  We propose and analyze an efficient spectral-Galerkin approximation for the
Maxwell transmission eigenvalue problem in spherical geometry. Using a vector
spherical harmonic expansion, we reduce the problem to a sequence of equivalent
one-dimensional TE and TM modes that can be solved individually in parallel.
For the TE mode, we derive associated generalized eigenvalue problems and
corresponding pole conditions. Then we introduce weighted Sobolev spaces based
on the pole condition and prove error estimates for the generalized eigenvalue
problem. The TM mode is a coupled system with four unknown functions, which is
challenging for numerical calculation. To handle it, we design an effective
algorithm using Legendre-type vector basis functions. Finally, we provide some
numerical experiments to validate our theoretical results and demonstrate the
efficiency of the algorithms.
