

\input amstex
\loadbold
\documentstyle{amsppt}
\pagewidth{32pc}
\pageheight{45pc}
\mag=1200
%\magnification=\magstephalf
\baselineskip=15 pt
\NoBlackBoxes
\TagsOnRight

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

%References
\def\borg{1}        % Borg
\def\dgs {2}        % del-Rio-Gesztesy-Simon
\def\gsun {3}       % Gesztesy-Simon, TAMS
\def\gsac {4}       % Gesztesy-Simon, ac paper
\def\gsmf {5}       % Gesztesy-Simon, m-function paper
\def\gsds {6}       % Gesztesy-Simon, ds paper
\def\levin {7}      % Levin
\def\lev {8}        % Levitan 68
\def\levbook {9}    % Levitan book
\def\lg {10}        % Levitan-Gasymov
\def\ls {11}        % Levitan-Sargsjan
\def\mar {12}       % Marchenko
\def\piv {13}       % Pivovarchik
\def\simon {14}     % Simon
\def\tit {15}       % Titchmarsh


\topmatter
\title On the Determination of a Potential from Three Spectra
\endtitle
\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. E-mail: fritz\@\linebreak 
math.missouri.edu 
\endthanks
\thanks
Partially supported by the National Science Foundation under 
Grant No.~DMS-9623121.
\endthanks
\thanks$^2$ Division of Physics, Mathematics, and Astronomy,
California Institute of Technology, Pasadena, CA~91125, USA.
E-mail: bsimon\@caltech.edu
\endthanks
\thanks To appear in the Birman Birthday Volume in
{\it{Advances in Mathematical Sciences}}, V.~Buslaev and 
M.~Solomyak (eds.), Amer.~Math.~Soc., Providence, RI.
\endthanks
\date August 19, 1997
\enddate
\dedicatory
Dedicated to M.S.~Birman on the occasion of his
seventieth birthday
\enddedicatory
\keywords Inverse spectral theory, Schr\"odinger operators, 
Weyl-Titchmarsh $m$-functions
\endkeywords
\subjclass Primary 34A55, 34B20; Secondary 34L05, 34L40
\endsubjclass
\abstract We prove that under suitable circumstances, the spectra 
of a Schr\"odinger operator on the three intervals $[0,1]$, $[0,a]$,
and $[a,1]$ for some $a\in (0,1)$ uniquely determine the potential 
$q$ on $[0,1]$.
\endabstract
\endtopmatter

\document

\vskip 0.1in
\flushpar{\bf \S 1. Introduction}
\vskip 0.1in

This is a paper in our series [\dgs,\gsac,\gsmf,\gsds]
on the use of Weyl-Titchmarsh $m$-function methods
to obtain information on what spectral information
uniquely determines the potential $q$
in a one-dimensional Schr\"odinger operator
$-\frac{d^2}{dx^2} + q$. Typical of our results is:

\proclaim{Theorem 1} Fix $c,d\in \Bbb R$ with $c<d$ and
$q\in L^1 ((c,d))$ real-valued. Let $S(c,d;q)$ denote
the set of eigenvalues
of $-\frac{d^2}{dx^2}+q$ on $L^2((c,d))$ with the boundary
conditions $u(c) = u(d)=0$. Suppose $q_1, q_2 \in L^1 ((0,1))$
are real-valued and there is some $a\in (0,1)$ so that
\roster
\item"\rom{(i)}" $S(0,1; q_1)=S(0,1; q_2)$, $S(0,a;q_1)=
S(0,a;q_2)$, and $S(a,1; q_1)=S(a,1;q_2)$.
\item"\rom{(ii)}" The sets $S(0,1; q_1)$, $S(0,a;q_1)$, and
$S(a,1; q_1)$ are pairwise disjoint.
\endroster
Then $q_1 = q_2$ a.e.~on $[0,1]$.
\endproclaim

Our immediate motivation for this result is a recent preprint of
Pivovarchik [\piv], who stated this result for $a=\frac12$
without condition (ii). As we will see in Section~5, there are
counterexamples if (ii) fails. After we pointed out the relevance 
of condition (ii) to Pivovarchik, he provided a corrected version. 
One of our goals here is to show that the methods of 
[\dgs,\gsmf,\gsds] provide a natural way to understand and extend 
this result.

