  We study the dynamics of a two-level crossing model with a parabolic
separation of the diabatic energies. The solutions are expressed in terms of
the tri-confluent Heun equations --- the generalization of the confluent
hypergeometric equations. We obtain analytical approximations for the state
populations in terms of Airy and Bessel functions. Applicable expressions are
derived for a large part of the parameter space. We also provide simple
formulas which connect local solution in different time regimes. The validity
of the analytical approximations is shown by comparing them to numerical
simulations.
