Uniqueness Results for Matrix-Valued Schrödinger, Jacobi, and Dirac-Type Operators

Abstract

Let $g(z,x)$ denote the diagonal Green's matrix of a self-adjoint $m\times m$ matrix-valued Schrödinger operator $H= -\f{d^2}{dx^2}I_m +Q(x)$ in $L^2(\mathbb{R}{)}^m$, $m\in \mathbb{N}$. One of the principal results proven in this paper states that for a fixed $x_0\in \mathbb{R}$ and all $z\in {\mathbb{C}}_{+}$, $g(z,x_0)$ and $g^{\prime }(z,x_0)$ uniquely determine the matrix-valued $m\times m$ potential $Q(x)$ for a.e.~$x\in \mathbb{R}$. 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 $H_j=-\f{d^2}{dx^2}I_m +Q_j(x)$ in $L^2(\mathbb{R}{)}^m$. Suppose that for fixed $a>0$ and $x_0\in \mathbb{R}$, $\Vert g_1(z,x_0)-g_2(z,x_0){\Vert }_{{\mathbb{C}}^{m\times m}}+\Vert g_1^{\prime }(z,x_0)-g_2^{\prime }(z,x_0){\Vert }_{{\mathbb{C}}^{m\times m}}\underset{|z|\to \infty }{=}O\left(e^{-2\Im (z^{1/2})a}\right)$ for $z$ inside a cone along the imaginary axis with vertex zero and opening angle less than $\pi /2$, excluding the real axis. Then $Q_1(x)=Q_2(x)$ for a.e.~$x\in [x_0-a,x_0+a]$. Analogous results are proved for matrix-valued Jacobi and Dirac-type operators.

LaTeX, 38 pages, this is a revised and updated version (to appear in Math. Nachr.)