There is a second motivation for this work. While not stated
in this language, we actually considered a problem very close
to a finite-difference analog of Theorem~1 in [\gsmf]. There we
considered a tridiagonal Jacobi matrix in $\Bbb C^N$
$$
A = \pmatrix b_1 & a_1 &  0 & \dots \\
a_1 & b_2 & a_2 & \dots \\
\dots & \dots & \dots & \dots \\
\dots & \dots & a_{N-1} & b_N \endpmatrix ,
$$
with $a_k >0$, $k=1,\dots,N-1$. Let $A^{[i,j]}$ be the submatrix 
of $A$ obtained by keeping rows and columns $i, i+1, \dots, j-1, 
j$. In [\gsmf] we considered to what extent $A$ is determined by 
$g(z,k)$, the $kk$ matrix element of $(A-z)^{-1}$ (for all $z\in
\Bbb C\backslash\text{spec}(A)$). We found that generically 
there were $\binom{N-1}{k-1}$ possible $A$'s consistent with 
a given $g(z,k)$.

The proof of this fact depends on the argument that looked at 
the eigenvalues of $A^{[1, k-1]}$ and $A^{[k+1, N]}$. The function 
$g(z,k)$ determined the union of these sets. Then $\binom{N-1}{k-1}$
possible values depended on the choice of which were actually
eigenvalues of $A^{[1, k-1]}$ and which of $A^{[k+1, N]}$. If one
a priori knows which are which (the hypothesis of Theorem~1), one
has uniqueness.

The non-generic case in [\gsmf] occurs precisely when $A^{[1,k]}$
and $A^{[k+1,N]}$ share an eigenvalue, in which case there is a
manifold of possible $A$'s consistent with $g(z,k)$.

In a sense, Theorem~1 can be thought of as a continuum analog
of a part of the result in [\gsmf].

We actually prove a more general result than Theorem~1. Let $h_c,
h_d\in\Bbb R \cup \{\infty\}$. We let $H(c,d; h_c, h_d; q)$ be 
the operator $-\frac{d^2}{dx^2}+q$ on $L^2((c,d))$ with boundary
conditions
$$
u'(c) + h_c u(c)= 0, \quad u'(d) + h_d u(d)=0,
$$
where $h_{x_0} =\infty$ is a shorthand notation for the Dirichlet
boundary condition at $x=x_0$ (i.e., $u(x_0)=0$). Let
$S(c,d; h_c, h_d; q)$ be the set of eigenvalues (i.e., the 
spectrum) of $H(c,d; h_c, h_d; q)$.

We will prove

\proclaim{Theorem 2} Fix $a\in (0,1)$ and $h_0, h_1, h_a \in
\Bbb R\cup \{\infty\}$. Suppose $q_1, q_2 \in L^1 ((0,1))$
are real-valued and
\roster
\item"\rom{(i)}" $S(0,1; h_0, h_1; q_1) =S(0,1; h_0, h_1; q_2)$,
$S(0,a; h_0, h_a; q_1)=S(0,a; h_0, h_a; q_2)$, and
$S(a,1; h_a, h_1; q_1)=S(a,1; h_a, h_1; q_2)$.
\item"\rom{(ii)}" The sets $S(0,1; h_0, h_1; q_1)$,
$S(0,a; h_0, h_a; q_k)$, and $S(a,1; h_a, h_1; q_k)$ are
pairwise disjoint.
\endroster
Then $q_1 = q_2$ a.e.~on $[0,1]$.
\endproclaim

\remark{Remark} The proof actually shows that not only is $q$
determined by $S(0,1)$, $S(0,a;h_a)$, and $S(a,1;h_a)$, but so 
are $h_0$ and $h_1$.
\endremark

\vskip 0.1in

The structure of this paper is as follows: In Section~2, we prove
several results which illustrate when Green's functions are 
determined by zeros, poles, and residues. In Section~3, we prove 
Theorem~2 when $h_a =\infty$ (including Theorem~1); and in 
Section~4, we prove Theorem~2 when $|h_a|< \infty$. In Section~5, 
we discuss the case where condition (ii) fails. In Section~6, we 
consider some cases where $q$ is defined on all of $\Bbb R$.

