%October 1999
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\begin{document}
\title[Local Borg-Marchenko Results]{On Local Borg-Marchenko
Uniqueness Results}
\author[F. Gesztesy and B. Simon]{Fritz Gesztesy$^{1}$ and
Barry Simon$^{2}$}
\date{October 15, 1999}
\subjclass{Primary: 34A55, 34B20; Secondary: 34L05, 47A10}
\keywords{Inverse spectral theory, Weyl-Titchmarsh
$m$-function, uniqueness theorems}
\footnotetext[1]{Department of Mathematics, University of
Missouri,
Columbia, MO 65211, USA. E-mail: fritz@math.missouri.edu}
\footnotetext[2]{Division of Physics, Mathematics, and
Astronomy,
253-37, California Institute of Technology, Pasadena,
CA~91125, USA.
E-mail: bsimon@caltech.edu. This material is based upon work
supported by the National Science Foundation under Grant
No.~DMS-9707661. The Government has certain rights in this
material.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract} We provide a new short proof of the
following
fact, first
proved by one of us in 1998: If two Weyl-Titchmarsh
$m$-functions,
$m_j(z)$,
of two Schr\"odinger operators $H_j = -\f{d^2}{dx^2} + q_j$,
$j=1,2$ in $L^2
((0,R))$, $00$, $q_j$
real-valued, $j=1,2$, be two self-adjoint operators in
$L^2 ([0,\infty))$ with a
Dirichlet boundary condition at $x=0_+$. Let $m_j(z)$,
$z\in\bbC\backslash\bbR$ be
the Weyl-Titchmarsh $m$-functions associated with $H_j$,
$j=1,2$. The principal
purpose of this note is to provide a short proof of the
following
uniqueness
theorem in the spectral theory of one-dimensional Schr\"odinger
operators,
originally obtained by Simon \cite{Si98} in 1998. (Actually,
Simon's result
\cite{Si98} was weaker; the result as stated is from
\cite{GS98}.)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\lb{t1.1} Let $a>0$, $0<\veps<\pi/2$ and
suppose that
\begin{equation} \lb{1.1}
|m_1 (z) - m_2(z)| \eqlim_{|z|\to\infty}
O(e^{-2\Ima (z^{1/2})a})
\end{equation}
along the ray $\arg(z) = \pi-\veps$. Then
\begin{equation} \lb{1.2}
q_1(x) = q_2 (x) \text{ for a.e. } x\in [0,a].
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For reasons of brevity we stated Theorem~\ref{t1.1} only in
the simplest possible case.
Extensions to finite intervals $[0,R]$ instead of the
half-line
$[0,\infty)$, a
discussion of boundary conditions other than Dirichlet at
$x=0_+$, and the case of
matrix-valued Schr\"odinger operators --- a new
result --- will be
provided in the main body of this paper.
Theorem~\ref{t1.1} should be viewed as a local (and
hence stronger)
version of the
following celebrated Borg-Marchenko uniqueness theorem,
published
by Marchenko
\cite{Ma50} in 1950. Marchenko's extensive treatise
on spectral
theory of
one-dimensional Schr\"odinger operators \cite{Ma52},
repeating
the proof of his
uniqueness theorem, then appeared in 1952, which
also marked the
appearance of
Borg's proof of the uniqueness theorem \cite{Bo52}
(apparently,
based on his lecture
at the 11th Scandinavian Congress of Mathematicians held at
Trondheim, Norway in
1949).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \lb{t1.2} \mbox{\rm (\cite{Bo52,Ma50,Ma52})}
Suppose
\begin{equation} \lb{1.3}
m_1(z) = m_2(z), \quad z\in\bbC\backslash\bbR,
\end{equation}
then
\begin{equation} \lb{1.4}
q_1(x) = q_2(x) \text{ for a.e. } x\in [0,\infty).
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Again, we emphasize that Borg and Marchenko also treat the
general case of non-Dirichlet
boundary conditions at $x=0_+$, whose discussion we defer to
Section~\ref{s2}. Moreover,
Marchenko simultaneously discussed the half-line and finite
interval case, also to be
deferred to Section~\ref{s2}.
As pointed out by Levitan \cite{Le87} in the Notes to Chapter~2,
Borg and
Marchenko were actually preceded by Tikhonov \cite{Ti49} in
1949, who
proved a special case of Theorem~\ref{t1.2} in connection
with the
string equation (and hence under certain additional
hypotheses on
$q_j$). Since Weyl-Titchmarsh functions $m(z)$ are uniquely
related to
the spectral measure $d\rho$ of a self-adjoint (Dirichlet)
Schr\"odinger
operator $H=-\f{d^2}{dx^2} + q$ in $L^2 ([0,\infty))$ by the
standard
Herglotz representation
\begin{equation} \lb{1.5}
m(z) = \Real(m(i)) + \int_\bbR d\rho(\lambda)
[(\lambda -z)^{-1} -
\lambda(1+\lambda^2)^{-1}], \quad z\in\bbC\backslash\bbR,
\end{equation}
Theorem~\ref{t1.2} is equivalent to the following statement:
Denote by $d\rho_j$ the
spectral measures of $H_j$, $j=1,2$. Then
\begin{equation} \lb{1.6}
d\rho_1 = d\rho_2 \text{ implies } q_1 = q_2 \text{ a.e.~on }
[0,\infty).
\end{equation}
In fact, Marchenko's proof takes the spectral measures
$d\rho_j$ as the point of
departure while Borg focuses on the Weyl-Titchmarsh functions
$m_j$.
