Adiabaticity and spectral splits in collective neutrino transformations

Neutrinos streaming off a supernova core transform collectively by neutrino-neutrino interactions, leading to ``spectral splits'' where an energy Eₒ₏₋₈ₓ divides the transformed spectrum sharply into parts of almost pure but different flavors. We present a detailed description of the spectral-split phenomenon which is conceptually and quantitatively understood in an adiabatic treatment of neutrino-neutrino effects. Central to this theory is a self-consistency condition in the form of two sum rules (integrals over the neutrino spectra that must equal certain conserved quantities). We provide explicit analytic and numerical solutions for various neutrino spectra. We introduce the concept of the adiabatic reference frame and elaborate on the relative adiabatic evolution. Violating adiabaticity leads to the spectral split being ``washed out. '' The sharpness of the split appears to be represented by a surprisingly universal function.