\vskip 0.1in

It is a great pleasure to dedicate this paper as a seventieth
birthday present to M.S.~Birman, whose work has long
inspired us. In our use of Green's functions and analytic
function theory, the reader will see echoes of his influence.

\vskip 0.1in
We thank V.~Pivovarchik for sending us his manuscript [\piv] prior 
to publication. F.G.~is indebted to A.~S.~Kechris and C.~W.~Peck 
for a kind invitation to Caltech for a month during the summer of 
1997. The extraordinary hospitality and support by the Department 
of Mathematics at Caltech are gratefully acknowledged. B.S.~would 
like to thank M.~Ben-Artzi for the hospitality of Hebrew 
University where some of this work was done. 

\vskip 0.3in
\flushpar{\bf \S 2. Some Uniqueness Theorems of Meromorphic
Herglotz Functions}
\vskip 0.1in

One could prove the basic result of this paper using the theorems
in [\dgs,\gsds] on the determination of an entire function by its
values on a set of suitable density. Instead we will use some
alternative theorems that allow ready extension to $q$'s on all
of $\Bbb R$,  a typical one being

\proclaim{Theorem 2.1} Let $0<w_1<z_1<w_2<z_2<\cdots$ with
$\lim_{n\to\infty} \, w_n =\infty$. Then
$$
g(z) = \lim_{n\to\infty} \prod_{j=1}^n (1-z/z_j) \bigg/
\prod_{j=1}^n (1-z/w_j)
$$
exists for any $z$ in $\Bbb C\backslash \{w_j\}_{j=1}^\infty$, 
with convergence uniform on compact subsets of \linebreak 
$\Bbb C\backslash
\{w_j\}_{j=1}^\infty$. $g(z)$ is a meromorphic function with
$$
\frac{\text{\rom{Im}}\, (g(z))}{\text{\rom{Im}}\, (z)} >0
\quad \text{\rom{for }} z\in\Bbb C \setminus \Bbb R \tag 2.1
$$
and hence a Herglotz function. Moreover, any meromorphic function
$f(z)$ satisfying {\rom{(2.1)}}
with zeros precisely at $\{z_j\}_{j=1}^\infty$ and poles precisely
at $\{w_j\}_{j=1}^\infty$ is a positive multiple of $g(z)$.
\endproclaim

\remark{Remarks} 1. Theorems of this genre can be found in
Levin [\levin].

2. This is a variant of the standard theorem on the convergence
of alternating series.

3. One can easily accommodate situations where there are also
zeros and poles alternating towards $-\infty$.

4. Any meromorphic Herglotz function (i.e., any meromorphic 
function satisfying (2.1)) can be seen to satisfy $f'(z)>0$ away 
from its polar singularities, so its zeros and poles are simple, 
its zeros and poles alternate, and residues at poles are negative. 
Thus Theorem~2.1 describes all meromorphic Herglotz functions
which are positive on $(-\infty, w_1)$ for some $w_1 >0$.
\endremark

\demo{Proof} Let $g_N(z)=\prod_{j=1}^N (1-z/z_j)/
\prod_{j=1}^{N+1} (1-z/w_j)$. Then $g_N$ has simple poles at
$w_1, w_2, \dots, w_{N+1}$ and because of the alternating nature
of the $z_j$'s and $w_j$'s, each residue is negative. Since
$g_N(z)\to 0$ as $|z|\to\infty$, it follows that $g_N(z) =
\sum_{j=1}^{N+1} \frac{\alpha_j^{(N)}}{w_j-z}$ with
$\alpha_j^{(N)}>0$, $j=1,\dots,N+1$.

