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\def\gap{\vskip 0.1in\noindent}
\def\ref#1#2#3#4#5#6{#1, {\it #2,} #3 {\bf #4} (#5), #6.}

%References
\def\am  {1}   %Agranovich-Marchenko
\def\akto {2}   % Aktosun
\def\aktosun {3}  %Aktosun
\def\akv {4}   %Aktosun, Klaus, van der Mee
\def\bo {5}   %Borg
\def\bsl {6}   %Braun, et al.
\def\ds {7}   %Davies-Simon
\def\gsdis {8}  %Gesztesy-Simon, discrete spectrum
\def\gsmfunc {9} %Gesztesy-Simon, m-functions
\def\ghs {10}    %Gesztesy-Holden-Simon
\def\gnp {11}   %Gesztesy-Nowell-Potz
\def \gw {12}   %Grebert-Weder
\def\hl {13}      %Hochstadt-Lieberman
\def\katz {14}   %Katznelson
\def\km {15}    %Kay-Moses
\def\ls {16}      % Levitan-Sargsjan
\def\ma {17}    %Marchenko
\def\rs {18}      % Rundell-Sacks
\def\sac {19}    % Sacks
\def\simon {20}  % Simon

\topmatter
\title Inverse Spectral Analysis With Partial
Information on
the Potential, I. The Case
of an A.C.~Component in the Spectrum
\endtitle
\rightheadtext{Inverse Spectral Analysis: The Case
of A.C.~Spectrum }
\author Fritz Gesztesy$^1$ and Barry Simon$^2$
\endauthor
\leftheadtext{F.~Gesztesy and B.~Simon}
\thanks$^1$ Department of Mathematics, University
of Missouri, Columbia, MO 65211,
USA. \linebreak E-mail: mathfg\@mizzou1.missouri.edu
\endthanks
\thanks$^2$ Division of Physics, Mathematics, and
Astronomy,
California Institute of
Technology, Pasade\-na, CA 91125, USA. E-Mail:
bsimon\@caltech.edu
\endthanks
\thanks
This material is based upon work supported by the National
Science Foundation
 under
Grant Nos.~DMS-9623121 and DMS-9401491.
\endthanks
\keywords Inverse scattering theory, Weyl $m$-functions,
Schr\"odinger operators
\endkeywords
\subjclass  Primary 34A55, 34B20, 34L25; Secondary 34L40
\endsubjclass
\thanks To appear in {\it{Helv.~Phys.~Acta}}
\endthanks
%\date August 13, 1996
%\enddate
\dedicatory Dedicated to Klaus Hepp and Walter
Hunziker on
the occasion of their
sixtieth birthdays
\enddedicatory
\abstract We consider operators
$-\frac{d^2}{dx^2} + V$ in
$L^2 (\Bbb R)$ with
 the sole
hypothesis that $V$ is limit point at $\pm\infty$
and that
$-\frac{d^2}{dx^2} +
 V$ in
$L^2 ((0,\infty))$ has some absolutely continuous
component
$S_+$ in its
 spectrum. We
prove that $V$ on $(-\infty, 0)$ is completely
determined
by knowledge of $V$ on
$(0, \infty)$ and by the reflection coefficient
$R_{+}(\lambda)$ for scattering
 from right
incidence and energies $\lambda \in S$, where
$S \subseteq S_+$ has positive
 Lebesgue
measure.
\endabstract
\endtopmatter

\document
\vskip 0.3in

It is well known [\km] that knowledge of the reflection
coefficient at positive
 energies does
not determine the potential $V$ of a Schr\"odinger
operator $-\frac{d^2}{dx^2} +
 V$
($V(x)\to 0$ sufficiently rapidly as $|x|\to\infty$), but
that one also needs
 bound state energies
and associated norming constants. This is most
dramatically
seen in one-soliton
 potentials
where $R_{+}(\lambda)\equiv 0$, $\lambda \geq 0$,
even
though there is a
 two-parameter
family of such potentials parametrized by the
center and
width of the soliton.

