Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis

We study quasinormal modes for massive scalar fields in Schwarzschild-anti-de Sitter black holes. When the mass-squared is above the Breitenlohner-Freedman bound we show that for large angular momenta, $\ell$, there exist quasinormal modes with imaginary parts of size $\exp(-\ell/C)$. We provide an asymptotic expansion for the real parts of the modes closest to the real axis and identify the vanishing of certain coefficients depending on the dimension.