To the best of our knowledge, the only alternative approaches
to Theorem~\ref{t1.2}
are based on the Gelfand-Levitan solution of the inverse
spectral problem published
in 1951 (see also Levitan and Gasymov \cite{LG64}) and
alternative
variants due to
M.~Krein \cite{Kr51}, \cite{Kr53}. In particular, it took over
45 years to improve
on Theorem~\ref{t1.2} and derive its local counterpart,
Theorem~\ref{t1.1}. While the
original proof of Theorem~\ref{t1.1} in \cite{Si98} relied on
the full power of a new
formalism in inverse spectral theory, relating $m(z)$ to finite
Laplace transforms of
the type
\begin{equation} \lb{1.7}
m(z) = iz^{1/2} - \int_0^a d\alpha\, A(\alpha) \,
e^{2\alpha iz^{1/2}} +
\widetilde O(e^{2\alpha iz^{1/2}})
\end{equation}
as $|z|\to\infty$ with $\arg(z)\in (\veps, \pi-\veps)$ for
some $0<\veps<\pi$ (with
$f=\widetilde O(g)$ if $g\to 0$ and for all $\delta >0$,
$(\f{f}{g}) |g|^\delta\to 0$),
we will present a short and fairly elementary argument in
Section~\ref{s2}. In fact,
as a corollary to our new proof of Theorem~\ref{t1.1}, we also
obtain an elementary
proof of a strengthened version of Theorem~\ref{t1.2}.
We should also mention some work of Ramm \cite{Ra99},
\cite{Ra99a}, who
provided a proof of Theorem~\ref{t1.2} under a very strong
additional
assumption, namely, that $q_1$ and $q_2$ are both of short
range. While
his result is necessarily weaker than the original
Borg-Marchenko result,
Theorem~\ref{t1.2}, his method of proof has elements in
common with parts
of our proof (namely, he uses
\eqref{2.27} below with $a=\infty$ and obtains a Volterra
integral
equation close to our \eqref{2.33}).
Finally, we have in preparation \cite{GS00} still
another alternate proof of the local Borg-Marchenko theorem.
Extensions to finite intervals and general
(i.e., non-Dirichlet)
boundary conditions
complete Section~\ref{s2}. Matrix-valued extensions of
Theorem~\ref{t1.1} are presented in Section~\ref{s3}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A New Proof of Theorem~1.1} \lb{s2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Throughout this section, unless explicitly stated otherwise,
potentials
$q$ are supposed to satisfy
\begin{equation} \lb{2.1}
q\in L^1 ([0,R]) \text{ for all } R>0,
\quad \text{$q$ real-valued}.
\end{equation}
Given $q$, we introduce the corresponding self-adjoint
Schr\"odinger operator $H$ in
$L^2 ([0,\infty))$ with a Dirichlet boundary condition at
$x=0_+$, by
\begin{align}
&H=-\f{d^2}{dx^2} + q, \no \\
&\dom(H) = \{g\in L^2 ([0,\infty)) \mid g,g'
\in \AC([0,R])\text{ for all }R>0; \lb{2.2} \\
&\hspace*{16mm} g(0_+)=0, \,\sabc\text{ at } \infty; \,
(-g''+qg) \in L^2
([0,\infty))\}. \no
\end{align}
Here ``s.-a.\,b.c." denotes a self-adjoint boundary condition
at $\infty$ (which
becomes relevant only if $q$ is in the limit circle case at
$\infty$, but should
be discarded otherwise, i.e., in the limit point case, where
such a boundary
condition is automatically satisfied). For example, an explicit
form of such a
boundary condition is
\begin{equation} \lb{2.3}
\lim_{x\uparrow \infty} W(f(z_0), g) (x) =0,
\end{equation}
where $f(z_0, x)$ for some fixed $z_0\in\bbC\backslash\bbR$,
satisfies
\begin{equation} \lb{2.4}
f(z_0, \dott)\in L^2 ([0\infty)), \quad -f'' (z_0, x)
+ [q(x)-z_0]
f(z_0, x) =0
\end{equation}
and $W(f,g)(x) = f(x)g'(x) - f'(x)g(x)$ denotes the Wronskian
of $f$ and $g$.
Since these possible boundary conditions hardly play a role in
the analysis
to follow, we will not dwell on them any further. (Pertinent
details can be
found in \cite{GS96} and the references therein.)
Next, let $\psi(z,x)$ be the unique (up to constant multiples)
Weyl solution associated
with $H$, that is,
\begin{equation} \lb{2.5}
\begin{align}
&\psi(z, \dott) \in L^2 ([0,\infty)),
\quad z\in\bbC\backslash\bbR, \notag \\
&\psi(z,x)\text{ satisfies the $\sabc$ of $H$ at $\infty$
(if any)}, \\
& -\psi'' (z,x) + [q(x) -z]\psi(z,x)=0. \notag
\end{align}
\end{equation}
Then the Weyl-Titchmarsh function $m(z)$ associated with $H$
is defined by
\begin{equation} \lb{2.6}
m(z) = \psi' (z,0_+) / \psi(z,0_+),
\quad z\in\bbC\backslash\bbR
\end{equation}
and for later purposes we also introduce the corresponding
$x$-dependent
version,
$m(z,x)$, by
\begin{equation} \lb{2.7}
m(z,x) = \psi'(z,x) / \psi(z,x), \quad z\in\bbC\backslash\bbR,
\quad x\geq 0.
