Landau-Zener problem in a three-level neutrino system with non-linear time dependence

We consider the level-crossing problem in a three-level system with non-linearly time-varying Hamiltonian (time-dependence $t^{-3}$). We study the validity of the so-called independent crossing approximation in the Landau-Zener model by making comparison with results obtained numerically in density matrix approach. We also demonstrate the failure of the so-called "nearest zero" approximation of the Landau-Zener level-crossing probability integral.