A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator
A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator
复制标题
计算分数阶薛定谔算子特征值间隙及其分布统计的雅可比谱方法
DOI：
10.1016/j.jcp.2020.109733
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发表时间：
2019-10
影响因子：
4.1
通讯作者：
Ying Ma
中科院分区：
文献类型：
--
作者：
Weizhu Bao;Lizhen Chen;Xiaoyun Jiang;Ying Ma
提出了一种利用Jacobi函数计算分数阶薛定谔算子（FSO）本征值间隙及其分布统计量的谱方法。在该问题中，为了得到可靠的间隙分布统计，我们必须准确有效地计算与FSO相关的特征值问题的大量特征值，例如高达数千甚至数百万的特征值。为了简单起见，我们从一维（1D）的FSO特征值问题开始，将其转化为变分公式，然后使用Jacobi谱方法对其进行离散。数值结果表明，本文提出的Jacobi谱方法与现有的有限差分法和有限元法相比，具有如下优点：（i）Jacobi谱方法是谱精确的，而有限差分法和有限元法仅为一阶精度;更重要的是（ii）在固定的自由度M下，Jacobi谱方法可以精确地计算出数量与M成正比的大量特征值，而当需要计算大量特征值时，FDM和FEM的性能较差。因此，建议的Jacobi谱方法是非常适合的，并要求离散的特征值问题时，需要计算大量的特征值。然后应用Jacobi谱方法数值研究了最近邻间隙、平均间隙、最小间隙、归一化间隙及其分布统计量的渐近性。基于我们的数值计算结果，给出了一维自由空间光本征能隙及其分布统计的一些有趣的数值观测（或描述）。最后，将Jacobi谱方法推广到高维的方向分数阶Schr dinger算子，得到了关于特征值间隙及其分布统计的大量数值结果.
We propose a spectral method by using the Jacobi functions for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator (FSO). In the problem, in order to get reliable gaps distribution statistics, we have to calculate accurately and efficiently a very large number of eigenvalues, e.g. up to thousands or even millions eigenvalues, of an eigenvalue problem related to the FSO. For simplicity, we start with the eigenvalue problem of the FSO in one dimension (1D), reformulate it into a variational formulation and then discretize it by using the Jacobi spectral method. Our numerical results demonstrate that the proposed Jacobi spectral method has several advantages over the existing finite difference method (FDM) and finite element method (FEM) for the problem: (i) the Jacobi spectral method is spectral accurate, while the FDM and FEM are only first order accurate; and more importantly (ii) under a fixed number of degree of freedomsM, the Jacobi spectral method can calculate accurately a large number of eigenvalues with the number proportional toM, while the FDM and FEM perform badly when a large number of eigenvalues need to be calculated. Thus the proposed Jacobi spectral method is extremely suitable and demanded for the discretization of an eigenvalue problem when a large number of eigenvalues need to be calculated. Then the Jacobi spectral method is applied to study numerically the asymptotics of the nearest neighbour gaps, average gaps, minimum gaps, normalized gaps and their distribution statistics in 1D. Based on our numerical results, several interesting numerical observations (or conjectures) about eigenvalue gaps and their distribution statistics of the FSO in 1D are formulated. Finally, the Jacobi spectral method is extended to the directional fractional Schrödinger operator in high dimensions and extensive numerical results about eigenvalue gaps and their distribution statistics are reported.
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DOI：
10.1112/s0025579318000165
发表时间：
2017-10
期刊：
arXiv: Number Theory
影响因子：
--
作者：
Z. Rudnick
通讯作者：
Z. Rudnick
DOI：
10.1088/0305-4470/30/6/009
发表时间：
1997-03
期刊：
Journal of Physics A
影响因子：
--
作者：
R. Connors;J. Keating
通讯作者：
R. Connors;J. Keating
影响因子：
3.9
作者：
B. Andrews;J. Clutterbuck
通讯作者：
B. Andrews;J. Clutterbuck
DOI：
--
发表时间：
2018-03
期刊：
arXiv: Numerical Analysis
影响因子：
--
作者：
Lizhen Chen;Zhiping Mao;Hui-yuan Li
通讯作者：
Lizhen Chen;Zhiping Mao;Hui-yuan Li
DOI：
10.1137/13091436x
发表时间：
2013-11
期刊：
SIAM J. Appl. Math.
影响因子：
--
作者：
W. Bao;H. Jian;N. Mauser;Yong Zhang
通讯作者：
W. Bao;H. Jian;N. Mauser;Yong Zhang