  We consider a $2\times 2$ system of 1D semiclassical differential operators
with two Schr\"odinger operators in the diagonal part and small interactions of
order $h^\nu$ in the off-diagonal part, where $h$ is a semiclassical parameter
and $\nu$ is a constant larger than $1/2$. We study the absence of resonance
near a non-trapping energy for both Schr\"odinger operators in the presence of
crossings of their potentials. The width of resonances is estimated from below
by $Mh\log(1/h)$ and the coefficient $M$ is given in terms of the directed
cycles of the generalized bicharacteristics induced by two Hamiltonians.