There has been a recent rash of papers [\akto,
\aktosun,
\akv, \bsl, \gw, \rs,
 \sac] showing that
if $V$ is known a.e.~on a half-line and vanishes
sufficiently fast as $|x| \to
 \infty$ in the sense
that at least its first moment on $\Bbb R$ exists,
then
the norming constants
 and even the
bound state energies are not needed (some of
these papers
are limited to the
 case where $V$
is assumed to vanish on the right half-line). Our
goal here
is to note that this
 is a special case
of a very general and very elementary phenomenon:
It is not
required that $V$
 has simple
asymptotics as $|x| \to \infty$. Rather, all that is
significant is that $V$ be
 known a.e.~on
$(0,\infty)$ and the Schr\"odinger operator $H_+$
associated with
 $-\frac{d^2}{dx^2} +
V$ in $L^2((0,\infty))$ and any self-adjoint boundary
condition at $0$, has some
 absolutely
continuous (a.c.) component in its spectrum.  Also,
rather
than require detailed
 manipulation
of the machinery of inverse problems and/or trace
formulas,
all that is required
 is a uniqueness
result to go from a Weyl $m$-function to a potential. In
particular, our
 $m$-function technique allows one to consider impurity
(defect) scattering in
 (half) crystals, scattering off potentials
with different spatial asymptotics at left and right
including asymptotically
 periodic potentials,
potential steps, and potentials diverging to
$+ \infty$
as $x \to -\infty$.

More subtle and deep is a comparison problem
concerning
knowledge of the
 potential on a
half-line where the spectrum is purely discrete
rather than
having an absolutely
 continuous
component. Here the paradigmal result is the
remarkable
theorem of Hochstadt and
 Lieberman
[\hl] that a knowledge of all the eigenvalues of
$-\frac{d^2}{dx^2} + V$ in $L^2
 ((0,1); dx)$
with (for example) Neumann boundary conditions
$u'(0) = u'(1) = 0$ and knowledge
 of the
potential on $(0, \frac12)$, uniquely determine
$V$ a.e.~on
all of $(0,1)$.  We
 will study these problems in two forthcoming
papers [\gsdis,
\gsmfunc]. Typical
 of our results is that a
knowledge of $V$ on $(0,\frac34)$ and of strictly
more than
half the eigenvalues
 uniquely
determines $V$ a.e.~on all of $(0,1)$.

Suppose that $V \in L^1_{\text{\rom{loc}}}
(\Bbb R)$ is
real-valued such that
 the differential
expression $-\frac{d^2}{dx^2} + V(x)$ is in the
limit point
case at $\pm\infty$.
 Then for any
$z$ with $\text{Im}\, (z) >0$, there is a unique
(up to
constant multiples)
 solution of
$$
-u'' + Vu = zu \tag 1
$$
which is $L^2$ at $+\infty$. Call it $\tilde \psi_+ (z,x)$.
Similarly, there is
 a solution
$\tilde\psi_- (z,x)$ which is $L^2$ at $-\infty$. The Weyl
$m$-functions $m_\pm$
 are
defined by
$$
m_\pm (z) = \pm \frac{\tilde \psi'_\pm (z,0)}
{\tilde \psi_\pm (z,0)}.
$$

It is a fundamental result of Marchenko [\ma] that
$m_\pm (z)$ uniquely
 determines $V$
a.e.~on $(0, \pm\infty)$. General principles (see, e.g.,
[\katz], Sect.~III.1;
 [\ls], Sect.~2.4;
[\simon]) imply that for a.e.~$\lambda \in \Bbb R$,
$\lim_{\epsilon\downarrow 0}
 m_\pm
(\lambda + i\epsilon):= m_\pm (\lambda+ i0)$ exists and
is finite. For such
 $\lambda \in
\Bbb R$, we'll define $\psi_\pm (\lambda,x)$ by requiring
that $\psi_{\pm}$
 satisfies (1)
(with $z=\lambda$) and the boundary conditions
$$
\psi_\pm (\lambda,0) = 1, \qquad \psi'_\pm (\lambda,0) =
\pm m_\pm (\lambda
 +i0).
\tag 2
$$

\example{Example} $V=0$. Then $m_\pm (z) =i\sqrt{z}$,
choosing the square root
 branch
with $\text{Im}\, (\sqrt{z}) > 0$ for $z \in
\Bbb C \backslash [0,\infty)$ and
 $\psi_\pm
(\lambda,x) = e^{\pm i \sqrt{\lambda}\, x}$ (where
$\sqrt{\lambda}> 0$) if
 $\lambda \geq 0$
and $\psi_\pm (\lambda,x)
=e^{\mp \sqrt{-\lambda}\, x}$
if $\lambda \leq 0$.
\endexample

