F. Gesztesy, A. Kiselev, and K. A. Makarov
Uniqueness Results for Matrix-Valued Schr\"odinger, Jacobi, and
Dirac-Type Operators
(136K, LaTeX)
ABSTRACT. Let $g(z,x)$ denote the diagonal Green's matrix of a self-adjoint
$m\times m$ matrix-valued Schr\"odinger operator $H=
-\f{d^2}{dx^2}I_m +Q(x)$ in $L^2 (\bbR)^{m}$, $m\in\bbN$. One
of the principal results proven in this paper states that for a
fixed $x_0\in\bbR$ and all $z\in\bbC_+$, $g(z,x_0)$ and
$g^\prime (z,x_0)$ uniquely determine the matrix-valued
$m\times m$ potential $Q(x)$ for a.e.~$x\in\bbR$. 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
Schr\"odinger operators $H_j=-\f{d^2}{dx^2}I_m +Q_j(x)$ in $L^2
(\bbR)^{m}$. Suppose that for fixed $a>0$
and $x_0\in\bbR$, $\|g_1(z,x_0)-g_2(z,x_0)\|_{\bbC^{m\times m}}+
\|g_1^\prime (z,x_0)-g_2^\prime (z,x_0)\|_{\bbC^{m\times m}}
\underset{|z|\to\infty}{=}O\big(e^{-2\Im(z^{1/2})a}\big)$ 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.
This is a revised and updated version of a previously archived file
(to appear in Math. Nachr.).