摘要：In this talk, I begin with the eigenvalue problem of the Laplacian and Schroedinger operators (LO/SO) on bounded domains with homogeneous Dirichlet boundary condition and present the Weyl's law and Weyl conjecture on the asymptotics of the eigenvalues. Then I discuss several cases that many (hundreds to thousands) eigenvalues need to be computed numerically and accurately. Spectral and spectral-element methods are presented to discretize the eigenvalue problem on simple and complicated geometries, respectively. Numerical results are reported for the eigenvalue asymptotics beyond the Weyl conjecture, with Rubin boundary condition, SO-LO gaps, Rubin-Neumann gaps, and for the fractional Schroedinger operators (FSO). Based on our numerical results, some conjectures are formulated. Then I review the fundamental gap conjecture -- difference between the first two smallest eigenvalues -- of the LO/SO. Again, based on our recent asymptotic and numerical results, we propose a gap conjecture on the fundamental gap of the FSO. In addition, different gaps of eigenvalues of the FSO are discussed and the ``unfolding'' gaps statistics of FSO is reported. Finally, fundamental gaps on energy and chemical potential of the Gross-Pitaevskii equation are studied asymptotically and numerically.