Thus, each $g_N$ is a Herglotz function and so $g_N$ maps 
$\Bbb C\backslash [0,\infty)$ to $\Bbb C \backslash 
(-\infty, 0]$. Let $H$ be a biholomorphic map of $\Bbb C
\backslash (-\infty, 0]$ to the open unit disk (e.g., $H(w)=
\frac{\sqrt w -1}{\sqrt w +1}$). By applying the Vitali 
convergence theorem (see, e.g., [\tit], Ch.~5) to $H \circ 
g_N$, we see it suffices to show $g_N (x)$ converges for each 
$x\in(-\infty,0)$ to conclude that $g_N(z)$ converges as 
$N\to\infty$ for $z\in \Bbb C\backslash (0,\infty)$.

Since $w_j < z_j$, we have $(1-x/z_j) / (1-x/w_j) <1$, and since
$w_{j+1} >z_j$, we have $(1-x/z_j) / (1-x/w_{j+1}) > 1$ assuming 
$x<0$. Thus $g_1 (x) < g_2 (x) < \cdots < g_N (x) < g_{N+1}(x)<1$, 
so $\lim_{N\to\infty} g_N (x)$ exists for $x<0$.

Once we have convergence on $\Bbb C \backslash (0,\infty)$, it
is easy to extend the argument to $\Bbb C \backslash
\{w_j\}_{j=1}^\infty$.

Finally, let $f(z)$ be a Herglotz function with the stated zeros
and poles. Then $f(z)/g(z)$ is an entire non-vanishing function,
and on $\Bbb C \backslash [0,\infty)$, $|\text{Im}\,
(\ln (f(z)/g(z)))| \leq 2\pi$ since \linebreak $|\text{Im}\,
(\ln (f(z)))| \leq \pi$ and $|\text{Im}\, (\ln (g(z)))|\leq\pi$
on $\Bbb C\setminus [0,\infty)$. It follows that $f(z)/g(z)$ is 
constant. \qed
\enddemo

In exactly the same way one infers

\proclaim{Theorem 2.2} Let $0 < z_1 < w_1 < z_2 < w_2 < \cdots$
with $\lim_{n\to\infty} w_n =\infty$. Then
$$
g(z) = \lim_{n\to\infty} \prod_{j=1}^n (1-z/z_j) \bigg/
\prod_{j=1}^n (1-z/w_j)
$$
exists for any $z$ in $\Bbb C \backslash \{w_j\}_{j=1}^\infty$
with convergence uniform on compact subsets of $\Bbb C
\backslash \{w_j\}_{j=1}^\infty$. $g(z)$ is a meromorphic
function with $\frac{\text{Im}\, (g(z))}{\text{Im}\, (z)} < 0$
for $z\in\Bbb C \setminus \Bbb R$.

Moreover, any meromorphic function $f(z)$ satisfying {\rom{(2.1)}}
with zeros precisely at $\{z_j\}_{j=1}^\infty$ and poles precisely
at $\{w_j\}_{j=1}^\infty$ is a negative multiple of $g(z)$.
\endproclaim

We also have theorems on asymptotics, poles, and residues
determining a meromorphic Herglotz function.

\proclaim{Theorem 2.3} Let $f_1(z), f_2(z)$ be two meromorphic
Herglotz functions with identical sets of poles and residues,
respectively. If
$$
f_1 (ix) - f_2 (ix) \to 0 \text{ as } x\to\infty, \tag 2.2
$$
then $f_1 = f_2$.
\endproclaim

\demo{Proof} By the Herglotz representation theorem, if $f(z)$
is a meromorphic Herglotz function with poles at
$\{ w_j\}_{j=1}^\infty$ in $\Bbb R$ and residues $-\alpha_k <0$
at $z=w_k$, then for some constants $A\geq 0$ and $B\in\Bbb R$,
$$
f(z) = Az + B + \sum_{j=1}^\infty \alpha_j
\biggl[ \frac{1}{w_j -z} - \frac{w_j}{1+w^2_j}\biggr],
$$
where the sum is absolutely convergent since $\sum_{j=1}^\infty
\frac{\alpha_j}{1+w^2_j} < \infty$.

Thus $f_1(z) - f_2(z) = {\tilde A}z - {\tilde B}$ for some
${\tilde A},{\tilde B} \in \Bbb R$, and
therefore, {\rom{(2.2)}} implies ${\tilde A}={\tilde B}=0$. \qed
\enddemo

In applications, either $f_1 (ix)$ and $f_2 (ix)$ are both 
$o(1)$ as $x\to\infty$ or else, $f_1 (ix)$ and $f_2 (ix)$ are 
both $\sqrt{ix} + o(1)$ as $x\to\infty$.

