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 \(2\times 2\) $2\times 2$ system of first-order h-differential operator with \(\mathcal {O}(\varepsilon )\) $\mathcal{O}(\epsilon )$ off-diagonal part is considered in 1D. Asymptotic behavior as \(\varepsilon h^{m/(m+1)}\rightarrow 0^+\) $\epsilon h^{m/(m+1)}\to 0^{+}$ 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.