\end{equation}
After these preliminaries we are now ready to state the main
ingredients used in
our new proof of Theorem~\ref{t1.1}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\lb{t2.1} \mbox{\rm(\cite{At81,Ev72})} Let
$\arg(z)\in(\veps, \pi -\veps)$
for some $0<\veps < \pi$. Then for any fixed $x\in [0,\infty)$,
\begin{equation} \lb{2.8}
m(z,x) \eqlim_{|z|\to\infty} iz^{1/2} + o(1).
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The following result shows that one can also get an estimate
uniform in $x$ as long as
$x$ varies in compact intervals.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\lb{t2.2} \mbox{\rm(\cite{GS98})} Let
$\arg(z)\in (\veps,\pi-\veps)$ for
some $0<\veps<\pi$, and suppose $\delta>0$, $a>0$. Then there
exists a $C(\veps,\delta,a)
>0$ such that for all $x\in [0,a]$,
\begin{equation} \lb{2.9}
|m(z,x) - iz^{1/2}| \leq C(\veps,\delta, a),
\end{equation}
where $C(\veps,\delta,a)$ depends on $\veps,\delta$, and
$\sup_{0\leq x\leq a}
(\int_x^{x+\delta} dy\, |q(y)|)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Theorems~\ref{t2.1} and \ref{t2.2} can be proved following
arguments of Atkinson
\cite{At81}, who studied the Riccati-type equation satisfied
by $m(z,x)$,
\begin{equation} \lb{2.10}
m'(z,x) + m(z,x)^2 = q(x)-z \text{ for a.e.}~x\geq 0
\text{ and all }
z\in\bbC\backslash\bbR.
\end{equation}
Next, let $q_j(x)$, $j=1,2$ be two potentials satisfying
\eqref{2.1},
with $m_j(z)$ the associated (Dirichlet) $m$-functions.
Combining the
{\it a priori}
bound \eqref{2.9} with the differential equation resulting
from \eqref{2.10},
\begin{align}
&[m_1(z,x) - m_2(z,x)]' \lb{2.11}\\
& =q_1 (x) - q_2(x) - [m_1 (z,x) + m_2(z,x)][m_1 (z,x) - m_2
(z,x)], \no
\end{align}
permits one to prove the following converse of
Theorem~\ref{t1.1}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\lb{t2.3} \mbox{\rm(\cite{GS98})} Let
$\arg(z)\in (\veps, \pi -\veps)$
for some $0<\veps<\pi$ and suppose $a>0$. If
\begin{equation} \lb{2.12}
q_1(x) = q_2(x)\text{ for a.e. } x\in [0,a],
\end{equation}
then
\begin{equation} \lb{2.13}
|m_1 (z) - m_2(z)| \eqlim_{|z|\to\infty}
O(e^{-2\Ima (z^{1/2})a}).
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\lb{l2.4} In addition to the hypotheses of
Theorem~\ref{t2.2}, {\rm(}resp.,
Theorem~\ref{t2.3}{\rm)}, suppose that $H$ {\rm(}resp.,
$H_j$, $j=1,2${\rm)} is bounded
from below. Then \eqref{2.9} {\rm(}resp., \eqref{2.13}{\rm)}
extends to all $\arg(z)
\in (\veps,\pi]$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof} Since $H_x \geq H$, where $H_x$ denotes the
Schr\"odinger operator
$-\f{d^2}{dx^2}+q$ in $L^2 ([x,\infty))$ with a Dirichlet
boundary condition at $x_+$
(and the same $\sabc$ at $\infty$ as $H$, if any), there is
an $E_0\in\bbR$ such that
for all $x\in [0,a]$, $m(z,x)$ is analytic in
$\bbC\backslash [E_0, \infty)$. Using
$\ol{m(z,x)} = m(\bar z,x)$, the estimate \eqref{2.9} holds on
the boundary of a
sector with vertex at $E_0 -1$, symmetry axis
$(-\infty, E_0 -1]$, and some opening
angle $0<\veps<\pi/2$. An application of the
Phragm\'en-Lindel\"of principle
(cf.~\cite[Part~III, Sect.~6.5]{PS72}) then extends
\eqref{2.9} to all of the
interior of that sector and hence in particular along the
ray $z\downarrow -\infty$.
Since \eqref{2.13} results from \eqref{2.9} upon integrating
(cf.~\eqref{2.11}),
\begin{align}
& m_1(z,x) - m_2(z,x)]' \no \\
&=-[m_1(z,x) + m_2(z,x)][m_1(z,x), - m_2(z,x)],
\quad x\in [0,a] \lb{2.14}
\end{align}
from $x=0$ to $x=a$, the extension of \eqref{2.9} to $z$
with $\arg(z)\in (\veps,\pi]$
just proven, allows one to estimate
\begin{equation} \lb{2.15}
|m_1(z,x) + m_2(z,x)| \eqlim_{|z|\to\infty} 2iz^{1/2} + O(1),
\quad \arg(z)\in (\veps, \pi],
\end{equation}
uniformly with respect to $x\in [0,a]$, and hence to extend
\eqref{2.13} to $\arg(z)\in
(\veps, \pi]$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\smallskip
Next, we briefly recall a few well-known facts on compactly
supported $q$. Hence we
suppose temporarily that
\begin{equation} \lb{2.15a}
\sup(\supp(q)) = \alpha <\infty.