\vskip 0.3in
\flushpar{\bf \S 3. The Case of a Dirichlet Boundary Condition 
$h_a=\infty$}
\vskip 0.1in

We want to prove Theorem~2 when $h_a = \infty$. If $h_0 < \infty$,
let $u_- (z,x; q)$ solve $-u'' + qu = zu$ with boundary conditions
$u_- (z,0; q) =1$, $u'_- (z,0; q)=-h_0$. If $h_0 = \infty$, let
it satisfy $u_- (z,0; q)=0$, $u'_- (z,0; q)=1$. As is well known
(see, e.g., [\ls], Ch.~1), $u_-$ is an entire function of $z$. 
Similarly, $u_+$ satisfies the $h_1$ boundary condition at $1$.

Let
$$
W(z;q) = u'_-(z,x;q) u_+(z,x;q) - u_-(z,x;q) u'_+(z,x;q),
$$
which is independent of $x$. The zeros of $W$ are precisely the
points $w_i$ of $S(0,1; h_0, h_1;q)$, that is, the eigenvalues of
$H := H(0,1; h_0, h_1; q)$.

Fix $a\in (0,1)$ and $q$. Let $g(z)=G(z,a,a)$ be the Green's
function of $H$ in $L^2 ((0,1))$ at $(a,a)$, that is, the
integral kernel of $(H-z)^{-1}$ at $(a,a)$. (We also use the 
notation $g(z;q)$ for $g(z)$ whenever the dependence of $g(z)$ 
on $q$ needs to be underscored.) Then, by a standard formula for 
the Green's function of $H$,
$$
g(z;q) = \frac{u_- (z,a; q) u_+ (z,a; q)}{W(z;q)}\, . \tag 3.1
$$
The zeros of $u_+ (z,a;q)$ are precisely the points of
$S(a,1; h_a =\infty, h_1;q)$ and the zeros of $u_- (z,a;q)$
are precisely the points of $S(0,a; h_0, h_a =\infty; q)$. The
hypothesis (ii) on disjointness of the $S$ sets in Theorem~2 says
that the poles of $g(z)$ are precisely the points of $S(0,1)$, 
and the zeros, the points of $S(0,a) \cup S(a,1)$. (If the sets 
are not disjoint, there are cancellations between zeros and 
poles.)

By Theorem~2.1 (adding a constant to $q$ if need be, we can 
assume all poles and zeros are positive), the zeros and poles
of $g(z)$ and the known asymptotics $g(-\kappa^2;q)=
(2\kappa)^{-1}$ ($1+o(1)$) as $\kappa\to\infty$ determine $g$,
that is, $g(z;q_1) = g(z; q_2)$.

Next we use the $m$-functions $m_\pm$ defined by  $m_\pm (z;q)
= \pm u'_\pm (z,a;q)/u_\pm (z,a;q)$. By (3.1),
$$
g(z;q) = -\frac{1}{[m_+ (z;q) + m_- (z;q)]}\, . \tag 3.2
$$
Moreover, the poles of $m_+$ (resp.~$m_-$) are precisely the
points $\lambda$ of $S(a,1; h_a =\infty, h_1; q)$ (resp.~$S
(0,a; h_0, h_a =\infty; q)$). And the residues of the poles
are determined by $g$. Explicitly, if $\lambda_0$ is a pole of
$m_+$, by hypothesis (ii) in Theorem~2, it is not a pole of 
$m_-$, and so its residue is $\left. -1/\frac{\partial g}
{\partial z}\right|_{z=\lambda_0}$.

By Theorem~2.2 and the asymptotics $m_\pm (-\kappa^2; q) =
-\kappa + o(1)$ as $\kappa\to\infty$, the poles and residues
determine $m_\pm$; that is, $m_\pm (z;q_1) = m_\pm (z;q_2)$.

