We explicitly determine the high-energy asymptotics for
Weyl--Titchmarsh matrices corresponding to matrix-valued
Schr\"odinger operators associated with general
self-adjoint $m\times m$ matrix potentials
$Q\in {L^1_{\text{loc}} ((x_0,\infty))}^{m\times m}$,
where $m\in{\mathbb N}$.
More precisely, assume that for some
$N\in {\mathbb N}$ and $x_0\in{\mathbb R}$,
$Q^{(N-1)}\in L^1([x_0,c))^{m\times m}$ for all
$c>x_0$, and that $x\geq x_0$ is a right Lebesgue point of
$Q^{(N-1)}$. In addition, denote by $I_m$ the $m\times m$
identity matrix and by $C_\varepsilon$ the open sector in
the complex plane with vertex at zero, symmetry axis along
the positive imaginary axis, and opening angle
$\varepsilon$, with $0<\varepsilon< \frac12\pi$. Then we
prove the following asymptotic expansion for
any point $M_+(z,x)$ of the unique limit point or a point
of the limit disk associated with the differential
expression $-I_m\frac{d^2}{dx^2}+Q(x)$ in
${L^2((x_0,\infty))}^m$ and
a Dirichlet boundary condition at $x=x_0$:
\begin{equation}
M_+(z,x)\underset{|z|\to\infty,\, z\in
C_\varepsilon}{=} i I_m z^{1/2}+
\sum_{k=1}^N m_{+,k}(x)z^{-k/2}+
o(|z|^{-N/2}), \quad \text{where }N\in{\mathbb N}.
\nonumber
\end{equation}
The expansion is uniform with respect to $\arg\,(z)$ for
$|z|\to \infty$ in $C_\varepsilon$ and uniform in $x$ as
long as $x$ varies in compact subsets of ${\mathbb R}$
intersected with the right Lebesgue set of $Q^{(N-1)}$.
Moreover, the $m\times m$ expansion coefficients
$m_{+,k}(x)$ can be computed recursively.
Analogous results hold for matrix-valued Schr\"odinger
operators on the real line. 2000 Mathematics Subject Classification:
34E05, 34B20, 34L40, 34A55.
