Non-adiabatic level crossing in (non-) resonant neutrino oscillations

We study neutrino oscillations and the level-crossing probability $P_{LZ}=\exp(-\gamma_n\F_n\pi/2)$ in power-law like potential profiles $A(r)\propto r^n$. After showing that the resonance point coincides only for a linear profile with the point of maximal violation of adiabaticity, we point out that the ``adiabaticity'' parameter $\gamma_n$ can be calculated at an arbitrary point if the correction function $\F_n$ is rescaled appropriately. We present a new representation for the level-crossing probability, $P_{LZ}=\exp(-\kappa_n\G_n)$, which allows a simple numerical evaluation of $P_{LZ}$ in both the resonant and non-resonant cases and where $\G_n$ contains the full dependence of $P_{LZ}$ on the mixing angle $\theta$. As an application we consider the case $n=-3$ important for oscillations of supernova neutrinos.