  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.