Finally, the uniqueness result of Borg [\borg] and Marchenko
[\mar] guarantees that $m_\pm (z;q)$ uniquely determine $g$ on
$[0,a]$ and $[a,1]$, so $q_1 = q_2$ a.e.~on $[0,1]$.

\vskip 0.3in
\flushpar{\bf \S 4. The Case $h_a\in\Bbb R$}
\vskip 0.1in

The changes in the proof when $|h_a| <\infty$ are minimal. Define
$u_\pm$ as in the last section, but instead of (3.1), define
$$
g(z;q) = \frac{[u'_- (z,a;q) + h_a u_- (z,a;q)]
[u'_+ (z,a; q) + h_a u_+ (z,a;q)]}{W(z;q)}\, .  \tag 4.1
$$
Since $W= (u'_- + h_a u_-) u_+ - u_- (u'_+ +h_a u_+)$, (3.2)
becomes
$$
g(z;q) = \frac{1}{\frac{1}{m_+(z;q) + h_a}
+ \frac{1}{m_-(z;q) - h_a}}\, . \tag 4.2
$$

The spectra determine the zeros and poles of $g$ which, together
with the asymptotics $g(-\kappa^2; q) = -\frac12 \kappa (1 +
o(1))$ as $\kappa\to\infty$, determine $g$ by Theorem~2.1 or 2.2.

By hypothesis (ii) of Theorem~2, the poles of 
$(m_\pm \pm h_a)^{-1}$ are distinct and so their residues 
are determined by (4.2) and the knowledge of $g$. The poles and 
residues of $-(m_\pm \pm h_a)^{-1}$ and the fact that 
$|m_\pm (ix)| \to \infty$ as $x\to\infty$ determine 
$(m_\pm \pm h_a)^{-1}$ by Theorem~2.3. The Borg-Marchenko 
uniqueness theorem then completes the proof.

\vskip 0.3in
\flushpar{\bf \S 5. Examples of Non-Uniqueness}
\vskip 0.1in

Our goal here is to show that if condition (ii) fails, then the
uniqueness result in Theorem~2 can also fail. We will take an
extreme case where $S(0,\frac12)=S(\frac12, 1)$ for simplicity;
but we have no doubt that a single point in common suffices to
construct counterexamples to the extension of Theorem~2 with
(ii) absent. We note that $S(0,1)\cap S(0,\frac12) = S(0,1) \cap
S(\frac12, 1)=S(0,\frac12) \cap S(\frac12, 1)$ so that if two 
$S$'s fail to be disjoint, each pair has non-zero intersection.

To begin we note

\proclaim{Lemma 5.1} Let $f$ be a continuous map of $Q:=[0,1] 
\times [0,1]$ to the unit circle. Then, there exists a pair 
of points $p_0, p_1 \in Q$ with $p_0\neq p_1$ and $f(p_0) = 
f(p_1)$.
\endproclaim

\demo{Proof} If $f(0,0) = f(1,1)$, we have the required points. 
If not, reparametrize the circle so that $f(0,0)=1$, $f(1,1) =
-1$. Consider the images $f(\gamma_j(t))$, $t\in[0,1]$, $j=
0,1,2$ of the three curves $\gamma_0, \gamma_1, \gamma_2$ 
given by $\gamma_j (t) = (t, t+ (j-1)\pi^{-1} \sin(\pi t))$, 
$t\in [0,1]$, $j=0,1,2$. If two of these images contain the 
point $(0,-1)$ on the unit circle, then that value is taken 
twice. If at most one of these images contains $(0, -1)$, then 
by the intermediate value theorem, two images must contain 
$(0,1)$. \qed
\enddemo

As explained in [\gsds], by results of Levitan [\lev], [\levbook],
Ch.~3 and Levitan-Gasymov [\lg], one can prove

