  In this paper, the asymptotic behaviors of the transition probability for
two-level avoided crossings are studied under the limit where two parameters
(adiabatic parameter and energy gap parameter) tend to zero. This is a
continuation of our previous works where avoided crossings are generated by
tangential intersections and obey a non-adiabatic regime. The main results
elucidate not only the asymptotic expansion of transition probability but also
a quantum interference caused by several avoided crossings and a coexistence of
two-parameter regimes arising from different vanishing orders.
