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We consider the Hamiltonian for neutrino oscillations in matter in the caseof arbitrary potentials including off-diagonal complex terms. We derive the'matter basis' Hamiltonian in terms of quantities one can derive from theflavor basis Hamiltonian and its derivative, for an arbitrary number ofneutrino flavors. We make our expressions explicit for the two-neutrino flavorcase and apply our results to the situation where the Hamiltonian includes bothcoupling to matter and to neutrinos, which describes neutrino propagation incore-collapse supernovae. We show that the neutrino flavour evolution dependson the mixing matrix derivatives involving not only the derivative of thematter mixing angles but also of the phases. In particular, we point out theimportant role of the phase derivatives, that appear due to theneutrino-neutrino interaction, and show how it can cause an oscillatingdegeneracy between the diagonal elements of the matter basis Hamiltonian. Ourresults also reveal, that the end of the synchronization regime is due to arapid increase of the phase derivative, and identify the condition to befulfilled for the onset of bipolar oscillations involving both the off-diagonalneutrino-neutrino interaction contributions and the vacuum terms.

The neutrino-neutrino interaction effects in supernovae: the point of view from the 'matter' basis