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\begin{document}
\title[Weyl-Titchmarsh Matrix]{Weyl-Titchmarsh
$M$-Function Asymptotics for Matrix-Valued Schr\"odinger
Operators}
\author[Clark and Gesztesy]{Steve Clark and Fritz Gesztesy}
\address{Department of Mathematics and Statistics, University
of Missouri-Rolla, Rolla, MO 65409, USA}
\email{sclark@umr.edu}
\urladdr{http://www.umr.edu/\~{ }clark}
\address{Department of Mathematics,
University of
Missouri, Columbia, MO
65211, USA}
\email{fritz@math.missouri.edu\newline
\indent{\it URL:}
http://www.math.missouri.edu/people/fgesztesy.html}
%\date{\today}
\subjclass{Primary 34E05, 34B20, 34L40; Secondary 34A55}
\thanks{{\it Proc. London Math. Soc.} {\bf 82}, 701--724 (2001).}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We explicitly determine the high-energy asymptotics for
Weyl-Titchmarsh matrices associated with general
matrix-valued Schr\"odinger operators on a half-line.\\
\noindent This is a revised and updated version of a
previously archived file.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\lb{s1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The high-energy asymptotics $z\to\infty$ of scalar-valued
Weyl-Titchmarsh functions $m_+(z,x_0)$ associated with general
half-line Schr\"odinger differential expressions of the type
$-\f{d^2}{dx^2}+q(x)$, with real-valued coefficients $q^{(n)}\in
L^1([x_0,c])$ for some $n\in\bbN_0(=\bbN\cup\{0\})$ and
all $c>x_0$, received enormous attention over the past three
decades as can be inferred, for instance from \cite{At81},
\cite{At88}, \cite{Be88}, \cite{Be89}, \cite{BM97}, \cite{DL91},
\cite{Ev72}, \cite{EH78}, \cite{EHS83}, \cite{Ha83}--\cite{Ha87},
\cite{Hi69}, \cite{HKS89}, \cite{KK86}, \cite{KK87}, \cite{Si98}
and the literature therein. Hence it may perhaps come as a surprise
that the corresponding matrix extension of this problem,
considering general matrix-valued differential expressions of the
type $-I_m\f{d^2}{dx^2}+Q(x)$, with $I_m$ the identity matrix in
$\bbC^m$, $m\in\bbN$, and $Q(x)$ a self-adjoint matrix satisfying
$Q^{(n)}\in L^1([x_0,c])^{m\times m}$ for some $n\in\bbN_0$ and
all $c>x_0$, received hardly any attention at all. In fact, we are
only aware of a total of two papers devoted to Weyl-Titchmarsh
$M$-function asymptotics for general half-line matrix
Schr\"odinger operators: one by Clark \cite{Cl92} in the case
$m=2$, $n=0$, and a second one, an apparently unpublished manuscript
by Atkinson \cite{At88} in the case $m\in\bbN$, $n=0$. Both authors
are solely concerned with determining the leading term in the
asymptotic high-energy expansion of the Weyl-Titchmarsh matrix
$M_+(z,x_0)$ as
$z\to\infty$ without investigating higher-order expansion
coefficients. (It should be noted that this observation discounts
papers in the special scattering theoretic case concerned with
short-range coefficients $Q^{(n)}\in L^1([x_0,\infty);
(1+|x|)dx)^{m\times m}$, where straightforward iterations of
Volterra-type integral equations yield the asymptotic high-energy
expansion of $M_+(z,x_0)$ to any order, cf.~Lemma~\ref{4.1}.) A
quick look at Atkinson's preprint reveals the nontrivial
nature of determining even the leading-order term of the
asymptotic expansion of $M_+(z,x_0)$ as $z\to\infty$,
\begin{equation}
M_+(z,x_0)\underset{z\to \infty}{=} iz^{1/2}I_m + o(|z|^{1/2})
\lb{1.1}
\end{equation}
for $z$ in sectors in the open upper half-plane due to a variety of
variable transformations, which finally reduce the
problem to the study of a managable Riccati system of differential
equations.
Our principal motivation in studying this problem stems from our
general interest in operator-valued Herglotz functions
(cf.~\cite{Ca76}, \cite{GKMT98}, \cite{GM99}, \cite{GM99a},
\cite{GMN98}, \cite{GMT98}, \cite{GT97},
\cite{Sh71}) and its possible applications in the areas of inverse
spectral theory and completely integrable systems. More precisely,
using higher-order expansions of the type
\eqref{1.1}, one can prove trace formulas for $Q(x)$ and certain
higher-order differential polynomials in
$Q(x)$, following an approach pioneered in \cite{GS96} (see also
\cite{GH97}, \cite{GHSZ95}). These trace formulas, in
turn, then can be used to prove various results in inverse spectral
theory for matrix-valued Schr\"odinger operators
$H=-I_m\f{d^2}{dx^2}+Q$ in $L^2(\bbR)^m$. For instance, using the
principal result of this paper, Theorem~\ref{t4.6}, and its
straightforward application to the asymptotic high-energy
expansion of the diagonal Green's matrix
$G(z,x,x)=(H-z)^{-1}(x,x)$ of $H$, the following matrix-valued
extension of a classical uniqueness result of Borg \cite{Bo46} has
been obtained in \cite{CGHL99}.
\begin{theorem} [\cite{CGHL99}] \lb{t1.1} Suppose $H$ is
reflectionless {\rm (}e.g., $Q$ is periodic and H has uniform spectral
multiplicity $2m${\rm )} and spectrum $[E_0,\infty)$ for some
$E_0\in\bbR$. Then
\begin{equation}
Q(x)=E_0I_m \text{ for a.e.~$x\in\bbR$}. \lb{1.2}
\end{equation}
\end{theorem}
For related results see, for instance,
\cite{AK92},
\cite{Ca98}, \cite{Ca98a}, \cite{Ca99}, \cite{Ch99}, \cite{Ch99a},
\cite{CS97}, \cite{CGR99}, \cite{De95}, \cite{JL97}, \cite{Kr83},
\cite{Ma94}, \cite{Ma98}, \cite{Pa95}, \cite{Sa94}. Incidentally,
the higher-order differential polynomials in $Q(x)$ just alluded to
represent the Korteweg-deVries (KdV) invariants (i.e., densities
associated with KdV conservation laws) and hence open the link to
infinite-dimensional completely integrable systems (cf.
\cite{AG98},
\cite{Ch96}, \cite{Di91}, \cite{DK97}, \cite{Du77}, \cite{Du83},
\cite{GD77}, \cite{GKS97}, \cite{Ma78}, \cite{Ma88}, \cite{MO82},
\cite{OMG81}, \cite{Sa94a}, \cite{Sc83} and the references therein).
In Section~\ref{s2} we briefly recall some of the basic notions
of Weyl-Titchmarsh theory for singular matrix Schr\"odinger
operators on a half-line as developed in detail by Hinton and
Shaw \cite{HS81}--\cite{HS86} (see also \cite{At64}, \cite{Hi69},
\cite{Jo87}, \cite{KR74}, \cite{KS88}, \cite{Kr89a}, \cite{Kr89b},
\cite{Or76},
\cite{Re96}, \cite{Ro60}, \cite{Ro69}, \cite{Sa94a},
\cite{We87}). In fact, most of these references deal with more
general singular Hamiltonian systems and hence we here specialize
this material to the matrix-valued Schr\"odinger operator case at
hand. The subsequent section is devoted to a proof of the
leading-order asymptotic high-energy expansion
\eqref{1.1} of $M_+(z,x_0)$, originally due to Atkinson. Since
the result is a fundamental ingredient for our principal
Section~\ref{s4}, and to the best of our knowledge, his manuscript
\cite{At88} remained unpublished, we provide a detailed
description of his approach in Section~\ref{s3}. Finally,
Section~\ref{s4} develops a systematic higher-order high-energy
asymptotic expansion of $M_+(z,x_0)$, combining Atkinson's result
in Section~\ref{s3} with matrix-valued extensions of some methods
based on an associated Riccati equation. More precisely,
following a technique in \cite{GS98} in the scalar-valued context,
we show how to derive the general high-energy asymptotic expansion
of
$M_+(z,x_0)$ as $z\to\infty$ by combining Atkinson's
leading-order term in \eqref{1.1} and the corresponding asymptotic
expansion of $M_+(z,x_0)$ in the special case where $Q$ has
compact support.
Analogous results for Dirac-type operators are in preparation
\cite{CG00}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Weyl-Titchmarsh Matrices}
\lb{s2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we briefly recall the Weyl--Titchmarsh
theory for matrix-valued Schr\"{o}dinger
operators on half-lines $(x_0,\infty)$ for some
$x_0\in\bbR$.
Throughout this note all $m\times m$ matrices
$M\in\bbC^{m\times m}$ will be considered over
the field of complex numbers $\bbC$. Moreover, $I_p$ denotes
the identity matrix in $\bbC^{p\times p}$ for $p\in\bbN$, $M^*$
the adjoint (i.e., complex
conjugate transpose), $M^t$ the transpose of the matrix
$M$, and $\AC([a,b])$ denotes the
set of absolutely continuous functions on $[a,b]$.
The basic assumption of this paper will be the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{hypothesis}\lb{h2.1}
Let $m\in\bbN$ and suppose
$Q=Q^*\in L^1([x_0,c])^{m\times m}$ for all $c>x_0$.
