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

Resonance free domain for a system of Schr\"odinger operators with energy-level crossings