In the dense-neutrino region above the neutrino sphere of a supernova ($r\lesssim 400\mathrm{km}$), neutrino-neutrino refraction causes collective flavor transformations. They can lead to “spectral splits” where an energy $E_{\mathrm{split}}$ splits the transformed spectrum sharply into parts of almost pure but different flavors. Unless there is an ordinary MSW resonance in the dense-neutrino region, $E_{\mathrm{split}}$ is determined by flavor-lepton number conservation alone. Spectral splits are created by an adiabatic transition between regions of large and small neutrino density. We solve the equations of motion in the adiabatic limit explicitly and provide analytic expressions for a generic example.

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