\end{hypothesis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given Hypothesis~\ref{h2.1} we consider the matrix-valued
Schr\"odinger operator $H_+$ in $L^2((x_0,\infty))^m$,
\begin{align}
&H_+=-I_m\f{d^2}{dx^2} + Q, \lb{2.1} \\
&\dom(H_+)=\{g\in L^2((x_0,\infty))^m \,|\,g,g'\in
AC([x_0,c])^m
\text{ for all } c>x_0; \lim_{x\downarrow x_0} g(x)=0, \no \\
& \hspace*{5.4cm} \text{s.-a.b.c. at }\infty; (-I_mg''+Qg)\in
L^2((x_0,\infty))^m \}, \no
\end{align}
where the abbreviation s.-a.b.c. denotes a fixed
self-adjoint boundary condition at $\infty$ throughout this
paper if the differential expression
$-I_m\f{d^2}{dx^2} + Q(x)$ is not in the limit point case at
$\infty$ (cf.~the paragraph following Definition~\ref{d2.7}) and is
disregarded in the case where
$-I_m\f{d^2}{dx^2} + Q(x)$ is in the limit point case at $\infty$.
Naturally associated with the operator $H_+$ is the matrix
Schr\"odinger equation
\begin{subequations}\lb{2.2}
\begin{equation}\lb{2.2a}
-\psi''(z,x) + Q(x)\psi(z,x) = z \psi(z,x), \quad
z\in\bbC, \, x\geq x_0,
\end{equation}
where $z$ represents a spectral parameter and $\psi(z,x)$
is assumed to satisfy
\begin{equation}\lb{2.2b}
\psi'(z,\cdot)\in AC([x_0,c])^{m\times r} \text{ for all } c>0,
\end{equation}
with the value of $r$ depending upon the context involved.
\end{subequations}
Equation \eqref{2.2a} may also be expressed as the system
\begin{subequations}\lb{2.3}
\begin{equation}\lb{2.3a}
J\varPsi (z,x)'=(zA(x)+B(x))\varPsi (z,x), \quad z\in\bbC, \, x\geq x_0,
\end{equation}
where
\begin{equation}\lb{2.3b}
\varPsi(z,x) = \begin{bmatrix}\psi(z,x)\\
\psi'(z,x) \end{bmatrix},\hspace{10pt}J=\begin{bmatrix}
0& -I_m\\ I_m
& 0 \end{bmatrix},
\end{equation}
\begin{equation}\lb{2.3c}
A(x) =\begin{bmatrix}I_m& 0\\0& 0\end{bmatrix},
\hspace{10pt} B(x)=\begin{bmatrix}-Q(x)& 0\\0& I_m\end{bmatrix},
\end{equation}
and where
\begin{equation}\lb{2.3d}
\varPsi(z,\cdot)\in AC([x_0,c])^{2m\times r} \text{ for all } c>0;
\end{equation}
again, with the value of $r$ depending upon the context involved.
\end{subequations}
Next we turn to Weyl-Titchmarsh theory associated
with \eqref{2.1} and briefly recall some
of the results developed by Hinton and Shaw in a series
of papers devoted to spectral theory of
(singular) Hamiltonian systems \cite{HS81}--\cite{HS86} (see
also \cite{At64}, \cite{Hi69},
\cite{Jo87}, \cite{KR74}, \cite{KS88}, \cite{Kr89a}, \cite{Kr89b},
\cite{Or76},
\cite{Re96}, \cite{Ro60}, \cite{Ro69}, \cite{Sa94a},
\cite{We87}). While they discuss
much more general systems, we here confine ourselves to
the special case of matrix-valued Schr\"{o}dinger
operators
governed by Hypothesis~\ref{h2.1}.
Let $\Psi(z,x,x_0)$ be a normalized fundamental matrix of
solutions of
\eqref{2.3} at some
$x_0\in\bbR$; that is, $\Psi(z,x,x_0)$ is of the
type
\begin{subequations}\lb{2.4}
\begin{align}
\Psi(z,x,x_0)&=[\psi_{j,k}(z,x,x_0)]_{j,k=1}^2 \notag\\
&= \begin{bmatrix}\Theta(z,x,x_0)& \Phi(z,x,x_0) \end{bmatrix} =
\begin{bmatrix}\theta(z,x,x_0) & \phi(z,x,x_0)\\
\theta'(z,x,x_0)& \phi'(z,x,x_0)\end{bmatrix}, \lb{2.4a} \\
\Psi(z,x_0,x_0)&=I_{2m}, \lb{2.4b}
\end{align}
where $\Theta(z,x,x_0),\ \Phi(z,x,x_0)\in \bbC^{2m\times m}$,
and where $\theta(z,x,x_0)$ and $\phi(z,x,x_0)$ represent a
fundamental system of $m\times m$
matrix-valued solutions of \eqref{2.2}, entire with
respect to $z\in\bbC$, and normalized according to
\eqref{2.4b}; that is,
\begin{equation}
\theta(z,x_0,x_0)=\phi'(z,x_0,x_0)=I_m, \quad
\theta'(z,x_0,x_0)=\phi(z,x_0,x_0)=0. \lb{2.4c}
\end{equation}
\end{subequations}
Let $\beta=[\beta_1\ \beta_2]$; with $\beta_1$, $\beta_2\in\bbC^{m\times
m}$. We assume
\begin{subequations}\lb{2.5}
\begin{equation}\lb{2.5a}
\ker{(\beta_1^*)}\cap\ker{(\beta_2^*)}=\{0\},
\text{ or equivalently that } \rank(\beta) = m,
\end{equation}
and that either
\begin{equation}\lb{2.5b}
\pm(1/2i)\beta J \beta^*=\pm\Im{(\beta_2\beta_1^*)> 0}, \quad\text{or}\quad
\Im{(\beta_2\beta_1^*) = 0}.
\end{equation}
\end{subequations}
Here we denote, as usual,
$\Im(M)=(M-M^*)/(2i)$ and $\Re(M)=(M+M^*)/2$.
We now prove the following result concerning $\Phi(z,x,x_0)$.
%%%%%%%%%%%%%%%%%%%lemma%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l2.2}
Let $\Phi(z,x,x_0)$ be defined in \eqref{2.4},
and let $\beta\in\bbC^{m\times 2m}$ satisfy
\eqref{2.5}. If $\beta\Phi(z,c,x_0)=\beta_1\phi(z,c,x_0)
+\beta_2\phi'(z,c,x_0)$ is singular for $c>x_0$ and
$\Im{(\beta_2\beta_1^*)}\gtrless 0$,
then $\Im{(z)}\lessgtr 0$. If $\beta\Phi(z,c,x_0)$ is singular for
$c>x_0$ and $\Im{(\beta_2\beta_1^*)}= 0$, then $\Im{(z)}= 0$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $\varPsi(z,x)$ satisfy \eqref{2.3}. Then,
\begin{subequations}\lb{2.6}
\begin{equation}\lb{2.6a}
\varPsi(z,c)^*J\varPsi(z,c) - \varPsi(z,x_0)^*J \varPsi(z,x_0)
=2i\Im{(z)}\int_{x_0}^c ds\, \varPsi(z,s)^*A(s)\varPsi(z,s).
\end{equation}
Let $\varPsi(z,x)=\Phi(z,x,x_0)$ and note that
\begin{equation}\lb{2.6b}
\varPsi(z,x_0)^*J\varPsi(z,x_0)=[0 \ I_m]J[0 \ I_m]^t=0.
\end{equation}
Moreover, note that
\begin{equation}\lb{2.6c}
\varPsi(z,c)^*J\varPsi(z,c)=
\phi'(z,c,x_0)^*\phi(z,c,x_0)-\phi(z,c,x_0)^*\phi'(z,c,x_0).
\end{equation}
\end{subequations}
Now suppose that $\beta\Phi(z,c,x_0)v =0$ for
$v\in\bbC^m$, $v\ne 0$. If $\Im{(\beta_2\beta_1^*)}\gtrless 0$, then
$\phi'(z,c,x_0)v=-\beta_2^{-1}\beta_1\phi(z,c,x_0)v$. Thus by
\eqref{2.6} we obtain,
\begin{align}
&v^*\phi(z,c,x_0)^*\beta_2^{-1}[\Im{(\beta_2\beta_1^*)}]
{\beta_2^*}^{-1}
\phi(z,c,x_0)v \no \\
&= -\Im{(z)} \int_{x_0}^c ds \ v^*
\phi(z,s,x_0)^*\phi(z,s,x_0)v,
\end{align}
by which we obtain the first part of the lemma. However, if
$ 0=\Im{(\beta_2\beta_1^*)}=(1/2i)\beta J \beta^*$ then by \eqref{2.5a}
there is a
$w\in \bbC^m$ such that
\begin{equation}
\Phi(z,c,x_0)v =J\beta^*w.
\end{equation}
For this case, by \eqref{2.6} we obtain
\begin{equation}
0=\Im{(z)} \int_{x_0}^c ds \ v^*
\phi(z,s,x_0)^*\phi(z,s,x_0)v,
\end{equation}
which implies that $\Im{(z)}=0$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\noindent One can also prove the following result.
%%%%%%%%%%%%%%%%%%%lemma%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\lb{l2.3}
Let $\Theta(z,x,x_0) $ and $\Phi(z,x,x_0)$ be defined in \eqref{2.4},
and let $\beta\in\bbC^{m\times 2m}$ satisfy
\eqref{2.5}. For $c>x_0$,
$\beta\Phi(z,c,x_0)$ is singular if and only if $z$ is an
eigenvalue for the regular boundary value
problem given
by \eqref{2.3} together with the boundary conditions
\begin{equation}
[I_m \ 0]\varPsi(z,x_0)=0, \quad \beta\varPsi(z,c)=0.
