Bi-maximal mixing of three neutrinos

We show that if the solar and atmospheric data are both described by maximal vacuum oscillations at the relevant mass scales then there exists a unique mixing matrix for three neutrino flavors. The solution necessarily conserves CP and automatically implies that there is no disappearance of atmospheric νe, consistent with indications from the Super-Kamiokande experiment. We also investigate the consequences for three-neutrino mixing if the solar and atmospheric oscillations exhibit mixing that is large but not maximal. For non-maximal mixing νe↔ντ and νe↔νμ oscillations are predicted that may be observable in future long-baseline experiments.