Microlocal scattering matrix for a coupled Schroedinger operator and application to the semiclassical resonances

 We consider a 2 by 2 matrix-valued operator with two 1D Schroedinger operators on the diagonals and small in h interactions on the off-diagonals. We compute the semiclassical connection formula of the microlocal solutions (microlocal scattering matrix) at a crossing point of the two characteristic sets, and apply it to the resonance asymptotics. We will see that the subprincipal term of the microlocal scattering matrix is reduced to an oscillatory integral whose critical point is the crossing point.