\proclaim{Proposition 5.2} Suppose that $x_0 < y_0 < x_1 < y_1
<\cdots$ so that for $n$ sufficiently large, $x_n = [(2n)\pi]^2$,
$y_n = [(2n+1)\pi]^2$. Then there exists a unique $h_1$ and a
$C^\infty$-function $q$ on $[\frac12, 1]$ so that
$$
-\frac{d^2}{dx^2} + q \text{ in } L^2 ((\tfrac12, 1));
\quad u'(\tfrac12)=0, \quad u'(1) + h_1 u(1) =0
$$
has eigenvalues $\{x_n\}_{n=0}^\infty$ and
$$
-\frac{d^2}{dx^2} + q \text{ in } L^2 ((\tfrac12, 1)];
\quad u(\tfrac12)=0, \quad u'(1) + h_1 u(1) =0
$$
has eigenvalues $\{y_n\}_{n=0}^\infty$. Moreover, if a finite
subset of $x$'s and $y$'s is varied, $h_1$ varies continuously
as a function of these numbers.
\endproclaim

Consider now fixing $y_n = [(2n+1)\pi]^2$ for all $n\in\Bbb N_0$
($=\Bbb N\cup\{0\}$) and $x_n = [(2n)\pi]^2$ for $n\geq 2$ and 
varying $(x_0, x_1)$ in $[0,1] \times [20,21]$. By Lemma~5.1 
and Proposition~5.2, we can find $(x^{(0)}_0, x^{(0)}_1) \neq
(x^{(1)}_0, x^{(1)}_1)$ so that the corresponding values of 
$h_1$ are equal. Set $\tilde q_0, \tilde q_1$ as the 
corresponding $q$'s and $h$ as the common value of $h_1$. 
Let $q_1, q_2$ be defined on $[0,1]$ by
$$\alignat2
q_1 (x) &= \tilde q_0 (1-x), \qquad && 0\leq x \leq \tfrac12, \\
&= \tilde q_1 (x), \qquad && \tfrac12 \leq x \leq 1, \\
q_2 (x) &= q_0 (1-x).
\endalignat
$$
Then $q_1 \neq q_2$ but $S(0,\frac12; h_0 =-h, h_{\frac12} =
\infty; q_1) = S(0,\frac12; h_0 = -h, h_{\frac12} = \infty; q_2)
= S(\frac12, 1; h_{\frac12} = \infty, h_1 = h; q_1) =
S(\frac12, 1; h_{\frac12}=\infty, h_1 =h; q_2) =
\{((2n+1)\pi)^2 \}_{n\in\Bbb N}$, and by reflection
symmetry:
$$
S(0,1; h_0 =-h, h_1 =h; q_1) = S(0,1; h_0 =-h, h_1 =h; q_2).
$$

Since $q_1 \neq q_2$, this provides the required counterexample. 
(There is no particular significance in our choice of $x_1 \in 
[20,21]$. Any interval of length one contained in $(y_0, y_1) 
= (\pi^2, 9\pi^2)$ would be admissible.)

As in the finite-difference case [\gsmf], we believe an analysis 
of the situation where $S(0,\frac12)\cap S(\frac12, 1)$ has 
$k$-points will yield $k$-parameter sets of $q$'s (as long as 
we are allowed to vary $h_0, h_1$ as well as $q$) consistent 
with the given sets of eigenvalues.


\vskip 0.3in
\flushpar{\bf \S 6. The Whole Line Case}
\vskip 0.1in

In this section, we will extend Theorem~2 to the situation where
$[0,1]$ is replaced by $\Bbb R$, but the spectrum of the 
corresponding Schr\"odinger operator $H$ in $L^2(\Bbb R)$
is purely discrete and 
bounded from below. Typical situations are, for instance, $q\in 
L^1_{\text{\rom{loc}}}(\Bbb R)$ real-valued with $q(x)\to\infty$ 
as $|x|\to\infty$ or, $q\in L^1_{\text{\rom{loc}}}(\Bbb R)$ 
real-valued, $q$ bounded from below, and $\lim_{x\to\pm\infty} 
\int_x^{x+a} dy \, q(y) =\infty$ for any $a>0$ (cf.~[\ls], 
Sect.~4.1). In this case, the maximal operator $H$ in $L^2 
(\Bbb R)$ associated with the differential expression 
$-\frac{d^2}{dx^2} + q$ on $\Bbb R$ (with domain $\Cal D(H) 
=\{f\in L^2(\Bbb R) \mid f,f' \text{ locally absolutely 
continuous on }\Bbb R; (-f'' + qf)\in L^2(\Bbb R)\}$) is 
self-adjoint. In [\gsds] our extensions required a hypothesis 
on $q$ that $q(x) \geq C |x|^{2+\varepsilon}+1$ for some 
$C,\varepsilon >0$. This was because we used results on densities 
of zeros. Here, because we rely on Theorems~2.1, 2.2, we note 
that the following result holds by the identical proof to 
Theorem~2:

