A microlocal Cauchy problem through a crossing point of Hamiltonian flows

In this paper, we consider $2\times 2$ matrix-valued pseudodifferential equations in which the two characteristic sets intersect with finite contact order. We show that the asymptotic behavior of its solution changes dramatically before and after the crossing point, and provide a precise asymptotic formula. This is a generalization of the previous results for matrix-valued Schr\"odinger operators and Landau-Zener models. The proof relies on a normal form reduction and a detailed analysis of a simple first-order system.