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Lemmas~\ref{l2.2} and \ref{l2.3} show that with appropriate values of
$z\in\bbC$ one may define a certain $m\times m$ matrix.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\lb{d2.4}
Let $\Theta(z,x) $ and $\Phi(z,x)$ be defined in \eqref{2.4}, and let
$\beta\in\bbC^{m\times 2m}$ satisfy \eqref{2.5}. Then, for $c>x_0$ and
for $\beta\Phi(z,c,x_0)$ nonsingular, let
\begin{equation}\lb{2.11}
M(z,c,x_0,\beta) =
-(\beta\Phi(z,c,x_0))^{-1}(\beta\Theta(z,c,x_0)).
\end{equation}
$M(z,c,x_0,\beta)$ is said to be the {\it Weyl-Titchmarsh M-function} for
the regular boundary value problem described in Lemma~\ref{l2.3}.
Also, let $D(z,c,x_0)$ denote the set of all $M(z,c,x_0,\beta)$ defined
in \eqref{2.11} together with the assumption \eqref{2.5b}.
\end{definition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
About $M(z,c,x_0,\beta)$, we prove the following result.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l2.5}
Let $c>x_0$. If $\Im{(\beta_2\beta_1^*)}>0$, then
$M(z,c,x_0,\beta) $ is well-defined if $\Im{(z)}\ge 0$; if
$\Im{(\beta_2\beta_1^*)}<0$, then $M(z,c,x_0,\beta) $ is well-defined for
$\Im{(z)}\le 0$; and if $\Im{(\beta_2\beta_1^*)}=0$ then
$M(z,c,x_0,\beta) $ is well-defined for $z\in \bbC\backslash\bbR$.
Moreover, if $\Im{(\beta_2 \beta_1^*)}\gtreqless 0 $ and
$ \Im{(z)}\gtrless 0 $, then
\begin{equation}\lb{2.12}
\Im{ (\pm M(z,c,x_0,\beta))} > 0.
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Except for the conditions given in \eqref{2.12}, the claims
made in the statement of this lemma are an immediate consequence of
Lemma~\ref{l2.2}.
To verify the conditions given in
\eqref{2.12}, we begin by letting
\begin{equation}\lb{2.13}
U(z,x,x_0)=
\begin{bmatrix}\Theta(z,x,x_0)& \Phi(z,x,x_0) \end{bmatrix}\begin{bmatrix}
I_m\\
M \end{bmatrix},
\end{equation}
where $\Theta(z,x,x_0) $ and $\Phi(z,x,x_0)$ are
defined in \eqref{2.4}, and where $M\in\bbC^{m\times m}$.
Now, let $\Psi(z,x)=U(z,x,x_0) $ in \eqref{2.6a} and note that
\begin{equation}\lb{2.14}
\Psi(z,x_0)^*J\Psi(z,x_0)= 2i\Im (M).
\end{equation}
Moreover, if $M=M(z,c,x_0,\beta)$, then also note that
\begin{equation}\lb{2.15}
\beta \Psi(z,c)=\beta U(z,c,x_0)=0.
\end{equation}
If $(1/2i)\beta J \beta^*=\Im{(\beta_2\beta_1^*)}=0$, then by \eqref{2.5a}
and \eqref{2.15}
there is a matrix $C\in\bbC^{m\times m}$ such that $\Psi(z,c)=
U(z,c,x_0)=J\beta^*C$. Thus,
\begin{equation} \lb{2.16}
\Psi(z,c)^*J\Psi(z,c)= C^* \beta J \beta^*C =0,
\end{equation}
and \eqref{2.12} follows immediately from \eqref{2.14}, \eqref{2.16} and
\eqref{2.6a} when $\Im{(z)}\ne 0$.
On the other hand, if $\Im{(\beta_2\beta_1^*)}\gtrless 0$, then
\eqref{2.15} implies that $u'(z,c) = -\beta_2^{-1}\beta_1 u(z,c)$. As a
result,
\begin{equation}\lb{2.17}
\Psi(z,c)^*J \Psi(z,c) =
-2iu(z,c)^*\beta_2^{-1}\Im{(\beta_2\beta_1^*)}{\beta_2^*}^{-1}u(z,c),
\end{equation}
and again \eqref{2.12} follows from \eqref{2.14}, \eqref{2.17}, and
\eqref{2.6a} when $\Im{(z)}\ne 0$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark}\lb{r2.6}
By Lemma~\ref{l2.5}, $M(z,c,x_0,\beta)$ is a Herglotz
function of rank $m$. Moreover, $D(z,c,x_0)$ is
contained in the standard {\em Weyl disk} of matrices associated
with \eqref{2.2} or
\eqref{2.3} for fixed values of $z$,
$c$, and $x_0$ (cf.~\cite{CG00}).
These disks nest with respect to increasing
values of $c$; that is, $D(z,c_2,x_0)\subseteq D(z,c_1,x_0)$
whenever $x_0 0, \quad z\in\bbC_+,
\lb{2.18} \\
M_+(\bar z,x_0)=M_+( z,x_0)^*,\lb{2.19} \\
\rank (M_+(z,x_0))=m, \lb{2.20} \\
\lim_{\varepsilon\downarrow 0} M_+(\lambda+
i\varepsilon,x_0) \text{ exists for a.e.\
$\lambda\in\bbR$}. \lb{2.21}
\end{gather}
Isolated poles of $M_+(z,x_0)$ and $-M_+(z,x_0)^{-1}$
are at most of first order,
are real, and have a nonpositive residue. \\
(ii) $M_+(z,x_0)$ admits the representations
\begin{align}
M_+(z,x_0)&=F_+(x_0)+\int_\bbR
d\Omega_+(\lambda,x_0) \,
\big((\lambda-z)^{-1}-\lambda(1+\lambda^2)^{-1}\big)
\lb{2.22} \\
&=\exp\bigg(C_+(x_0)+\int_\bbR d\lambda \, \Xi_+
(\lambda,x_0)
\big((\lambda-z)^{-1}-\lambda(1+\lambda^2)^{-1}\big)
\bigg), \lb{2.23}
\end{align}
where
\begin{align}
F_+(x_0)&=F_+(x_0)^*, \quad \int_\bbR \f{\Vert
d\Omega_+(\lambda,x_0)\Vert}{1+\lambda^2}<\infty,
\lb{2.24} \\
C_+(x_0)&=C_+(x_0)^*, \quad 0\le\Xi_+(\dott,x_0)
\le I_m \, \rm{ a.e.}
\lb{2.25}
\end{align}
Moreover,
\begin{align}
\Omega_+((\lambda,\mu],x_0)&=\lim_{\delta\downarrow
0}\lim_{\varepsilon\downarrow 0}\f1\pi
\int_{\lambda+\delta}^{\mu+\delta} d\nu \, \Im(
M_+(\nu+i\varepsilon,x_0)), \lb{2.26} \\
\Xi_+(\lambda,x_0)&=\lim_{\varepsilon\downarrow 0}
\f1\pi\Im(\ln(M_+(\lambda+i\varepsilon,x_0)) \text{ for
a.e.\ $\lambda\in\bbR$}.\lb{2.27}
\end{align}
(iii) With $\phi(z,x,x_0)$ and $\theta(z,x,x_0)$ given in \eqref{2.4},
define the $m\times m$ matrices
\begin{equation}
u_+(z,x,x_0)=\theta(z,x,x_0) + \phi(z,x,x_0) M_+(z,x_0),
\quad x,x_0\in\bbR, \, z\in\bbC\backslash\bbR.
\lb{2.28}
\end{equation}
Then
\begin{equation}
u_+(z,\cdot,x_0)\in L^2((x_0,\infty))^{m\times m},
\lb{2.29}
\end{equation}
$u_+(z,x,x_0)$ is invertible, $u_+(z,x,x_0)$
satisfies the boundary condition of
$H_+$ at infinity (if any), and
\begin{equation}
\Im(M_+(z,x_0))=\Im(z)\int_{x_0}^{\infty}dx\,
u_+(z,x,x_0)^* u_+(z,x,x_0).\lb{2.30}
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Atkinson's Leading-Order Argument}\lb{s3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The purpose of this section is to prove the following result
due to Atkinson.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \mbox{\rm (Atkinson \cite{At88}.)} \lb{t3.1}
Assume Hypothesis~\ref{h2.1}
and denote by $C_\varepsilon\subset
\bbC_+$ the open sector with vertex at zero, symmetry
axis along the positive imaginary axis, and opening angle
$\varepsilon$, with $0<\varepsilon< \pi/2$. Let $M_+(z,x_0)$
be either the unique limit point or a point
of the limit disk $D_+(z,x_0)$
associated with \eqref{2.2}. Then
\begin{equation}\lb{3.1}
M_+(z,x_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} iz^{1/2}I_m + o(|z|^{1/2}).
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
More precisely, Atkinson proved the analog of \eqref{3.1} in
\cite{At88} for
more general Sturm-Liouville-type situations rather
than matrix-valued Schr\"odinger operators. To the best
of our knowledge, Atkinson's manuscript
\cite{At88} remains unpublished. Since the result is
crucial for our principal Section~\ref{s4}, we will
specialize his arguments to the simplified situation at hand
and provide a complete account of his approach (at times going
beyond some of the arguments sketched in \cite{At88}).
