Local scattering matrix for a degenerate avoided-crossing in the non-coupled regime

A Landau-Zener type formula for a degenerate avoided-crossing is studied in the non-coupled regime. More precisely, a $2\times2$ system of first order $h$-differential operator with $\mathcal{O}(\varepsilon)$ off-diagonal part is considered in 1D. Asymptotic behavior as $\varepsilon h^{m/(m+1)}\to0^+$ of the local scattering matrix near an avoided-crossing is given, where $m$ stands for the contact order of two curves of the characteristic set. A generalization including the cases with vanishing off-diagonals and non-Hermitian symbols is also given.