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\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ H = -\frac{d^2}{dx^2} I_{m} + Q \; {\rm in} \; L^2 {(\mathbb R}^m), \; m \in {\mathbb N}. $$ \end{document}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\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ H_j = -\frac{d^2}{dx^2} I_{m} + Q_j \; {\rm in} \; L^2 {(\mathbb R}^m). $$ \end{document}Suppose that for fixed a > 0 and x 0 ∈ ℝ,\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$$ \parallel g_1(z, x_0) - g_2(z, x_0) \parallel _{{(\mathbb C}^{m \times m}} + \parallel g{^{\prime}_1} (z, x_0) - g{^{\prime}_2} (z, x_0) \parallel _{{(\mathbb C}^{m \times m}} {{=} \atop {\mid z \mid \rightarrow \infty}} O ( e^{-2{\rm Im}(z^{1/2})a} \parallel $$ \end{document}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.