One of our tools in studying $M_+(z,x_0)$ will be to relate it
to a matrix Riccati-type equation with solution $M_+(z,x)$
defined as follows. We first note that
\begin{equation}
M_+(z,x_0)=\ti u_+^\prime(z,x_0) \ti u_+(z,x_0)^{-1},
\quad z\in\bbC\backslash\bbR, \lb{3.2}
\end{equation}
where $\ti u_+(z,\dott)\in L^2([x_0,\infty))^{m\times m}$
is a nonnormalized solution of \eqref{2.2} satisfying
\begin{equation}
\ti u_+(z,x)=u_+(z,x,x_0)C_+, \lb{3.3}
\end{equation}
with $u_+ $ defined in \eqref{2.28}, and with $C_+\in\bbC^{m\times m}$
a nonsingular matrix. Varying the reference point $x_0$ we may define
\begin{equation}
M_+(z,x)=\ti u_+^\prime(z,x) \ti u_+(z,x)^{-1},
\quad z\in\bbC\backslash\bbR, \, x\geq x_0. \lb{3.4}
\end{equation}
Then $M_+(z,x)$ is independent of the chosen normalization
$C_+$ and is well-known (see, e.g.,
\cite{CGHL99}, \cite{GH97}, \cite{Jo87}) to satisfy
\eqref{3.5} below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\lb{l3.2}
Assume Hypothesis~\ref{h2.1}, suppose
$z\in\bbC\backslash\bbR$,
$x\in [x_0,\infty)$, and define $M_+(z,x)$ as in
\eqref{3.4}. Then $M_+(z,x)$
satisfies the standard Riccati-type equation,
\begin{equation}
M_+^\prime(z,x)+M_+(z,x)^2=Q(x)-z I_m. \lb{3.5}
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Differentiating \eqref{3.4} with respect to $x$ and
inserting
$\ti u_+^{''}(z,x)+(Q(x)-z)\ti u_+(z,x)=0$
immediately yields \eqref{3.5}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Utilizing
\begin{equation}
\Psi(z,x,x_1)=\Psi(z,x,x_0)\Psi(z,x_0,x_1) \text{ and }
\Psi(z,x_0,x_1)\Psi(z,x_1,x_0)=I_{2m},
\end{equation}
(cf.~\eqref{2.4}) one computes for all $x_1\in\bbR$
\begin{align}
&\theta(z,x,x_1)+\phi(z,x,x_1)M_+(z,x_1) \no \\
&=u_+(z,x,x_0)
[\theta(z,x_0,x_1)+\phi(z,x_0,x_1)M_+(z,x_1)], \quad
z\in\bbC\backslash\bbR. \lb{3.7}
\end{align}
Since $u_+(z,\cdot,x_0)\in L^2([R,\infty))^{m\times m}$
for all $R\in\bbR$, the left-hand side of \eqref{3.7} is
in $L^2([R,\infty))^{m\times m}$. Thus, for $M_+(z,x_1)$
defined according to \eqref{3.4}, one concludes that
\begin{equation}
M_+(z,x_1)\in D_+(z,x_1) \text{ for all }x_1\in\bbR
\end{equation}
since $M_+(z,x_0)\in D_+(z,x_0)$ by hypothesis. This
justifies our choice of notation $M_+(z,x)$ when compared
with the notation employed in Definition~\ref{d2.7}.
Atkinson's proof of Theorem~\ref{t3.1}, which we follow very
closely, is first concerned with the relationship between
$D(z,c,x_0)$,
described in Definition~\ref{d2.4},
and $D^\calR (z,c,x_0)$, a {\em Riccati disk}, which we
now define.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%definition%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{definition}\lb{d3.3}
With $Q(x)$ given in Hypothesis~\ref{h2.1} and
$z\in \bbC_+$,
$D^\calR (z,c,x_0)$ is defined to be the set of all
$M(z,x_0)\in\bbC^{m\times m}$ where
$\Im{(M(z,x_0))} > 0$, and such that the solution of
\begin{subequations}\lb{3.9}
\begin{align}\lb{3.9a}
\vT '(z,x)= \frac{1}{2}\begin{bmatrix} I_m+\vT (z,x)&
I_m-\vT (z,x)\end{bmatrix}
&\begin{bmatrix}-i|z|^{-1/2}(zI_m -Q(x))& 0\\0& i|z|^{1/2}I_m
\end{bmatrix}\times \no \\
&\times\begin{bmatrix} I_m+\vT (z,x)\\ I_m-\vT (z,x)\end{bmatrix},
\end{align}
\begin{align} \lb{3.9b}
\vT (z,x_0)&= (I_m+i|z|^{-1/2} M(z,x_0))(I_m-i|z|^{-1/2} M(z,x_0))^{-1}
\no \\
&= (i|z|^{1/2}I_m -M(z,x_0))(i|z|^{1/2}I_m +M(z,x_0))^{-1},
\end{align}
\end{subequations}
satisfies
\begin{equation}\lb{3.10}
\vT (z,x)^* \vT (z,x)\le I_m \text{ for all } x\in[x_0,c].
\end{equation}
\end{definition}
Note that there is a correspondence between elements of
$D^\calR (z,c,x_0)$
and certain solutions of \eqref{3.9a}. If $\vT (z,x) $ satisfies
\eqref{3.9a} and
\eqref{3.10} then note that a matrix $M(z,x_0)\in
D^\calR (z,c,x_0)$ is defined by the initial data $\vT (z,x_0) $
if $\vT (z,x_0)^*\vT (z,x_0)< I_m$.
Note also that
\eqref{3.9a} can be written as
\begin{align}
\vT'=&\frac{1}{2}\begin{bmatrix}I_m+\vT & I_m-\vT\end{bmatrix}
\begin{bmatrix}I_m&0\\0&i|z|^{1/2}I_m\end{bmatrix}
(zA+B)
\begin{bmatrix}-i|z|^{-1/2} I_m& 0\\0& I_m \end{bmatrix}
\times \no \\
&\times \begin{bmatrix}I_m+\vT \\ I_m-\vT \end{bmatrix}.
\end{align}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%remark%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r3.4}
We emphasize that \eqref{3.9a}
formally results
from the Cayley-type transformation,
\begin{equation}
\vT (z,x)= (i|z|^{1/2}I_m -M(z,x))(i|z|^{1/2}I_m
+M(z,x))^{-1}, \lb{3.12}
\end{equation}
where $M(z,x)$ satisfies the Riccati-type equation \eqref{3.5}.
In the scalar context this corresponds to a conformal
mapping of the complex upper half-plane to the unit disk. Moreover, we
note that Atkinson proves the asymptotic result of
Theorem~\ref{t3.1}
for elements of $D^\calR (z,c,x_0)$. Recall that $D_+(z,x_0)\subseteq
D(z,c,x_0)$ (cf.~Remark~\ref{r2.6}). Theorem~\ref{t3.1} then
follows upon showing that $D(z,c,x_0)\subseteq D^\calR (z,c,x_0)$ \ \
for
$c>x_0$. This containment is shown in Theorem~\ref{t3.6}.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%lemma%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\lb{l3.5}
If $u(z,x)$ is defined by \eqref{2.13} with $M\in
D(z,c,x_0)$, then $u(z,x) - i|z|^{-1/2} u'(z,x)$ is invertible for
all $z\in\bbC_+$, and all $x\in [x_0 , c]$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
With $M\in D(z,c,x_0)$ and with $U(z,c)$ defined in
\eqref{2.13}, $U(z,c)$ satisfies \eqref{2.15}.
If $\Im{(\beta_2\beta_1^*)}> 0$, then $u'(z,c)= -\beta_2^{-1}\beta_1
u(z,c)$. $U(z,c)$ has rank $m$, hence $u(z,c)$ is nonsingular. As a
result,
\begin{equation}
\Im{(u(z,c)^* u'(z,c))}= u(z,c)^*\beta_2^{-1}\Im{(\beta_2\beta_1^*)}
{\beta_2^{*}}^{-1} u(z,c)>0.
\end{equation}
However, if $\Im{(\beta_2\beta_1^*)}=0 $, then
\begin{equation}\lb{3.14}
0=\beta_1\beta_2^* - \beta_2\beta_1^* =
\begin{bmatrix}\beta_1& \beta_2
\end{bmatrix}\begin{bmatrix}\beta_2^*\\ -\beta_1^* \end{bmatrix}.
\end{equation}
By \eqref{2.5a}, $\rank [ \beta_1 \ \beta_2 ] =
\rank [ \beta_2 \ -\beta_1 ]^* = m$; thus, by \eqref{2.15}
and \eqref{3.14} there is a $w\in\bbC^{m\times m}$ such
that $U(z,c)= [
\beta_2 \ -\beta_1]^* w$. From this, we obtain
\begin{equation}\lb{3.15}
\Im{(u(z,c)^*u'(z,c))}=w^*\Im{(\beta_2 \beta_1^*)}w=0.
\end{equation}
Hence for $\beta$ which satisfy \eqref{2.5},
\begin{equation}\lb{3.16}
\Im{(u(z,c)^*u'(z,c))\ge 0} .
\end{equation}
For all $x\in [x_0, c]$ and $z\in \bbC_+$, note that
\begin{equation}\lb{3.17}
(\Im{(u(z,x)^*u'(z,x))})' = - \Im{(z)} u(z,x)^*u(z,x) \le 0.
\end{equation}
Together, \eqref{3.16} and \eqref{3.17} imply that
\begin{equation}\lb{3.18}
\Im{(u(z,x)^*u'(z,x))\ge 0} \text{ for all }
x\in [x_0, c], \, z\in \bbC_+ .