\end{equation}
In this case, the Jost solution $f(z,x)$ associated with
$q(x)$ satisfies
\begin{align}
f(z,x) &= e^{iz^{1/2}x} - \int_x^\alpha dy \,
\f{\sin(z^{1/2}(x-y))}{z^{1/2}}\,
q(y)\, f(z,y) \lb{2.16} \\
&= e^{iz^{1/2}x} + \int_x^\alpha dy K(x,y)\, e^{iz^{1/2}y},
\quad \Ima(z^{1/2}) \geq 0, \ x\geq 0, \lb{2.17}
\end{align}
where $K(x,y)$ denotes the transformation kernel satisfying
(cf.~\cite[Sect.~3.1]{Ma86})
\begin{align}
K(x,y)&= \f12 \int_{(x+y)/2}^\alpha dx' \, q(x')
- \int_{(x+y)/2}^\alpha \int_0^{(y-x)/2}
dx'' \, q(x'-x'')\times \notag \\
&\hspace*{3.9cm} \times K(x'-x'', x' + x''), \quad x\leq y,
\lb{2.18}
\\ K(x,y)&= 0, \quad x>y, \lb{2.19} \\
|K(x,y) &\leq \f12 \int_{(x+y)/2}^\alpha dx' \, |q(x')|
\exp \left(\int_x^\alpha
dx'' \, x''|q(x'')| \right). \lb{2.20}
\end{align}
Moreover, $f(z,x)$ is a multiple of the Weyl solution,
implying
\begin{equation} \lb{2.21}
m(z,x) = f'(z,x) / f(z,x), \quad z\in\bbC\backslash\bbR,
\ x\geq 0,
\end{equation}
and the Volterra integral equation \eqref{2.16} immediately
yields
\begin{align}
|f(z,x)| &\leq Ce^{-\Ima(z^{1/2})x}, \quad \Ima(z^{1/2})
\geq 0, \ x\geq 0, \lb{2.22} \\
f(z,x) &\eqlim_{\substack{|z|\to\infty \\
\Ima (z^{1/2})\geq 0}}
e^{iz^{1/2}x}(1+O(|z|^{-1/2}), \quad x\geq 0, \lb{2.23}
\end{align}
Our final ingredient concerns the following result on finite
Laplace transforms.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\lb{l2.5} \mbox{\rm($=$ Lemma~A.2.1 in
\cite{Si98})}
Let $g\in L^1 ([0,a])$ and
assume that $\int_0^a dy\, g(y)e^{-xy}
\underset{x\uparrow\infty}{=}
O(e^{-xa})$. Then $g(y) = 0$ for a.e.~$ y\in [0,a]$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Given these facts, the proof of Theorem~\ref{t1.1} now becomes
quite simple.
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof of Theorem 1.1] By Theorem~\ref{t2.3} we
may assume, without loss
of generality, that $q_1$ and $q_2$ are compactly supported
such that
\begin{equation} \lb{2.24}
\supp (q_j) \subseteq [0,a], \quad j=1,2,
\end{equation}
and by Lemma~\ref{l2.4} we may suppose that \eqref{1.1} holds
along the ray
$z\downarrow -\infty$, that is,
\begin{equation} \lb{2.25}
|m_1(z) - m_2(z)| \eqlim_{z\downarrow -\infty}
O(e^{-2|z|^{1/2}a}).
\end{equation}
Denoting by $m_j(z,x)$ and $f_j(z,x)$ the $m$-functions and
Jost solutions associated
with $q_j$, $j=1,2$, integrating the elementary identity
\begin{equation} \lb{2.26}
\f{d}{dx}\, W(f_1(z,x), f_2(z,x)) = -[q_1(x) - q_2(x)]
f_1(z,x) f_2(z,x)
\end{equation}
from $x=0$ to $x=a$, taking into account \eqref{2.21}, yields
\begin{align}
&\int_0^a dx\, [q_1(x) - q_2(x)] f_1(z,x)f_2(z,x) \no \\
&= f_1(z,x) f_2(z,x) [m_1(z,x) - m_2(z,x)]\bigg|_{x=0}^a.
\lb{2.27}
\end{align}
By \eqref{2.8}, \eqref{2.22}, and \eqref{2.25}, the right-hand
side of \eqref{2.27} is
$O(e^{-2|z|^{1/2}a})$ as $z\downarrow -\infty$, that is,
\begin{equation} \lb{2.28}
\int_0^a dx\, [q_1(x) - q_2(x)] f_1(z,x) f_2(z,x)
\eqlim_{z\downarrow -\infty} O(e^{-2|z|^{1/2}a}).
\end{equation}
Denoting by $K_j (x,y)$ the transformation kernels associated
with $q_j$, $j=1,2$,
\eqref{2.17} implies
\begin{equation} \lb{2.29}
f_1(z,x) f_2(z,x) = e^{2iz^{1/2}x} + \int_x^a dy \, L(x,y)
\, e^{2iz^{1/2}y},
\end{equation}
where
\begin{align}
L(x,y) &= 2[K_1(x, 2y-x) + K_2(x, 2y-x)] \notag \\
&\quad + 2 \int_x^{2y-x} dx'\, K_1(x, x') K_2(x, 2y-x'),
\quad x\leq y,
\lb{2.30} \\ L(x,y)&= 0, \quad x > y \quad \text{or}
\quad y>a. \lb{2.31}
\end{align}
Insertion of \eqref{2.29} into \eqref{2.28}, interchanging
the order of integration in
the double integral, then yields
\begin{align}
&\int_0^a dx [q_1 (x) - q_2(x)] f_1(z,x) f_2(z,x) \no \\
&= \int_0^a dy\left\{ [q_1(y) - q_2(y)] + \int_0^y dx \, L(x,y)
[q_1(x) - q_2(x)]\right\} \, e^{-2|z|^{1/2}y} \no \\
& \eqlim_{z\downarrow -\infty} O(e^{-2|z|^{1/2}a}). \lb{2.32}
\end{align}
An application of Lemma~\ref{l2.5} then yields
\begin{equation} \lb{2.33}
[q_1(y) - q_2(y)] + \int_0^y dx\, L(x,y) [q_1(x) - q_2(x)] = 0
\quad\text{for a.e. } y\in [0.a].
