Asymptotic expansion of the scattering matrix associated with matrix Schroedinger operator

摘要

In this paper, we study the scattering theory for a 2×2 matrix Schrodinger operator P = -h~2 d~2/dx~2I_2+v(x)+hR(x,hD_x) on L~2(R) ⊕ L~2(R), where V(x) is a real diagonal matrix, the eigenvalues of which are never equal. Under some assumptions of analyticity and decay at infinity of V, we describe the asymptotic behavior of the scattering matrix S = (sij)1≤i,j≤4 associated with P when the semi-classical parameter h goes to zero. Moreover, we obtain the estimate where S_(12)and S_(21) are the two off-diagonal elements of S and 5 > 0 is a constant which is explicitly related to the behavior of V(x) in the complex domain.