Scattering estimates for a time dependant matricial Klein-Gordon operator.

In this paper, we study the scattering theory for a time dependent 2 x 2 matricial Klein-Gordon operator, of the type P = (root 1 - h(2)Delta(x))I-2 + V(t, x) + hR(t, x) on L-2(R-n) + L-2(R-n), where V(t, x) is a real diagonal matrix, the eigenvalues of which are never equals when (t,x) varies in Rn+1. One also assumes that V and R extend holomorphically in a complex strip around Rn+1, and satisfy to some decay properties at infinity. Then, denoting S = (S-i,S-j)(1 less than or equal to i,j less than or equal to 2) the scattering operator associated to P, we show that its off-diagonal coefficients S-1,S-2 and S-2.1 have an exponentially small norm as h tends to 0(+). More precisely, we obtain an estimate of the type O(e(-Sigma/h)), where Sigma >0 is a constant which is explicitely related to the behaviour of V(t, x) in the complex domain.