  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.