\end{equation}
Since \eqref{2.33} is a homogeneous Volterra integral
equation with a continuous
integral kernel $L(x,y)$, one concludes $q_1 = q_2$
a.e.~on $[0,a]$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In particular, one obtains the following strengthened version
of the original
Borg-Marchenko uniqueness result, Theorem~\ref{t1.2}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\lb{c2.6} Let $0<\veps <\pi/2$ and suppose
that for all $a>0$,
\begin{equation} \lb{2.34}
|m_1(z) - m_2(z)| \eqlim_{|z|\to\infty}
O(e^{-2\Ima (z^{1/2})a})
\end{equation}
along the ray $\arg(z) = \pi -\veps$. Then
\begin{equation} \lb{2.35}
q_1(x) = q_2(x) \text{ for a.e. } x\in [0,\infty).
\end{equation}
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r2.7} The Borg-Marchenko uniqueness result,
Theorem~\ref{t1.2}
(but not our strengthened version, Corollary~\ref{c2.6}),
under the
additional condition of short-range potentials $q_j$
satisfying
$q_j\in L^1 ([0,\infty);
(1+x)\, dx)$, $j=1,2$, can also be proved using Property~C,
a device
recently used by Ramm \cite{Ra99,Ra99a} in a variety of
uniqueness
results. In this case, \eqref{2.27} for $z=\lambda>0$ becomes
\begin{align}
&\int_0^\infty dx\, [q_1(x) - q_2(x)] f_1(\lambda,x)
f_2(\lambda, z) \no
\\
&= -f_1(\lambda, 0) f_2(\lambda, 0) [m_1(\lambda +i0) -
m_2(\lambda+i0)] = 0,
\quad \lambda >0 \lb{2.36}
\end{align}
since $m_1(z) = m_2(z)$, $z\in\bbC_+$ extends to
$m_1(\lambda+i0) = m_2(\lambda+i0)$,
$\lambda >0$ by continuity in the present short-range case.
By definition, Property~C
stands for completeness of the set
$\{f_1(\lambda, x)f_2(\lambda, x)\}_{\lambda>0}$
in $L^1 ([0,\infty); (1+x)\, dx)$ (this extends to
$L^1 ([0,\infty))$)
and hence
\eqref{2.36} yields $q_1 = q_2$ a.e.~on $[0,\infty)$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the remainder of this section, we consider a variety of
generalizations of the
result obtained.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r2.8} The ray $\arg(z) = \pi -\veps$,
$0<\veps < \pi/2$ chosen in
Theorem~\ref{t1.1} and Corollary~\ref{2.6} is of no particular
importance. A limit
taken along any non-self-intersecting curve $\calC$ going to
infinity in the sector
$\arg(z)\in (\pi/2+\veps, \pi -\veps)$ will do as we can apply
the Phragm\'en-Lindel\"of
principle (\cite[Part~III, Sect.~6.5]{PS72}) to the region
enclosed by $\calC$ and its
complex conjugate $\bar\calC$ (needed in connection with
Lemma~\ref{l2.4} in order to
reduce the general case to the case of spectra bounded from
below).
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r2.9} For simplicity of exposition, we only
discussed the
Dirichlet boundary condition
\begin{equation} \lb{2.37}
g(0_+)=0
\end{equation}
in the definition of $H$ in \eqref{2.2}. Next we replace
\eqref{2.37} by the general
boundary condition
\begin{equation} \lb{2.38}
\sin(\alpha) g'(0_+) + \cos(\alpha) g(0_+) = 0, \quad
\alpha\in [0,\pi)
\end{equation}
in \eqref{2.2}, denoting the resulting Schr\"odinger operator
by $H_\alpha$, while
keeping the boundary condition at infinity (if any) identical
for all $\alpha\in
[0,\pi)$. Denoting by $m_\alpha (z)$ the Weyl-Titchmarsh
function associated with
$H_\alpha$, the well-known relation (cf.~e.g., Appendix~A of
\cite{GS96} for precise
details on $H_\alpha$ and $m_\alpha(z)$)
\begin{equation} \lb{2.39}
m_\alpha(z) = \f{-\sin(\alpha) + \cos(\alpha) m(z)}{\cos(\alpha)
+ \sin(\alpha) m(z)}\, ,
\quad \alpha\in [0,\pi), \ z\in \bbC\backslash\bbR
\end{equation}
reduces the case $\alpha\in (0,\pi)$ to the Dirichlet case
$\alpha =0$.
In particular, Theorem~\ref{t1.1} and Corollary~\ref{c2.6}
remain valid
with $m_j (z)$ replaced by $m_{j,\alpha}(z)$,
$\alpha\in [0,\pi)$.
