0 v 1 [ m at h . SP ] 1 9 A pr 2 00 0 UNIQUENESS RESULTS FOR MATRIX - VALUED SCHRÖDINGER , JACOBI , AND DIRAC - TYPE OPERATORS

Let g(z, x) denote the diagonal Green's matrix of a self-adjoint m × m matrix-valued Schrödinger operator H = − d 2 dx 2 Im + Q(x) in L 2 (R) m , m ∈ N. One of the principal results proven in this paper states that for a fixed x 0 ∈ R and all z ∈ C + , g(z, x 0) and g ′ (z, x 0) uniquely determine the matrix-valued m × m potential Q(x) for a.e. x ∈ 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 Schrödinger operators H j = − d 2 dx 2 Im + Q j (x) in L 2 (R) m. Suppose that for fixed a > 0 and x 0 ∈ R, g 1 (z, x 0) − g 2 (z, x 0) C m×m + g ′ 1 (z, x 0) − g ′ 2 (z, x 0) C m×m = |z|→∞ O(e −2Im(z 1/2)a) for z inside a cone along the imaginary axis with vertex zero and opening angle less than π/2, excluding the real axis. Then Q 1 (x) = Q 2 (x) for a.e. x ∈ [x 0 − a, x 0 + a]. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators .