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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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\gap
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\gap
\end{document}