\end{equation}
If for some $\xi \in [x_0, c] $ there is an $f\in\bbC^m$,
$f\ne 0$, such that $u(z,\xi)f = i|z|^{-1/2} u'(z,\xi)f$,
then
\begin{equation}\lb{3.19}
-i|z|^{1/2}f^*u(z,\xi)^*u(z,\xi)f = f^*u(z,\xi)^*u'(z,\xi)f.
\end{equation}
Together, \eqref{3.18} and \eqref{3.19} imply that
$f^*u(z,\xi)^*u(z,\xi)f\le 0$, and hence that
$u(z,\xi)f=u'(z,\xi)f=0$. Given the uniqueness of
solutions for
system \eqref{2.3}, we conclude that $0=u(z,x_0)f=f$;
thereby
producing a contradiction which completes the proof.
\end{proof}
%%%%%%%%%%%%%%%%%%%%theorem%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\lb{t3.6}
$D(z,c,x_0)\subseteq D^\calR (z,c,x_0)$ \ for all $c>x_0$
and $z\in\bbC_+$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%proof%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
With $z\in \bbC_+$ and $M=M(z,c,x_0,\beta) \in D(z,c,x_0)$,
we see that $\Im{(M)}>0 $ by \eqref{2.12}. With
$u(z,x)$ defined
in \eqref{2.13}, then by Lemma~\ref{l3.5}, we can
define
\begin{equation}\lb{3.20}
\vT(z,x)= (u(z,x)+i|z|^{-1/2} u'(z,x))(u(z,x)-i|z|^{-1/2}
u'(z,x))^{-1}
\end{equation}
for $x\in [x_0,\ c]$. As defined, $\vT (z,x)$ satisfies \eqref{3.9b}.
Then for $x\in [x_0\ c]$, \eqref{3.18} implies that
\begin{equation}\lb{3.21}
\begin{split}
&2i|z|^{-1/2} \{u(z,x)^*u'(z,x) -{u^*}'(z,x)u(z,x) \} \\
&= -4|z|^{-1/2} \Im{(u^*(z,x)u'(z,x))}\le 0.
\end{split}
\end{equation}
This is equivalent to
\begin{equation}\lb{3.22}
\begin{split}
&(u(z,x)^* -i|z|^{-1/2} u'(z,x)^*) (u(z,x)
+i|z|^{-1/2} {u'}(z,x)) \\
&\le (u(z,x)^* +i|z|^{-1/2} u'(z,x)^*)(u(z,x)
-i|z|^{-1/2} {u'}(z,x)).
\end{split}
\end{equation}
Given the invertibility of $u(z,x) -i|z|^{-1/2}
{u'}(z,x)$ shown in Lemma~\ref{l3.5}, we infer
that
$\vT(z,x)$ satisfies \eqref{3.10}. The proof of
this theorem will be completed upon showing that $\vT(z,x)$
satisfies \eqref{3.9a}.
First, observe that
\begin{equation}\lb{3.23}
\vT(z,x)(u(z,x)-i|z|^{-1/2} u'(z,x)) = u(z,x)
+ i|z|^{-1/2} u'(z,x),
\end{equation}
and hence that
\begin{equation}\lb{3.24}
I_m + \vT = 2u(u-i|z|^{-1/2} u')^{-1},\,\,\,
I_m - \vT = -2i|z|^{-1/2} u'(u-i|z|^{-1/2} u')^{-1}.
\end{equation}
Upon differentiating \eqref{3.23} and using the fact
that $u(z,x)$ satisfies \eqref{2.2}, we obtain
\begin{equation}
\vT'(u-i|z|^{-1/2} u') = (I_m- \vT)u' -i|z|^{-1/2}
(I_m+\vT)(zI_m -Q)u.
\end{equation}
By \eqref{3.24}, it follows that
\begin{equation}
\vT' = -\frac{1}{2i|z|^{-1/2}}(I_m -\vT)^2
- \frac{i|z|^{-1/2}}{2}(I_m +\vT)(zI_m -Q) (I_m +\vT),
\end{equation}
which is precisely \eqref{3.9a}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The following relation holds for the sets in
Theorem~\ref{t3.6}, that is,
\begin{equation}
\overline{D(z,c,x_0)}=D^\calR (z,c,x_0) \text{ for all }c>x_0
\text{ and }z\in\bbC_+.
\end{equation}
This will be discussed in detail in \cite{CG00}.
An associated system for \eqref{3.9} is obtained by
introducing a change
of independent variable: With $z\in \bbC_\varepsilon$,\
$c\in [x_0 , \infty)$, and\ $x\in [x_0 , c]$, let
\begin{equation}\lb{3.28}
t\ = \ (x-x_0)|z|^{1/2}.
\end{equation}
Introducing
\begin{equation}
\vP (z,t) = \vT (z,x), \lb{3.29}
\end{equation}
with $x$ and $t$ related as in \eqref{3.28}, \eqref{3.9} becomes
\begin{subequations} \lb{3.30}
\begin{align}\lb{3.30a}
\vP '(z,t)= &\frac{1}{2}\begin{bmatrix} I_m+\vP (z,t)&
I_m-\vP (z,t)\end{bmatrix}
\begin{bmatrix}-i|z|^{-1}(zI_m -\hatt Q(t))& 0\\0& iI_m
\end{bmatrix} \times \no \\
&\times \begin{bmatrix} I_m+\vP (z,t)\\ I_m
-\vP (z,t)\end{bmatrix},
\end{align}
\begin{equation}\lb{3.30b}
\vP (z,0)= (i|z|^{1/2}I_m -M(z,x_0))(i|z|^{1/2}I_m +M(z,x_0))^{-1},
\end{equation}
\end{subequations}
and \eqref{3.10} becomes
\begin{equation}\lb{3.31}
\vP (z,t)^*\vP (z,t)\le I_m \text{ for all } t \in
[0,(c-x_0)|z|^{1/2}].
\end{equation}
Note that in \eqref{3.30a}
\begin{equation}\lb{3.32}
\hatt Q(t)=Q(x_0 + t|z|^{-1/2}).
\end{equation}
In \eqref{3.30} and \eqref{3.31}, we have a set
of conditions equivalent to \eqref{3.9} and \eqref{3.10}
for $D^\calR (z,c,x_0)$ given in Defintion~\ref{d3.3}.
Now consider a sequence, $z_n \in \bbC_{\varepsilon}$, such
that $|z_n| \to \infty$ as $n\to \infty$ and such that
\begin{equation}\lb{3.33}
0< \varepsilon < \delta_n = \arg{(z_n)} < \pi - \varepsilon.
\end{equation}
By choosing an appropriate subsequence, we may assume that
\begin{equation}\lb{3.34}
\delta_n \to \delta \in [\varepsilon, \pi - \varepsilon].
\end{equation}
Let $\vP (z_n ,t)$ denote a corresponding sequence of functions
that satisfy \eqref{3.30a} and \eqref{3.31},
with initial
data, $\vP (z_n ,0)$, defined by \eqref{3.30b} for a
sequence of
points $M(z_n,x_0)$, $n\in\bbN $, where each $M(z_n,x_0)$ is chosen to
lie in the
disk $D^\calR (z_n,c,x_0)$. Note that as $z_n\to \infty$, the
intervals
described in
\eqref{3.31} eventually cover all compact
subintervals of
$ \bbR_+$. Given the uniform boundedness of
$\vP_n(t)=\vP (z_n ,t)$ described in \eqref{3.31},
we assume, upon passing to an appropriate subsequence
still denoted by $\vP_n (0)$, that
\begin{equation}\lb{3.35}
\vP_n(0) = \vP (z_n,0) \rightarrow \vP_0(\delta), \text{ as }
n\rightarrow \infty,
\end{equation}
and as a consequence, that
\begin{equation}
{\vP_0(\delta)}^* \vP_0(\delta) \le I_m.
\end{equation}
With $\vP_0(\delta)$ defined in \eqref{3.35} as
$|z_n|\to\infty$, we
consider a limiting system associated with \eqref{3.30}:
\begin{subequations}\lb{3.37}
\begin{align}
\vE '(\delta,t)= &\frac{1}{2}
\begin{bmatrix}
I_m+ \vE (\delta,t) & I_m-\vE (\delta,t)
\end{bmatrix}
\begin{bmatrix}
-ie^{i\delta}I_m & 0\\ 0 & iI_m
\end{bmatrix}\times\no\\
&\times\begin{bmatrix}
I_m+\vE (\delta,t)\\ I_m- \vE (\delta,t)
\end{bmatrix},
\quad t\geq 0, \lb{3.37a}
\end{align}
\begin{equation}
\vE (\delta,0)= \vP_0(\delta).
\end{equation}
\end{subequations}
%%%%%%%%%%%%%%%%%%%%%%theorem%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\lb{t3.7}
The solution $\vE(\delta,t) $ of \eqref{3.37} satisfies
\begin{equation}\lb{3.38}
\vE (\delta,t)^* \vE(\delta,t) \le I_m
\end{equation}
for $ 0\le t < \infty$. Moreover, the
solutions $\vP _n (t)=\vP (z_n,t) $
of \eqref{3.30} converge to $\vE (\delta,t) $ uniformly on
$[0,T]$ for every $T>0$, as $n\to \infty $.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%proof%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Let $T\in \bbR_+$ be the greatest value such that
\eqref{3.38} holds for
$t\in [0,\ T] $. We show that \eqref{3.38} must hold
for some $[0,\ T'] $ with $ T' > T $, thus proving $T=\infty$.
