Uniqueness Results for Matrix - Valuedschr

Let g(z; x) denote the diagonal Green's matrix of a self-adjoint m m matrix-valued Schrr odinger operator H = ? d 2 dx 2 Im + Q(x) in L 2 (R) m , m 2 N. One of the principal results proven in this paper states that for a xed x 0 2 R and all z 2 C + , g(z; x 0) and g 0 (z; x 0) uniquely determine the matrix-valued m m potential Q(x) for a.e. x 2 R. We also prove the following local version of this result. Let g j (z; x), j = 1; 2 be the diagonal Green's matrices of the self-adjoint Schrr odinger operators H j = ? d 2 dx 2 Im + Q j (x) in L 2 (R) m. Suppose that for xed a > 0 and x 0 2 R, kg 1 (z; x 0) ? g 2 (z; x 0)k C mm + kg 0 1 (z; x 0) ? g 0 2 (z; x 0)k C mm = jzj!1 O ? e ?2Im(z 1=2)a for z inside a cone along the imaginary axis with vertex zero and opening angle less than =2, excluding the real axis. Then Q 1 (x) = Q 2 (x) for a.e. x 2 x 0 ? a; x 0 + a]. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators. This is a revised and updated version of a previously archived le.