Indeed, $|m_{1,\alpha}(z) - m_{2,\alpha}(z)|
\underset{|z|\to\infty}{=}
O(e^{-2\Ima (z^{1/2})a})$ along the ray $\arg(z) =
\pi -\veps$ is easily
seen to imply, for all sufficiently small $\delta >0$,
\begin{equation} \lb{2.40}
\begin{aligned}
|m_{1,0}(z) - m_{2,0}(z)| &\eqlim_{|z|\to\infty} O(|z|\,
e^{-2\Ima(z^{1/2})a}) \\
&\eqlim_{|z|\to\infty} O(e^{-2\Ima (z^{1/2})(a-\delta)})
\end{aligned}
\end{equation}
along the ray $\arg(z) = \pi - \veps$. Hence one infers
from Theorem~\ref{t1.1}
that for all $0<\delta 0$ can be chosen arbitrarily small, one
concludes $q_1 = q_2$
a.e.~on $[0,a]$. In fact, more is true. Since
$m_\alpha(z)\underset{|z|\to\infty}{\to}\cot(\alpha)$
along the ray, one
concludes that
$|m_{1,\alpha_1} - m_{2,\alpha_2}(z)|
\underset{|z|\to\infty}{=}
O(e^{-2\Ima (z^{1/2})a})$ along a ray implies $\alpha_1
= \alpha_2$
and $q_1 = q_2$ a.e.~on $[0, a]$.
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{remark} \lb{r2.10} If one is interested in a finite
interval
$[0,b]$ instead of the half-line $[0,\infty)$ in
Theorem~\ref{t1.1}, with
$00$, $0<\veps<\pi/2$ and
suppose that
\begin{equation} \lb{2.46d}
\|\calM_1 (z) - \calM_2(z)\|_{\bbC^{2\times 2}}
\eqlim_{|z|\to\infty}
O(e^{-2\Ima (z^{1/2})a})
\end{equation}
along the ray $\arg(z) = \pi-\veps$. Then
\begin{equation} \lb{2.46e}
q_1(x) = q_2 (x) \text{ for a.e. } x\in [-a,a].
\end{equation}
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
We denote by $m_{j,\pm}(z)$ the half-line (Dirichlet)
$m$-functions
associated with $H_j$ on $[0,\pm\infty)$, $j=1,2$. Then a
straightforward combination of \eqref{2.8} and
\eqref{2.46d} yields
\begin{equation}
|m_{1,\pm}(z)-m_{2,\pm}(z)|\underset{|z|\to\infty}{=}
O(|z|e^{-2\Ima (z^{1/2})a}) \lb{2.46f}
\end{equation}
and hence \eqref{2.46e}, applying Theorem~\ref{t1.1}
separately
to the two half-lines $[0,\infty)$ and $(-\infty,0]$
(and using the
argument following \eqref{2.40}.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Finally, the reader might be interested in the analog of
Theorem~\ref{t1.1} in the case of
second-order difference operators, that is, Jacobi operators.
Let $A$ be a bounded
self-adjoint Jacobi operator in $\ell^2 (\bbN_0)$ ($\bbN_0
= \bbN\cup\{0\}$) of the
type
\begin{equation} \lb{2.47}
A = \left(
\begin{matrix}
b_0 & a_0 & 0 & 0 & \dots & \dots\\
a_0 & b_1 & a_1 & 0 & \dots & \dots \\
0 & a_1 & b_2 & a_2 & \dots & \dots\\
0 & 0 & a_2 & \ddots & \ddots \\
\vdots & \vdots & \vdots & \ddots & \ddots & \ddots \\
\vdots & \vdots & \vdots & {} & \ddots & \ddots
\end{matrix} \right), \quad a_k >0, \ b_k\in\bbR, \ k\in\bbN_0.
\end{equation}
The corresponding $m$-function of $A$ is then defined by
\begin{equation} \lb{2.48}
m(z) = (\delta_0, (A-z)^{-1}\delta_0)
= \int_\bbR d\rho(\lambda) (\lambda -z)^{-1},
\quad z\in\bbC\backslash\bbR,
\end{equation}
where $\delta_0 = (1,0,0,\dots)$. The analog of
Theorem~\ref{t1.1} in the discrete case
then reads as follows. Denote by $m_j(z)$ the $m$-functions
for two self-adjoint Jacobi
operators $A_j$, $j=1,2$, denoting the matrix elements of
$A_j$ by $a_{j,k}$, $b_{j,k}$,
$j=1,2$, $k\in \bbN_0$. Then
\begin{equation} \lb{2.49}
|m_1(z) - m_2(z)| \eqlim_{|z|\to\infty} O(|z|^{-N}),
\end{equation}
for some $N\in\bbN$, $N\geq 3$, if and only if
\begin{equation} \lb{2.50}
a_{1,k} = a_{2,k}, \quad b_{1,k} = b_{2,k},
\quad 0\leq k \leq \f{N-4}{2}
\quad\text{if $N$ is even} \quad (N\geq 4)
\end{equation}
and
\begin{equation} \lb{2.51}
\begin{aligned}
a_{1,k} &= a_{2,k}, \quad 0\leq k \leq \f{N-5}{2}\, , \\
b_{1,k} &= b_{2,k}, \quad 0\leq k \leq \f{N-3}{2}
\quad\text{if $N$ is odd}.
\end{aligned}
\end{equation}
The proof is clear from \eqref{2.48} and the well-known
formulas
(cf.~\cite[Sect.~VII.1]{Be68}).
\begin{equation} \lb{2.52}
a_k = \int_\bbR d\rho(\lambda) \, \lambda P_k(\lambda)
P_{k+1}(\lambda),
\quad b_k = \int_\bbR d\rho(\lambda)\, \lambda P_k(\lambda)^2,
\quad
k\in\bbN_0,
\end{equation}
where $\{P_k(\lambda)\}_{k\in\bbN_0}$ is an orthonormal
system of polynomials with
respect to the spectral measure $d\rho$, with $P_k(z)$ of
degree $k$ in $z$, $P_0(z) =1$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Matrix-Valued Schr\"odinger Operators} \lb{s3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In our final section we extend Theorem~\ref{t1.1} to
matrix-valued potentials
(cf., \cite[Ch.~III]{CL90}, \cite{KS88}, \cite{La84} and
the references therein).
