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 freedoms M, the Jacobi spectral method can calculate accurately a large number of eigenvalues with the number proportional to M, 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.

我们提出了一种使用Jacobi函数来计算特征值间隙及其分数Schrödinger算子（FSO）的分布统计量的频谱方法。在问题中，为了获得可靠的缺口分布统计数据，我们必须准确而有效地计算与FSO相关的特征值问题的大量特征值，例如，多达数千甚至数百万个特征值。为简单起见，我们从一维（1D）FSO的特征值问题开始，将其重新构造为变分公式，然后使用Jacobi谱方法将其离散化。我们的数值结果表明，与现有的有限差分法（FDM）和有限元方法（FEM）相比，所提出的Jacobi谱方法具有以下优点：（i）Jacobi频谱方法是频谱准确的，而FDM和FEM只是一阶的；更重要的是（ii）在一定数量的自由度下M，雅可比谱方法可以精确计算大量与M成正比的特征值。，而当需要计算大量特征值时，FDM和FEM表现不佳。因此，当需要计算大量特征值时，提出的Jacobi谱方法非常适合特征值问题的离散化。然后，利用Jacobi谱方法对一维最近邻间隙，平均间隙，最小间隙，归一化间隙及其分布统计量的渐近性进行数值研究。根据我们的数值结果，提出了一些有趣的数值观测（或猜想），它们涉及特征值差距及其在一维中FSO的分布统计。最后，