  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.
