  We study a nonlinear generalization of the Landau-Zener resonance-crossing
problem relevant to coherent photo- and magneto-association of ultracold atoms.
Due to the structure of the corresponding classical phase space, the adiabatic
theorem breaks down even at very small sweep rates, and the adiabatic
approximation diverges because of the crossing of a separatrix. First, by
introducing a complex term into the Hamiltonian of the system, we eliminate
this divergence and construct a valid zero-order approximation. Further, taking
into account that the molecular conversion efficiency and the change of the
classical adiabatic invariant at the separatrix crossing are related
quantities, we calculate the change of the action for the situation when the
system starts from the all-atomic state that corresponds to the case of zero
initial action. The absolute error of the presented formula for the change in
the action is of the order of or less than 10^-4.
