A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator

Abstract

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.