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