  We study the propagation coherence for neutrino oscillations in media with
different density profiles. For each profile, we find the dependence of the
coherence length, $L_{coh}$, on neutrino energy and address the issue of
correspondence of results in the distance and energy-momentum representations.
The key new feature in matter is existence of energy ranges with enhanced
coherence around the energies $E_0$ of "infinite coherence" at which $L_{coh}
\rightarrow \infty$. In the configuration space, the infinite coherence
corresponds to equality of the (effective) group velocities of the eigenstates.
In constant density medium, there is a unique $E_0$, which coincides with the
MSW resonance energy of oscillations of mass states and is close to the MSW
resonance energy of flavor states. In the case of massless neutrinos or
negligible masses in a very dense medium the coherence persists continuously.
In the adiabatic case, the infinite coherence is realized for periodic density
change. Adiabaticity violation changes the shape factors of the wave packets
(WPs) and leads to their spread. In a medium with sharp density changes
(jumps), splitting of the eigenstates occurs at crossing of each jump. We study
the increase of the coherence length in a single jump and periodic density
jumps - castle-wall (CW) profiles. For the CW profile, there are several $E_0$
corresponding to parametric resonances. We outlined applications of the results
for supernova neutrinos. In particular, we show that coherence between two
shock wave fronts leads to observable oscillation effects, and our analysis
suggests that the decoherence can be irrelevant for flavor transformations in
the central parts of collapsing stars.
