4 NH case. Briefly, the high-energy splits at ≈ 20 MeV re- main, but mixed spectra appear below [49]. One should also investigate the sensitivity of the answer to the de- tails of the spectra and what happens in full multiangle calculations. Finally, numerous applications await ex- ploration: how the collective oscillations affect the signa- tures of the shock and turbulence, whether they change the r -process and the diffuse supernova neutrino back- ground, etc . We concur with the conclusions of [48]: the physics of supernova neutrino conversion continues to surprise us with its richness. I thank J. Carlson, V. Cirigliano, and S. Reddy for feedback. I am especially grateful to H. Duan for nu- merous helpful discussions and for bringing to my atten- tion Refs. [27] and [48]. This work was supported by the LDRD program of the Los Alamos National Laboratory. different. Indeed, it is. In the two-flavor case, the final state has a swap ν$_{2}$ ↔ ν$_{3}$ on the interval 6 ≲ E$_{ν}$ ≲ 22 MeV[54]. In contrast, the final state of the three- flavor calculation is significantly more complicated, as illustrated in Figs. 4 and 5 [55]. The pattern is the same only up to E$_{ν}$ ∼ 10 MeV, with the third state ν$_{1}$ being a spectator, ν$_{1}$ → ν$_{1}$ . For 10 ≲ E$_{ν}$ ≲ 20 MeV we instead observe a cyclical permutation : ν$_{2}$ → ν$_{3}$ , ν$_{3}$ → ν$_{1}$ , ν$_{1}$ → ν$_{2}$ ; while for E$_{ν}$ ≳ 20 MeV it is the ν$_{3}$ state that is a spectator and the other two swap spectra: ν$_{1}$ ↔ ν$_{2}$ , ν$_{3}$ → ν$_{3}$ . Since the initial spectra of ν$_{1}$ and ν$_{3}$ are identical, no second split is seen at 1000 km. See [49] for more details. Figure 2 also shows another important role of Δ m 2 $_{⊙}$: as it nears its physical value, the spectrum at high energy becomes closer to the emitted ν$_{x}$ spectrum. This means the neutrinos at the end of the collective oscillations are “put” into the Hamiltonian eigenstates. This happens when the evolution is adiabatic [39, 41, 42]. Observe that adiabaticity is broken even when Δ m 2 ⊙ is 20-50% of its physical values. This means that adiabaticity for physical Δ m 2 ⊙ is only marginal in this channel. Indeed, let us compare the neutrino vacuum oscillation length to the scale height of the neutrino-neutrino poten- tial. The latter is a power law ∝ r − $^{4}$, so the scale height is | d ln H$_{νν}$/dr | − 1 ∼ r/ 4 ∼ 75 − 100 km for r ∼ 300 − 400 km. The atmospheric splitting for E ∼ 15 MeV gives a characteristic scale 2 E/ Δ m 2 atm ∼ 2 km, so a high degree of adiabaticity is expected (and seen for the low-energy ν$_{e}$ split). In contrast, for the solar splitting, 2 E/ Δ m 2 ⊙ ∼ 77 km, so the evolution is only borderline adiabatic [56]. The weakness of adiabaticity also helps to understand the ¯ ν$_{e}$ spectra: the antineutrinos are not placed into mass eigenstates. To increase adiabaticity, one can artificially increase the value of Δ m 2 $_{⊙}$. We reran the calculation with Δ m 2 ⊙ 5 and 10 times larger than its actual value. The resulting spectra, shown in Fig. 3, indeed exhibited a more “conventional” split, centered around E$_{¯}$ ν ∼ 19 MeV, although still fairly broad ( ∼ 5 MeV and ∼ 3 MeV half-widths correspondingly). The mixed ¯ ν$_{e}$ spec- trum found for physical Δ m 2 ⊙ can be thought of as an extremely broad split, with a width comparable to the entire range of the antineutrino energies. (The width of the split is related to the degree of adiabaticity, [42].) 6. Conclusions . – The late-time spectra provide an in- teresting physical system for studying collective transfor- mations. Two- and three-flavor calculations of the phe- nomenon yield qualitatively different results. 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