  We formulate a perturbative approach for studying a class of multi-level
time-dependent quantum systems with constant off-diagonal couplings and
diabatic energies being odd functions of time. Applying this approach to a
general multistate Landau-Zener (MLZ) model with all diabatic levels crossing
at one point (named the one-crossing MLZ model), we derive analytical formulas
of all its transition probabilities up to $4$th order in the couplings. These
analytical asymptotic solutions at the diabatic limit can serve as reliable
benchmarks for future studies of any unsolved one-crossing MLZ models.