The solution of \eqref{3.37}, $\vE (\delta,t)$,
presumed
to be defined
on $[0,\ T]$, can be continued onto some $[0,\ T']$
with $T' > T$;
$\vE (\delta,t)$ then satisfies
\begin{equation}\lb{3.39}
\vE (\delta,t)^* \vE (\delta,t) \le k^2 I_m
\end{equation}
for $0\le t \le T'$ and for some $k\ge 1$.
For brevity, let $\vP'_n = G_n(\vP_n,t) $ denote
\eqref{3.30a} with
$z=z_n$, and let $\vE'= H(\vE,t) $ denote \eqref{3.37a}
in
the following. Integrating \eqref{3.37} and
\eqref{3.30}, we obtain
\begin{align}
\vP_n (t) -\vE (\delta,t) &= \vP_n(0) -\vP_0(\delta)
+ \int_0^t ds \{
G_n(\vE,s) - H(\vE,s)\} \no \\
&\quad + \int_0^t ds \{ G_n(\vP_n,s) - G_n(\vE,s)\}. \lb{3.40}
\end{align}
Note that
\begin{align}
G_n(\vE ,s) - H(\vE ,s) &= \frac{1}{2}i(e^{i\delta} -
e^{i\delta_n}) (I_m +\vE (\delta,s))^2 \no \\
&\quad + \frac{1}{2}i|z_n|^{-1}(I_m
+ \vE (\delta,s))\hatt Q(s)(I_m + \vE (\delta,s)),
\end{align}
and by \eqref{3.39}, as $n\to\infty$ and for $t \in [0,T']$, that
\begin{equation}
\frac{1}{2}i(e^{i\delta} - e^{i\delta_n}) \int_0^t ds\ (I_m + \vE
(\delta,s) )^2 = O (\delta - \delta_n),
\end{equation}
and by \eqref{3.28}, \eqref{3.32}, and \eqref{3.39}, that
\begin{align}
&\frac{1}{2}i|z_n|^{-1}\int_0^t ds\ (I_m + \vE (\delta,s))
\hatt Q(s)(I_m + \vE (\delta,s)) \no \\
&= |z_n|^{-1/2}O \left ( \int_{x_0}^{x_0
+t|z_n|^{-1/2}}dx\ || Q(x)
|| \right ) \lb{3.43} \\
&= o (|z_n|^{-1/2}) \text{ as } n\to\infty. \lb{3.44}
\end{align}
Together with \eqref{3.35}, we observe that as $n \to \infty$
\eqref{3.40} is equivalent to
\begin{equation}
\vP_n (t) -\vE (\delta,t) = o (1) + \int_0^t ds
(G_n(\vP_n,s) - G_n(\vE,s))
\end{equation}
uniformly so, for $0\le t \le T' $.
Since $|| \vP _n || \le 1 $ and $|| \vE || \le k$
for $ t\in [0, \ T' ] $,
\begin{equation}
|| G_n ( \vP _n,s) - G_n( \vE , s) || \le
\frac{1}{2}(3 + k)|| \vP _n (s)-\vE (\delta,s)||
(2 + |z_n|^{-1}\ ||\hatt Q (s)||).
\end{equation}
Thus \eqref{3.40} yields
\begin{equation}\lb{3.47}
\begin{split}
&|| \vP_n (t) -\vE (\delta,t) || \\
&\le o(1) + \int_0^t ds\
\frac{1}{2}(3 + k)|| \vP _n (s)-\vE (\delta,s) ||
(2 + |z_n|^{-1}\ ||\hatt Q(s) ||).
\end{split}
\end{equation}
In light of \eqref{3.44}, an application of Gronwall's
inequality to \eqref{3.47} yields
\begin{equation}
\vP_n(t) - \vE (\delta,t) \to 0 \text{ as } n\to\infty
\lb{3.48}
\end{equation}
uniformly for $0\le t\le T'$. For $n$ sufficiently
large and $t\in [0, \ T']$, $\vP_n (t) $ satisfies
\eqref{3.31}
with $z=z_n$, and $ \vE (\delta,t) $ satisfies \eqref{3.39},
hence $ \vE (\delta,t) $ satisfies \eqref{3.38} for
$t\in [0,\ T']$, where $T'>T$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The following computation identifies $\vP_0 (\delta)$
and proves
that $\vE (\delta,t)$ is constant with respect to $t\geq 0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary} \lb{c3.8}
\begin{equation} \lb{3.49}
\vE
(\delta,t)=\vE(\delta,0)=\vP_0(\delta)
=\frac{1-\exp(i\delta/2)}{1+\exp(i\delta/2)}I_m,
\quad t\geq 0.
\end{equation}
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Utilizing the standard connection between the explicit
exponential solutions of the second-order Schr\"odinger equation
\eqref{2.2a} and the Riccati equation \eqref{3.5} (in
the special case
$Q(x)=0$, cf.~\eqref{3.2} and Lemma~\ref{l3.2}), performing a
conformal map of the type
\eqref{3.12}, and the variable transformations \eqref{3.28} and
\eqref{3.29} then yields the following solution for
\eqref{3.30a},
\begin{align}
\vP (z,t)=&\big(-(i|z|^{1/2}-iz^{1/2})(M_0(z)+iz^{1/2}I_m)+
\exp(-2it\exp(i\delta/2))\times \no \\
&\times (i|z|^{1/2}+iz^{1/2})(M_0(z)-iz^{1/2}I_m)\big)\times
\no \\
&\times \big(-(i|z|^{1/2}+iz^{1/2})(M_0(z)+iz^{1/2}I_m)+
\exp(-2it\exp(i\delta/2))\times \no \\
&\times (i|z|^{1/2}-iz^{1/2})
(M_0(z)-iz^{1/2}I_m)\big)^{-1}, \lb{3.50}
\end{align}
associated with the general initial condition
\begin{equation}
\vP (z,0)=\big(i|z|^{1/2}I_m -M_0(z)\big)
\big(i|z|^{1/2}I_m +M_0(z) \big)^{-1} \lb{3.51}
\end{equation}
for some $M_0(z)\in\bbC^{m\times m}$ with $\Im (M_0(z))>0$,
$z\in\bbC_+$.
Since by hypothesis $0<\delta<\pi$, the exponential terms in
\eqref{3.50} enforce
\begin{equation}
\|\vP (z,t)\| > 1 \text{ as } t\uparrow\infty
\end{equation}
unless
\begin{equation}
M_0(z)=iz^{1/2}I_m,
\end{equation}
implying \eqref{3.49}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given these facts we proceed to the
\vspace*{1mm}
\noindent {\it Conclusion of the proof of Theorem~\ref{t3.1}.}
\vspace*{1mm}
With $M(z_n,x_0)\in D^{\calR}(z,c,x_0)$, and for $\delta$ as in
\eqref{3.34} let
\begin{equation}
C(\delta)=\frac{1-\exp(i\delta/2)}{1+\exp(i\delta/2)}. \lb{3.54}
\end{equation}
From \eqref{3.30b}, \eqref{3.35},\eqref{3.48}, and
\eqref{3.49}, we infer that
\begin{equation}
(I_m +i|z_n|^{-1/2} M(z_n,x_0))(I_m-i|z_n|^{-1/2} M(z_n,x_0))^{-1}
\to C(\delta) I_m \text{ as } n\to \infty.
\end{equation}
As a consequence,
\begin{align}
i|z_n|^{-1/2} M(z_n,x_0) &= \frac{C(\delta)-1
+o(1)}{C(\delta)+1 +o(1)}I_m=
\frac{C(\delta)-1}{C(\delta)+1}(1+o(1))I_m \no \\
&=-e^{i\delta/2}(1+o(1))I_m. \lb{3.56}
\end{align}
Thus, by \eqref{3.54} and \eqref{3.56} we conclude that
\begin{equation}\lb{3.57}
M(z_n,x_0)= i z_n^{1/2}(1+o(1)) I_m,
\end{equation}
and hence by Theorem~\ref{t3.6}, that \eqref{3.1} holds.
\hspace*{5.33cm} $\square$\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In \eqref{3.1} an asymptotic expansion is given that is uniform with
respect to $\arg(z)$ for $|z| \to \infty$ in $C_{\varepsilon}$. However,
we observe that the proof just completed shows more: Allowing the
reference point $x_0$ to vary, the asymptotic expansion given in
\eqref{3.1} is also uniform in $x_0$ for $x_0$ in a compact subset of
$\bbR$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t3.9}
Assume Hypothesis~\ref{h2.1}, let $z\in\bbC_+$,
$x\in \bbR$, and denote by $C_\varepsilon\subset
\bbC_+$ the open sector with vertex at zero, symmetry
axis along the positive imaginary axis, and opening angle
$\varepsilon$, with $0<\varepsilon< \pi/2$. Let $M_+(z,x)$,
$x\geq x_0$,
be as in Definition~\ref{d2.7}. Then
\begin{equation}
M_+(z,x)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} iz^{1/2}I_m + o(|z|^{1/2}) \lb{3.58}
\end{equation}
uniformly with respect to $\arg\,(z)$ for $|z|
\to \infty$ in $C_\varepsilon$ and uniformly in $x$ as long as $x$
varies in compact subsets of $[x_0,\infty)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Though stated for elements of the Weyl disk $D_+(z,x_0)$, the
proof of
Theorem~\ref{t3.1} shows that the asymptotic expansion given
in \eqref{3.1}
holds, uniformly with respect to $\arg\,(z)$ for $|z|
\to \infty$ in $C_\varepsilon$, for all elements of the Riccati disk
$D^{\calR}(z,c,x_0)$ as noted in \eqref{3.57}.