Let $m\in\bbN$ and denote by $I_m$ the identity matrix
in $\bbC^m$. Assuming
\begin{equation} \lb{3.1}
Q=Q^* \in L^1 ([0,R])^{m\times m} \text{ for all } R>0,
\end{equation}
we introduce the corresponding matrix-valued self-adjoint
Schr\"odinger operator
$H$ in $L^2 ([0,\infty))^m$ with a Dirichlet boundary
condition at $x=0_+$, by
\begin{equation}
\begin{align}
&H=-\f{d^2}{dx^2}\,I_m + Q, \lb{3.2} \\
&\dom(H) = \{g\in L^2 ([0,\infty))^m \mid g,g'
\in \AC([0,R])^m\text{ for all }R>0; \no \\
&\hspace*{17mm} g(0_+)=0, \,\sabc\text{ at }
\infty; \, (-g''+Qg)
\in L^2 ([0,\infty))^m\}. \no
\end{align}
\end{equation}
Here ``$\sabc$ at $\infty$" again denotes a self-adjoint
boundary condition at $\infty$
(if $Q$ is not in the limit point case at $\infty$). For
more details about the limit
point/limit circle and all the intermediate cases, see
\cite{CG99,HS81,HS83,HS84,KR74,
Kr89a,Kr89b,Or76,Ro69} and the references therein.
Next, let $\Psi(z,x)$ be the unique (up to right multiplication
of non-singular constant
$m\times m$ matrices) $m\times m$ matrix-valued Weyl solution
associated with $H$,
satisfying
\begin{align}
&\Psi(z,\dott) \in L^2 ([0,\infty))^{m\times m}, \quad
z\in\bbC\backslash\bbR, \lb{3.3} \\
&\Psi(z,x)\text{ satisfies the $\sabc$ of $H$ at
$\infty$ (if any)},
\lb{3.4} \\
&-\Psi''(z,x) + [Q(x) - zI_m]\Psi(z,x) = 0. \lb{3.5}
\end{align}
The $m\times m$ matrix-valued Weyl-Titchmarsh function
$M(z)$ associated with $H$ is then
defined by
\begin{equation} \lb{3.6}
M(z) = \Psi' (z, 0_+) \Psi(z, 0_+)^{-1},
\quad z\in\bbC\backslash\bbR
\end{equation}
and similarly, we introduce its $x$-dependent version,
$M(z,x)$, by
\begin{equation} \lb{3.7}
M(z,x) = \Psi' (z,x) \Psi(z,x)^{-1},
\quad z\in\bbC\backslash\bbR, \, x\geq 0.
\end{equation}
The matrix Riccati equation satisfied by $M(z,x)$, the
analog of
\eqref{2.10}, then reads
\begin{equation} \lb{3.8}
M'(z,x) + M(z,x)^2 = Q(x) - zI_m \text{ for a.e.}~x\geq 0
\text{ and all } z\in\bbC\backslash\bbR.
\end{equation}
Next, let $Q_j(x)$, $j=1,2$ be two self-adjoint matrix-valued
potentials satisfying
\eqref{3.1}, and $M_j(z)$, $M_j(z,x)$ the Weyl-Titchmarsh
matrices associated with the
corresponding (Dirichlet) Schr\"odinger operators. Then
the analog of \eqref{2.11} is
of the form
\begin{align}
&[M_1(z,x) - M_2(z,x)]' \lb{3.9} \\
&=Q_1 (x) - Q_2(x) - \tfrac{1}{2} [M_1 (z,x) + M_2(z,x)]
[M_1 (z,x) - M_2 (z,x)] \no \\
& \quad - \tfrac{1}{2} [M_1 (z,x)
- M_2(z,x)][M_1 (z,x) + M_2
(z,x)]. \no
\end{align}
Combining \eqref{3.9} with the elementary fact that any
$m\times m$ matrix-valued
solution $U(x)$ of
\begin{equation} \lb{3.10}
U'(x)=B(x) U(x) + U(x)B(x)
\end{equation}
is of the form
\begin{equation} \lb{3.11}
U(x) = V(x) CW(x),
\end{equation}
where $C$ is a constant $m\times m$ matrix and $V(x)$,
respectively,
$W(x)$, is a fundamental system of solutions of $R'(x)
= B(x)R(x)$,
respectively, $S'(x) = S(x)B(x)$, one can prove the
analogs of
Theorems~\ref{t2.1}--\ref{t2.3} in the present matrix
context. More
precisely, the matrix analogs of Theorems~\ref{t2.1} and
\ref{t2.2}
follow from Theorem~4.8 in \cite{CG99}. The corresponding
analog of
Theorem~\ref{t2.3} follows from Theorem~4.5 and
Remark~4.7 in
\cite{CG99}. Moreover, in the case that $H$ is bounded
from below,
Lemma~\ref{l2.4} generalizes to the matrix-valued context
and hence
permits one to take the limit $z\downarrow -\infty$ in
the matrix
analog of
\eqref{2.13}. While the scalar case treated in detail in
\cite{GS98} is based on
Riccati-type identities such as \eqref{2.11} and an
{\it a priori} bound
of the type
\eqref{2.9} inspired by Atkinson's 1981 paper \cite{At81},
the matrix-valued case
discussed in depth in \cite{CG99} is based on corresponding
Riccati-type identities
such as \eqref{3.9} and an {\it a priori} bound of the type
\begin{equation} \lb{3.12}
M(z,x) = iz^{1/2} I_m + o(|z|^{1/2})
\end{equation}
first obtained by Atkinson in an unpublished manuscript
\cite{At??}.
