Non-perturbative asymptotics of the eigenvalues of the spheroidal equation

New non-perturbative results on the eigenvalues of the spheroidal equation are presented. The results, found using an all orders WKB analysis, include a perturbative/non-perturbative (P/NP) relation as well as the first exponential correction to the perturbative series which is valid in certain regions of parameters. The quantum periods are also computed.