Asymptotic expansion of the scattering matrix associated with matrix Schrödinger operator

Abstract

In this paper, we study the scattering theory for a 2×2 matrix Schrödinger operator

P=−h²d²/dx²I₂+V(x)+hR(x,hD_x)

on L²(R)⌖L²(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=(s_ij)_1≤i,j≤4 associated with P when the semi-classical parameter h goes to zero. Moreover, we obtain the estimate

‖S₁₂‖+‖S₂₁‖=O(e^−δ/h),

where S₁₂ and S₂₁ are the two off-diagonal elements of S and δ>0 is a constant which is explicitly related to the behavior of V(x) in the complex domain.