Note that the system \eqref{3.37} is independent of the reference
point $x_0$. Recall that $\delta$, defined in \eqref{3.34}, is
determined by our apriori choice of the sequence $z_n$ subject only to
$z_n$ being in $C_\varepsilon$ (c.f. \eqref{3.33}). Next note that
$\vP_0(\delta)$, which is defined as a limit in \eqref{3.35}, which is
described explicitly in Corollary~\ref{c3.8}, and which gives
solutions of
\eqref{3.37} which satisfy \eqref{3.38} for $0\le t<\infty$, is
also independent of the reference point $x_0$. Thus, had we chosen
$x_0'\ne x_0$ as our reference point at the start, the asymptotic
analysis begun in Theorem~\ref{t3.7} and continued in
\eqref{3.54}--\eqref{3.57} would remain the same after the variable
change in \eqref{3.28} except for the integral expression present in
\eqref{3.43} in which $x_0$ would be replaced by $x_0'$. However,
given the local integrability assumption on
$Q(x)$ present in Hypothesis~\ref{h2.1}, we see that the integral
expression in \eqref{3.43} is uniformly continuous for $x_0$ in
a compact
subset of $\bbR$. Thus \eqref{3.44} and consequently \eqref{3.48} are
uniform for $t$, and for $x_0$, in compact subsets of $\bbR$.
As a consequence, we see that \eqref{3.57} holds for elements of the
Riccati disk $D^{\calR}(z,c,x_0)$, that this asymptotic expansion is
uniform with respect to $\arg\,(z)$ for $|z|\to \infty$ in
$C_\varepsilon$
and that it is uniform in $x_0$ as long as $x_0$ varies in
compact subsets
of $\bbR$. The asymptotic expansion described in \eqref{3.58}
then follows by Theorem~\ref{t3.6}
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We will see in the next section that the remainder term
$o(|z|^{1/2})$ in \eqref{3.58} can be improved to $o(1)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Higher-Order Asymptotic Expansions}\lb{s4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we shall prove our principal result, the
asymptotic high-energy expansion of $M_+(z,x)$ to
arbitrarily high orders in sectors of the type
$C_\varepsilon\subset\bbC_+$ as defined in
Theorem~\ref{t3.1}.
Throughout this section we choose $\Im(z^{1/2})>0$ for
$z\in\bbC_+$. We also recall the following notion: $x\in [a,b)$ (resp.,
$x\in (a,b]$) is called a right (resp., left) Lebesgue point of an
element
$q\in L^1 ((a,b))$, $a**0$, $z\in\bbC_+)$
\begin{equation}
\widetilde M_+(z,x)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=}
i I_m z^{1/2}+\sum_{k=1}^N \tilde m_{+,k}(x)z^{-k/2}+
o(|z|^{-N/2}), \quad N\in\bbN. \lb{4.1}
\end{equation}
The expansion \eqref{4.1} 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
subintervals of $[x_0,\infty)$ intersected with the right Lebesgue set
of $\widetilde Q^{(N-1)}$. The expansion coefficients $\tilde
m_{+,k}(x)$ can be recursively computed from
\begin{align}
\tilde m_{+,1}(x)&=\f1{2i} \widetilde Q(x),
\quad \tilde m_{+,2}(x)= \f1{4} \widetilde Q^\prime(x),
\no \\
\tilde m_{+,k+1}(x)&=\f{i}2\bigg(\tilde m_{+,k}^\prime(x)+
\sum_{\ell=1}^{k-1}\tilde m_{+,\ell}(x)
\tilde m_{+,k-\ell}(x) \bigg),
\quad k\ge 2. \lb{4.2}
\end{align}
\noindent If one merely assumes
$\wti Q\in L^1([x_0,\infty))^{m\times m}$ with compact support in
$[x_0,\infty)$, then
\eqref{4.1} holds with $N=0$ {\rm (}interpreting $\sum_{k=1}^0 \cdot
=0${\rm )}, uniformly with respect to $\arg (z)$ for $|z|\to \infty$ in
$C_\varepsilon$ and uniformly in $x$ as long as $x$ varies in compact
subsets of $[x_0,\infty)$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
In the following let $z\in\bbC_+$, $\Im(z^{1/2}) > 0$, and $x\geq x_0$.
The existence of an expansion of the type \eqref{4.1} is shown as
follows. First one considers a matrix Volterra integral equation of the
type (cf.~\cite[Ch. I]{AM63}, \cite{Co77}, \cite{NJ55},
\cite{Sc83}, \cite{WK74})
\begin{equation}
\tilde u_+(z,x)=\exp(iz^{1/2}x)I_m - \int_x^\infty dx'\,
\f{\sin(z^{1/2}(x-x'))}{z^{1/2}}\widetilde Q(x')\tilde u_+(z,x'),
\lb{4.3}
\end{equation}
and observes that the solution
$\tilde u_+(z,x)$ of \eqref{4.3} satisfies
$\tilde u_+(z,\cdot)\in L^2([x_0,\infty))^{m\times m}$ in
accordance with \eqref{3.3} and \eqref{3.4}. Next, introducing
\begin{equation}
\tilde v_+(z,x)=\exp(-iz^{1/2}x)\tilde u_+(z,x), \lb{4.3a}
\end{equation}
one rewrites \eqref{4.3} in the form
\begin{equation}
\tilde v_+(z,x)=I_m - \f{1}{2iz^{1/2}}\int_x^\infty dx'\,
[1-\exp(-2iz^{1/2}(x-x'))]\widetilde Q(x')\tilde v_+(z,x'),
\lb{4.3b}
\end{equation}
and thus infers,
\begin{equation}
\widetilde M_+(z,x)=\tilde u_+'(z,x)\tilde u_+(z,x)^{-1}
=iz^{1/2}I_m + \tilde v_+'(z,x)\tilde v_+(z,x)^{-1}. \lb{4.4}
\end{equation}
Iterating \eqref{4.3b} then yields
\begin{equation}
\|\tilde v_+(z,x) \| \leq C, \quad z\in\bbC_+, \,\, \Im(z^{1/2}) > 0,
\,\, x\geq x_0 \lb{4.4c}
\end{equation}
for some $C>0$ depending on $\widetilde Q$.
Finally, we need one more ingredient, proven in \cite[Lemma~3]{Ry99}
using appropriate maximal functions. Let $q\in L^1 ([x_0,\infty))$,
$\supp(q)\subseteq [x_0,R]$, for some $R>0$, and suppose $x\in [x_0,R]$
is a right Lebesgue point of $q$. Then
\begin{equation}
\int_x^R dx^\prime \, \exp(2iz^{1/2}(x^\prime -x))q(x^\prime)
\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=}-\f{q(x)}{2iz^{1/2}} + o(|z|^{-1/2}). \lb{4.3d}
\end{equation}
An alternative proof of \eqref{4.3d} follows from
\cite[Theorem~I.13]{Ti86}, which implies
\begin{equation}
\lim_{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}z^{-1/2} \int_x^R dx^\prime \, \exp(2iz^{1/2}(x^\prime
-x)) |q(x^\prime) -q(x)| = 0 \lb{4.3e}
\end{equation}
for any right Lebesgue point $x$ of $q$.
Given these facts, one iterates \eqref{4.3b} and its $x$-derivative,
integrates by parts, applies \eqref{4.3d} to $q=\|Q_{j,k}\|$ for all
$1\leq j,k\leq m$, and estimates $\|\tilde v_+(z,x^\prime)\|$ by
\eqref{4.4c}. Inserting the expansions for $\tilde v_+'(z,x)$ and
$\tilde v_+(z,x)^{-1}$into \eqref{4.4} (using a geometric series
expansion for $\tilde v_+(z,x)^{-1}$) then yields the existence of an
expansion of the type \eqref{4.1}. The actual expansion coefficients and
the associated recursion relation \eqref{4.2} then follow upon inserting
expansion \eqref{4.1} into the Riccati-type equation \eqref{3.5}. The
assertion following \eqref{4.2} is an immediate consequence of
\eqref{4.3b} and its derivative with respect to $x$, \eqref{4.4},
and the Riemann-Lebesgue lemma.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The corresponding scalar case $m=1$ (for a sufficiently smooth
coefficient $q$) was treated in this manner in \cite{GHSZ95}. The
original version of our preprint assumed
$\widetilde Q^{(N)}\in L^1([x_0,\infty))^{m\times m}$ in
Lemma~\ref{l4.1}. Prompted by a recent
preprint by A.~Rybkin \cite{Ry99}, who
used an $m$-function-type approach to trace formulas for scalar
Schr\"odinger operators and succeeded in removing any smoothness
hypotheses on $q$, we reconsidered our original approach and extended
the asymptotic expansion \eqref{4.1} to the case
$\widetilde Q^{(N-1)}\in L^1 (\bbR)$ and $x$ a right Lebesgue point of
$\widetilde Q^{(N-1)}$.