It is also known [\ls, \simon] that if $H_+$ is
associated with
 $-\frac{d^2}{dx^2}+ V$ in
$L^2 ((0, \infty))$ and Dirichlet boundary conditions
$u(0)=0$ (or equivalently,
 any other
self-adjoint boundary condition at $0$ of the type
$u'(0)+\beta u(0)=0$, $\beta
 \in \Bbb R$),
then the essential support of the a.c.~spectrum of
$H_+$ is precisely $S_+ :=
 \{\lambda \in
\Bbb R \mid \text{Im}\, [m_+ (\lambda + i0)] >0\}$.
For $\lambda \in S_+$,
 $\psi_+
(\lambda,x)$ is not a multiple of a real solution, so
$\overline{\psi_+
 (\lambda,x)}$ is always
a linearly independent solution of (1). As a result we
can expand,
$$
\psi_- (\lambda,x) = A(\lambda) \,
\overline{\psi_+ (\lambda,x)} \,
+ B(\lambda) \psi_+(\lambda,x), \, \,
\lambda \in S_+. \tag 3
$$

\definition{Definition} For $\lambda \in S_+$,
$R_+(\lambda) :=
 B(\lambda)/A(\lambda)$
denotes the (relative) reflection coefficient (from
right incidence).
\enddefinition

\remark{Remarks} 1. Suppose that $V=0$ on $(0,\infty)$
so $\psi_+ (\lambda,x) =
 e^{i\lambda^{1/2}x}$ and that for some $\epsilon >0$,
$V=0 (|x|^{-1-\epsilon})$
 at $-\infty$
so $\psi_- (\lambda,x) \sim C e^{-i\lambda^{1/2}x}$
near $-\infty$. Then the
 usual reflection
coefficient is $B/A$ and the usual transmission
coefficient $C/A$. Thus, this
 very general definition agrees with the usual one if
$V=0$ on $(0,\infty)$.

2.  If $V=0 (|x|^{-1-\epsilon})$ at $\pm\infty$, then
$\psi_+ (\lambda,x)\sim
 D(\lambda)
3.  e^{i\lambda^{1/2}x}$ at $+\infty$ (note we chose
a particular normalization
 of
$\psi_+(\lambda,x)$ in (2)). In this case, the usual
reflection coefficient is
 {\it{not}} $B/A$
but is $(B/A)(D/ \overline D) =\tilde R_+$. However, if
$V$ is explicitly known
 on
$[0, \infty)$, so is $D$, and thus knowing $R_+$ is the
same as knowing $\tilde
 R_+$.

3. (2) and (3) let us solve for $A,B$ and $R$ in terms
of $m_\pm$, viz.,
$$\align
A(\lambda) & = \frac{m_+ (\lambda + i0) + m_- (\lambda
+ i0)}{2i \,
 \text{Im}(m_+
(\lambda + i0))}, \\
B(\lambda) &= -\frac{\overline m_+ (\lambda +i0)
+ m_- (\lambda +i0)}
{2i  \, \text{Im}(m_+ (\lambda +i0))}, \\
R_+(\lambda) &= -\frac{\overline m_+ (\lambda +i0)
+ m_- (\lambda +i0)}
{m_+ (\lambda +i0) + m_- (\lambda +i0)}, \quad \lambda
\in S_+, \tag 4
\endalign
$$
(see also the corresponding discussions in [\gnp]). In
particular, since
 $\text{Im}\, (m_+)$,
$\text{Im} \, (m_-) \geq 0$, we have
$|R_+(\lambda)|\leq 1$. Also, since
 $\text{Im}\,
[m_+ (\lambda +i0)] >0$ for a.e.~$\lambda \in S_+$,
the essential support of
$\sigma_{\text{\rom{ac}}} (H_+)$,
$$
R_+(\lambda) \neq -1 \quad \text{for a.e. }
\lambda \in S_+. \tag 5
$$
\endremark

