有Robin边界条件的Schrödinger算子，提出了谱方法和谱元方法，用于精确计算其众多特征值。由于这些方法在逼近对应于高指标特征值的特征函数（通常高度振荡）时具有谱级精度，所提方法在高效精确计算数千个特征值方面具有出色的分辨率特性，即具有合理精度的特征值数量与自由度数量成正比。使用一个配备128 GB内存的标准HPC节点，我们能够在二维（2D）的不同复杂域上数值获得超过5,000个可靠特征值，相对误差低于$10^{-8}$。基于这些计算得到的特征值，我们首先确认了最近由Rudnick等人[《Comm. Math. Phys.》388 (2021)]研究的二维Laplacian算子Robin-to-Neumann（RtN）间隙的一些理论结果。随后，我们系统地研究了Schrödinger算子的RtN间隙及其收敛速率。基于我们大量的数值结果，我们提出了关于Schrödinger算子RtN间隙累积平均值的统一猜想。
英文摘要：
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.