Next we recall an elementary result on finite-dimensional
evolution equations essentially taken from \cite{MPS90}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \mbox{\rm (\cite{MPS90}.)} \lb{l4.2}
Let $A\in L^1_{\loc}(\bbR)^{m\times m}$. Then any
$m\times m$ matrix-valued solution $X(x)$ of
\begin{equation}
X'(x)=A(x)X(x)+X(x)A(x) \text{ for ~a.e. } x\in\bbR, \lb{4.5}
\end{equation}
is of the type
\begin{equation}
X(x)=Y(x)CZ(x), \lb{4.6}
\end{equation}
where $C$ is a constant $m\times m$ matrix and $Y(x)$ is a
fundamental system of solutions of
\begin{equation}
\Psi'(x)=A(x)\Psi(x) \lb{4.7}
\end{equation}
and $Z(x)$ is a fundamental system of solutions of
\begin{equation}
\Phi'(x)=\Phi(x)A(x). \lb{4.8}
\end{equation}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Clearly \eqref{4.6} satisfies \eqref{4.5} since
\begin{equation}
X'=Y'CZ+YCZ'=AYCZ+YCZA=AX+XA. \lb{4.9}
\end{equation}
Conversely, let $X$ be a solution of \eqref{4.5} and $Y$ a
fundamental matrix of solutions of \eqref{4.7}. Define
\begin{equation}
K(x)=Y(x)^{-1}X(x), \text{ that is, } X(x)=Y(x)K(x).
\lb{4.10}
\end{equation}
Then
\begin{equation}
X'=Y'K+YK'=AYK+YK'=AX+YK'=AX+XA \lb{4.11}
\end{equation}
implies
\begin{equation}
YK'=XA, \quad K'=Y^{-1}XA=KA. \lb{4.12}
\end{equation}
Thus, there exists a constant $m\times m$ matrix $C$
(possibly singular), such that
\begin{equation}
K(x)=CZ(x), \lb{4.13}
\end{equation}
with $Z$ a fundamental matrix of solutions of \eqref{4.8}.
Hence, $X=YCZ$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The next result provides the proper extension of
Proposition~2.1 in \cite{GS98} to the matrix-valued case.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma} \lb{l4.3}
Suppose $Q_1,Q_2 \in L^1_{\loc} (\bbR)^{m\times m}$ with
$Q_1(x)=Q_2(x)$ for~a.e. $x\in [x_0,x_1]$, $x_00$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Define for $x\in [x_0,x_1]$, $z\in\bbC\backslash\bbR$
\begin{align}
X_+(z,x)&=M_{1,+}(z,x)-M_{2,+}(z,x), \\
A_+(z,x)&=-(1/2)(M_{1,+}(z,x)+M_{2,+}(z,x)).
\end{align}
By Lemma~\ref{l4.3},
\begin{equation}
X_+'=A_+X_++X_+A_+
\end{equation}
and hence by Lemma~\ref{l4.2},
\begin{equation}
X_+(z,x)=Y_+(z,x)X_+(z,x_1)Z_+(z,x), \lb{4.30}
\end{equation}
where $Y_+(z,x)$ and $Z_+(z,x)$ are fundamental matrices of
\begin{equation}
\Psi'(z,x)=A_+(z,x)\Psi(z,x) \text{ and }
\Phi'(z,x)=\Phi(z,x)A_+(z,x),
\end{equation}
respectively, with
\begin{equation}
Y_+(z,x_1)=I_m, \quad Z_+(z,x_1)=I_m.
\end{equation}
By Lemma~\ref{l4.4},
\begin{equation}
\|Y_+(z,x_0)\|, \|Z_+(z,x_0)\| \leq
\exp(-(x_1-x_0)\Im(z^{1/2})(1+o(1))) \lb{4.33}
\end{equation}
as $|z|\to\infty$, $z\in C_\varepsilon$. Thus, as
$|z|\to\infty$, $z\in C_\varepsilon$,
\begin{align}
\|X_+(z,x_0)\|&\leq
\|X_+(z,x_1)\|\,\|Y_+(z,x_0)\|\,\|Z_+(z,x_0)\| \no \\
&\leq C(1+|z|^{1/2})\exp(-2(x_1-x_0)\Im(z^{1/2})(1+o(1)))
\lb{4.34}
\end{align}
for some constant $C>0$ by \eqref{3.58}, \eqref{4.30}, and
\eqref{4.33}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The result \eqref{4.26} can be slightly improved as will be
discussed in Remark~\ref{r4.6a}.
Given these preparations we can now drop the compact
support assumption on $Q$ in Lemma~\ref{l4.1} and hence arrive at
the principal result of this paper.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t4.6}
Fix $x_0\in\bbR$. In addition to Hypothesis~\ref{h2.1}
suppose that for some $N\in\bbN$,
$Q^{(N-1)}\in L^1([x_0,c])^{m\times m}$ for all $c>x_0$, and that $x_0$ is
a right Lebesgue point of $Q^{(N-1)}$. Let $M_+(z,x_0)$
be either the unique limit point or a point of the limit disk
$D_+(z,x_0)$ associated with \eqref{2.2}. Then, as
$\abs{z}\to\infty$ in $C_\varepsilon$, $M_+(z,x_0)$ has an asymptotic
expansion of the form $(\Im(z^{1/2})>0$, $z\in\bbC_+)$
\begin{equation}
M_+(z,x_0)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=} i I_m z^{1/2}+\sum_{k=1}^N m_{+,k}(x_0)z^{-k/2}+
o(|z|^{-N/2}), \quad N\in\bbN. \lb{4.35}
\end{equation}
The expansion \eqref{4.35} is uniform with respect to $\arg\,(z)$ for $|z|
\to \infty$ in $C_\varepsilon$. The expansion coefficients $m_{+,k}(x_0)$
can be recursively computed from \eqref{4.2} {\rm (}replacing $\tilde
m_{+,k}(x)$ by $m_{+,k}(x)${\rm )}.
\noindent If one merely assumes Hypothesis~\ref{h2.1}, then \eqref{4.35}
holds with $N=0$ {\rm (}interpreting $\sum_{k=1}^0 \cdot =0${\rm )},
uniformly with respect to $\arg (z)$ for $|z|\to \infty$ in
$C_\varepsilon$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
Define
\begin{equation}
\widetilde Q(x)=\begin{cases} Q(x) &\text{ for }
x\in [x_0,x_1], \quad x_0x_0$, and that $x$ is
a right Lebesgue point of $Q^{(N-1)}$. Let $M_+(z,x)$, $x\geq x_0$,
be either the unique limit point or a point of the limit disk $D_+(z,x)$
associated with \eqref{2.2}. Then, as $\abs{z}\to\infty$ in
$C_\varepsilon$, $M_+(z,x)$ has an asymptotic expansion of the form
$(\Im(z^{1/2})>0$, $z\in\bbC_+)$
\begin{equation}
M_+(z,x)\underset{\substack{\abs{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 N\in\bbN. \lb{4.37}
\end{equation}
The expansion \eqref{4.37} 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 $[x_0,\infty)$ intersected with the right
Lebesgue set of $Q^{(N-1)}$. The expansion coefficients $m_{+,k}(x)$ can
be recursively computed from \eqref{4.2} {\rm (}replacing $\tilde
m_{+,k}(x)$ by $m_{+,k}(x)${\rm )}.
\noindent If one merely assumes Hypothesis~\ref{h2.1}, then \eqref{4.37}
holds with $N=0$ {\rm (}interpreting $\sum_{k=1}^0 \cdot =0${\rm )},
uniformly with respect to $\arg (z)$ for $|z|\to \infty$ in
$C_\varepsilon$ and uniformly in $x$ as long as $x$ varies in compact
subsets of $[x_0,\infty)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r4.8}
Our asymptotic results in Theorem~\ref{t4.7} are not
necessarily confined to $M_+(z,x_0)$-matrices associated
with half-line Schr\"odinger operators $H_+$ on $[x_0,\infty)$.
In fact, introducing in addition the analogous Weyl-Titchmarsh
matrix $M_-(z,x_0)$ associated with a half-line Schr\"odinger
operator $H_-$ on $(-\infty,x_0]$, and noticing that the
diagonal Green's matrix $G(z,x,x)=(H-z)^{-1}(x,x)$ of a
matrix-valued Schr\"odinger operator $H=-I_m\f{d^2}{dx^2}+Q$
in $L^2(\bbR)^m$ is given by
\begin{equation}
G(z,x,x)=(M_-(z,x)-M_+(z,x))^{-1}, \lb{4.38}
\end{equation}
Theorem~\ref{t4.6} then yields an analogous
asymptotic expansion for $G(z,x,x)$ in $C_\varepsilon$ of the form
\begin{equation}
G(z,x,x)\underset{\substack{\abs{z}\to\infty\\ z\in
C_\varepsilon}}{=}
\begin{cases}
(i/2) I_m z^{-1/2}+o(|z|^{-1}) & \text{for $N=0$}, \\
(i/2) \sum_{k=0}^N G_k(x) z^{-k-1/2}+o(|z|^{-N-1/2}) &
\text{for $N\in\bbN$}, \lb{4.39}
\end{cases}
\end{equation}
where
\begin{equation}
G_0(x)=I_m, \quad G_1(x)=\f12 Q(x), \text{ etc.} \lb{4.40}
\end{equation}
The expansion \eqref{4.39} is uniform with respect to
$\arg\,(z)$ for $|z|\to \infty$ in
$C_\varepsilon$ and uniform in $x\in\bbR$ as long as
$x$ varies in compact intervals interesected with the
Lebesgue set of $Q^{(N-1)}$ (if $N\in\bbN$). We refer to
\cite{CGHL99} for further details and applications of this fact
in connection with trace formulas and Borg-type uniqueness
results for $Q(x)$, $x\in\bbR$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{3mm}
\noindent {\bf Acknowledgments.}
We would like to thank Don Hinton, Helge Holden, Boris
Levitan, Mark Malamud, Alexei Rybkin, and Barry Simon for helpful
discussions and hints regarding the literature.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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