\proclaim{Theorem} Assume that $V \in
L^1_{\text{\rom{loc}}}(\Bbb R)$ is
 real-valued and
$-\frac{d^2}{dx^2} + V(x)$ is in the limit point case
at $\pm \infty$. Suppose
 that $V$ is
known a.e.~on $(0,\infty)$ and that  $R_+(\lambda)$ is
known a.e.~on a set $S
 \subseteq
S_+$ of positive Lebesgue measure inside the essential
support $S_+$ of
$\sigma_{\text{\rom{ac}}} (H_+)$. Then $V$ is uniquely
determined a.e.~on
 $(-\infty, 0)$
and hence a.e.~on $\Bbb R$.
\endproclaim

\demo{Proof} By (4),
$$
m_- (\lambda + i0) = - \, \frac{ m_+ (\lambda+i0)
R_+(\lambda) +
 \overline{m_+(\lambda +i0)}}
{(1+ R_+(\lambda))} \quad \text{for a.e. } \lambda
\in S. \tag 6
$$
By (5), $m_-$ is well defined for a.e.~$\lambda \in S$.
Thus knowing
 $R_+(\lambda)$ a.e.~on
$S$ and knowing $m_+$ a.e.~on $S$ (since we know $V$
a.e.~on $(0,\infty)$), we
 know
$m_-(\lambda+i0)$ a.e.~on $S$. But $m_-$ is the
boundary
value of a Herglotz
 function and
such functions are determined uniquely by their
boundary
values on any set of
 positive
Lebesgue measure, and so on $S$. By Marchenko's
uniqueness theorem [\ma], $m_-$
 uniquely
determines $V$ a.e.~on $(-\infty,0)$. \qed
\enddemo

\remark{Remarks} 1. The principal strategy behind our
theorem and the results in
[\gsdis, \gsmfunc] is extremely simple and may be
summarized as follows:
 Consider a
Schr\"odinger operator $-\frac{d^2}{dx^2} + V$ on an
interval $(a,b) \subseteq
 \Bbb R$
with fixed separated boundary conditions (if any)
at $a$
and $b$. Suppose $x_0
 \in (a,b)$
and denote by $m_{+,x_0}$ and $m_{-,x_0}$ the Weyl
$m$-functions associated with
 the
intervals $(x_0,b)$ and $(a,x_0)$, respectively. By
Marchenko's uniqueness
 theorem [\ma],
$m_{+,x_0}$ and $m_{-,x_0}$ uniquely determine $V$
a.e.~on $(x_0,b)$ and
 $(a,x_0)$.
Hence, if $V$ (and thus $m_{+,x_0}$) is known on
$(x_0,b)$, one only needs to
 specify
$m_{-,x_0}$ in order to determine $V$ uniquely
a.e.~on $(a,b)$. The issue thus
 becomes
determination of $m_{-,x_0}$ from knowledge of
$m_{+,x_0}$ and additional
 spectral
(e.g., scattering) data associated with
$-\frac{d^2}{dx^2} + V$ on $(a,b)$. For
 instance,
if $(a,b)= \Bbb R$, $x_0 =0$, and $-\frac{d^2}{dx^2}
+ V$ restricted to $(0,
 \infty)$ has
an a.c.~component in its spectrum as considered in this
paper, the reflection
 coefficient $R_+$
from right incidence together with $m_+$ determine
$m_-$ and hence $V$ on $\Bbb
 R$. If, on
the other hand, $-\frac{d^2}{dx^2} + V$ on $(a,b)$
has purely discrete spectrum
 as considered
in [\gsdis], then a certain portion of the eigenvalues
of $-\frac{d^2}{dx^2} +
 V$ on $(a,b)$,
the portion depending on $x_0$, together with
$m_{+,x_0}$ will again determine
 $m_{-,x_0}$
and hence $V$ on all of $(a,b)$ as long as the size of
the interval $(x_0,b)$ is
 ``sufficiently
large" compared to the size of $(a,x_0)$. The fact that
$m_{\pm,x_0}$ are
 Herglotz functions
(and in the discrete spectrum case also meromorphic)
then considerably aids in
 determining
$m_{-,x_0}$. This comment also underscores that our
approach is by no means
 restricted to
Schr\"odinger operators on $\Bbb R$. It applies as well
to one-dimensional
 Dirac-type operators,
second-order finite difference (Jacobi) operators
[\gsmfunc], and $n \times n$
 matrix-valued
Schr\"odinger operators [\am] (in this case
$m_{\pm,x_0}$, $R_+$, etc., are $n
 \times n$
matrices) on arbitrary intervals $(a,b)$. In particular,
it applies to
 three-dimensional
Schr\"odinger operators with spherically symmetric
potentials $v(x)=V(|x|)$, $x
 \in \Bbb R^3$
upon decomposition with respect to angular momenta
and restriction to the
 angular momentum
channel $\ell =0$.

