A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schr\"{o}dinger operator

TL;DR

This paper introduces a Jacobi spectral method for efficiently computing a large number of eigenvalues of the fractional Schrödinger operator, enabling detailed statistical analysis of eigenvalue gaps in 1D and high dimensions.

Contribution

The paper develops a Jacobi spectral method tailored for large-scale eigenvalue problems of the fractional Schrödinger operator, facilitating accurate gap statistics analysis.

Findings

Numerical observations and conjectures on eigenvalue gaps in 1D FSO

Extension of the method to high-dimensional fractional Schrödinger operators

Extensive numerical results on eigenvalue distribution statistics

Abstract

We propose a spectral method by using the Jacobi functions for computing eigenvalue gaps and their distribution statistics of the fractional Schr\"{o}dinger 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. 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 Schr\"{o}dinger operator in high dimensions and extensive numerical results about eigenvalue gaps and their distribution statistics are reported.