  We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For
ellipsoids that are close to the sphere, we use analytic perturbation theory to
estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids
sufficiently close to the sphere, the first $N$ eigenvalues have multiplicity
at most two, and characterize those that are simple. For the triaxial
ellipsoids sufficiently close to the sphere that are not biaxial, we show that
at least the first sixteen eigenvalues are all simple.
  We also give the results of various numerical experiments, including
comparisons to our results from the analytic perturbation theory, and
approximations for the eigenvalues of ellipsoids that degenerate into infinite
cylinders or two-dimensional disks. We propose a conjecture on the exact number
of nodal domains of near-sphere ellipsoids.