2. In some cases, one only needs to know
$m_- (\lambda +i0)$ on a smaller set
 than one of
positive measure. For example, if it is known a priori
that for some $\alpha
 >0$, $|V(x)| \leq
e^{-\alpha |x|}$ near $x=-\infty$, then $m_-$ is known
to be analytic in a
 neighborhood of
$\Bbb R$, and so it suffices that $R_+(\lambda)$ (and
so $m_- (\lambda +i0)$) is
 known on
a set of points with a finite limit point. Or if the
restriction of $V$ to
 $(-\infty, 0]$ is known
to have compact support, then $m_-$ is a ratio of
entire functions of order
 $\frac12$ and
known type (depending on the size of the support
in $(-\infty, 0]$), so $m_-$ is
 uniquely
determined by a sequence of values $\lambda_j \to
\infty$ of sufficient density.

3. All the results of [\akto, \aktosun, \akv, \bsl,
\gw, \rs, \sac] are
 consequences of our theorem
save that in [\rs], which follows from the extension
indicated at the end of
 Remark~1. (For
those results where one only supposes $V(x)$ vanishes
in $(b, \infty)$ rather
 than $(0, \infty)$,
we use the fact that $b$ can be determined from
$R_+$ [\akto], and then the
 problem can be
translated to one with $V$ vanishing on $(0, \infty)$.)

4. An example of a totally new result is a situation
where $V(x)\to \infty$ as
 $x\to -\infty$ in
which case $|R_+(\lambda)|=1$. By a result of Borg [\bo],
it suffices, for
 example, to consider
$V(x) = 0$, $x>0$, $V(x)\geq 0$ for $x <0$,
$V(x)\to\infty$ at $-\infty$ and to
 then know
those energies $\lambda_j$ with $R_+(\lambda_j) = -1$
and those $\lambda_k$ with
$R_+(\lambda_k)=+1$.

5. Other situations of interest in physics, covered
by our theorem but not
 addressed by previous
results in this context, concern impurity (defect)
scattering in (half) crystals
 and charge
transport in mesoscopic quantum-interference devices
associated with (possibly
 different)
asymptotically periodic potentials as
$x \to \pm \infty$. The interested reader
 might consult
[\ds, \ghs, \gnp] and the literature cited therein.
\endremark

\vskip 0.5in

\example{Acknowledgments} We would like to thank
Tuncay Aktosun and Alexei
 Rybkin
for discussions and pertinent hints to the literature.
F.G.~is indebted to
 A.~Kechris and
C.W.~Peck for a kind invitation to Caltech during the
summer of 1996 where some
 of this
work was done. The extraordinary hospitality and support
by the Department of
 Mathematics
at Caltech are gratefully acknowledged. B.S.~would
like to thank M.~Ben-Artzi of
 the
Hebrew University where some of this work was done.

\vskip 0.3in
\example{Dedication}  It is an enormous pleasure to
dedicate this paper in honor
 of the
sixtieth birthdays of Klaus Hepp and Walter Hunziker.
During his mathematical
 physics
phase, Klaus made important contributions to quantum
field theory. Walter has
 been a
major figure in multiparticle quantum theory for more
than thirty years, and we
 have
learned much from him.
\endexample

\vskip 0.3in
\Refs
\endRefs

\vskip 0.1in

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Inverse Problem of
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\gap
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 Schr\"odinger
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{6231--6236}
\gap
\item{\aktosun.} \ref{T.~Aktosun}{Inverse Schr\"odinger
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\gap
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\gap
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\gap
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\gap
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\gap
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 line that
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\gap
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\gap

\end{document}