In the special case of short-range matrix-valued potentials
$Q(x)$, $m\times m$ matrix
analogs of the Jost solution $F(z,x)$ as well as the
transformation kernel $K(x,y)$
associated with $H$ as in \eqref{2.16}--\eqref{2.20}
(replacing $|\dott|$ by an
appropriate matrix norm $\|\dott\|_{\bbC^{m\times m}}$,
have been
discussed in great detail in the classical 1963 monograph
by Agranovich
and Marchenko
\cite[Ch.~I]{AM63}. Moreover,
\eqref{2.21}--\eqref{2.23} trivially extend to the matrix case.
Given these preliminaries, the analog of Theorem~\ref{t1.1}
and Corollary~\ref{c2.6}
reads as follows in the matrix-valued context.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\lb{t3.1} Let $a>0$, $0<\veps<\pi/2$ and
suppose
\begin{equation} \lb{3.13}
\|M_1(z) - M_2(z)\|_{\bbC^{m\times m}}
\eqlim_{|z|\to\infty} O(e^{-2\Ima
(z^{1/2})a})
\end{equation}
along the ray $\arg(z) = \pi - \veps$. Then
\begin{equation} \lb{3.14}
Q_1(x) = Q_2(x) \text{ for a.e. } x\in [0,a].
\end{equation}
In particular, if \eqref{3.13} holds for all $a>0$,
then $Q_1 = Q_2$ a.e.~on $[0, \infty)$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Sketch of Proof] As in the scalar case, we
may assume without loss of
generality that
\begin{equation} \lb{3.15}
\supp (Q_j) \subseteq [0,a], \quad j=1,2.
\end{equation}
The fundamental identity \eqref{2.26}, in the present
non-commutative case, needs to
be replaced by
\begin{equation} \lb{3.16}
\f{d}{dx}\, W(F_1 (\bar z, x)^*, F_2(z,x)) =
-F_1(\bar z,x)^* [Q_1(x) - Q_2(x)]
F_2 (z,x),
\end{equation}
where $F_j(z,x)$ denote the $m\times m$ matrix-valued Jost
solutions associated with
$Q_j$, $j=1,2$, and $W(F,G)(x) = F(x) G'(x) - F'(x)G(x)$
the matrix-valued
Wronskian of $m\times m$ matrices $F$ and $G$. Identity
\eqref{2.27} then becomes
\begin{align}
&\int_0^a dx\, F_1(\bar z,x)^* [Q_1(x) - Q_2(x)]
F_2 (z,x) \no \\
&= F_1(\bar z, x)^* [M_1(z,x) - M_2(z,x)]
F_2(z,x)\bigg|_{x=0}^a\, ,
\lb{3.17}
\end{align}
utilizing the fact
\begin{equation} \lb{3.18}
M_1(\bar z,x)^* = M_1(z,x).
\end{equation}
$F_j$ obeys a transformation kernel representation
\begin{align}
&F_j (z,x) = e^{iz^{1/2}x} I_m + \int_x^a dy\, K_j (x,y)\,
e^{iz^{1/2}y} I_m \, , \lb{3.19} \\
& \hspace*{3cm} \Ima(z^{1/2}) \geq 0, \, x\geq 0, \,
j=1,2. \no
\end{align}
From this, \eqref{3.12}, and the hypothesis of \eqref{3.13},
one concludes by \eqref{3.17} that
\begin{equation} \lb{3.20}
\int_0^a dx\, F_1 (\bar z, x) [Q_1(x) - Q_2(x)] F_2 (z,x)
\underset{z\downarrow -\infty}{=} O(e^{-2\Ima(z^{1/2})a}).
\end{equation}
Now let $R_A$ be right multiplication by $A$ on $n\times n$
matrices and $L_B$ be
left multiplication by $B$. Then
\begin{align}
&\text{LHS of \eqref{3.20}} \lb{3.21} \\
&= \int_0^a dx\, \left\{Q_1(y) - Q_2(y) + \int_0^y dx\,
\calL(x,y) [Q_1(x) - Q_2(x)]\right\} e^{-2z^{1/2}y}, \no
\end{align}
where $\calL$ is an operator on $n\times n$ matrices which
is a sum of a left
multiplication (by $2K_j(x, 2y-x)$), a right multiplication
(by $2K_2 (x, 2y-x)$), and
a convolution of a left and right multiplication.
It follows by Lemma~\ref{l2.5}, \eqref{3.20}, and
\eqref{3.21} that
\begin{equation} \lb{3.22}
Q_1(y) - Q_2(y) + \int_0^y dx \, \calL(x,y) [Q_1 (x)
- Q_2(x)] = 0.
\end{equation}
This is a Volterra equation and the same argument based on
\[
\int_0^y dx_1 \int_0^{x_1} dx_2 \cdots \int_0^{x_{n-1}}
dx_n = \f{y^n}{n!}
\]
that a Volterra operator has zero spectral radius applies to
operator-valued Volterra equations. Thus, \eqref{3.22}
mplies
$Q_1(y) - Q_2(y) = 0$ for a.e.~$y\in [0,a]$.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Extensions of Theorem~\ref{t3.1} in the spirit of
Remarks~\ref{r2.8}--\ref{r2.10} and Theorem~\ref{t2.11}
can be made, but
we omit the corresponding details at this point.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{2mm}
\noindent {\bf Acknowledgments.} F.~G.~thanks
T.~Tombrello for the hospitality of Caltech where this work
was done.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}
\end{document}
\begin{equation} \lb{}
\end{equation}