It has recently been realized that supernova neutrino signals may be affected
by shock propagation over a time interval of a few seconds after bounce. In the
standard three-neutrino oscillation scenario, such effects crucially depend on
the neutrino level crossing probability P_H in the 1-3 sector. By using a
simplified parametrization of the time-dependent supernova radial density
profile, we explicitly show that simple analytical expressions for P_H
accurately reproduce the phase-averaged results of numerical calculations in
the relevant parameter space. Such expressions are then used to study the
structure of P_H as a function of energy and time, with particular attention to
cases involving multiple crossing along the shock profile. Illustrative
applications are given in terms of positron spectra generated by supernova
electron antineutrinos through inverse beta decay.Comment: Major changes both in the text and in the figures in order to include
the effect of a step-like shock front density profile; final version to
appear in Physical Review