\proclaim{Theorem 3} Suppose $q\in L^1_{\text{loc}}(\Bbb R)$ is
real-valued and $H$ in $L^2(\Bbb R)$ is bounded from below with 
purely discrete spectrum $S(-\infty, \infty; q)$. Let 
$S(-\infty, 0; h_0;q)$ denote the spectrum of the corresponding 
\rom(maximally defined\rom) operator in $L^2((-\infty, 0))$ 
with $u'(0) + h_0u(0)=0$ boundary conditions, and similarly 
for $S(0,\infty; h_0;q)$. Suppose that $q_1, q_2$ are given 
and we have a fixed $h_0\in\Bbb R\cup \{0\}$ so that
\roster
\item"\rom{(i)}" $S(-\infty,\infty; q_1) = S(-\infty, \infty; q_2)$,
$S(-\infty,0;h_0; q_1) = S(-\infty, 0; h_0; q_2)$, and \linebreak  
$S(0,\infty; h_0; q_1) = S(0, \infty; h_0; q_2)$
\item"\rom{(ii)}" The sets $S(-\infty, \infty; q_1)$,
$S(-\infty, 0; h_0; q_1)$, and $S(0, \infty;h_0; q_1)$ are pairwise
disjoint.
\endroster
Then $q_1 = q_2$ a.e.~on $\Bbb R$.
\endproclaim

As noted in Remark~2 following Theorem~2.1, this result extends to 
Schr\"odinger operators $H$ with purely discrete spectra 
accumulating at $+\infty$ and $-\infty$. In particular, it extends 
to cases where $H$ is in the limit circle case at $+\infty$ and/or 
$-\infty$ as long as the corresponding (separated) boundary 
condition at $+\infty$ and/or $-\infty$ is kept fixed for all 
three operators on $\Bbb R$, $(-\infty, 0)$, and $(0,\infty)$.

The reader might want to contrast Theorem~3 with Corollary~3.4 
in [\gsun], where we obtained uniqueness of $q$ from three 
(discrete) spectra of operator realizations of $-\frac{d^2}
{dx^2}+q$ on $\Bbb R$. There one of the three spectra is 
$S(-\infty,\infty;q)$ as above in Theorem~3; the other two, 
$S(-\infty,\infty;\beta_j, q)$, $j=1,2$, are associated with 
$-\frac{d^2}{dx^2}+q$ on $\Bbb R$ and the boundary conditions 
$\lim_{\varepsilon\downarrow 0} [u' (\pm\varepsilon) + \beta_j 
u(\pm\varepsilon)]=0$, where $\beta_j\in\Bbb R\cup \{\infty\}$, 
$j=1,2$, $\beta_1\neq\beta_2$, $(\beta_1,\beta_2)\neq (0,\infty)$, 
$(\infty, 0)$.

\vskip 0.3in
\Refs
\endRefs

\vskip 0.1in

\item{\borg.} G.~Borg, {\it{Uniqueness theorems in the spectral theory of
 $y''+(\lambda -q(x))y=0$}}, Proc.~11th Scandinavian Congress of
Mathematicians, Johan Grundt Tanums Forlag, Oslo, 1952, 276--287.
\gap
\item{\dgs.} R.~del Rio, F.~Gesztesy and B.~Simon, {\it{Inverse
spectral analysis with partial information on the potential, III.
Updating boundary conditions}}, Intl.~Math. Research Notices,
to appear.
\gap
\item{\gsun.} \ref{F.~Gesztesy and B.~Simon}{Uniqueness
theorems in inverse spectral theory for one-dimensional
Schr\"odinger operators}{Trans.~Amer.~Math.~Soc.}{348}{1996}
{349--373}
\gap
\item{\gsac.} \ref{F.~Gesztesy and B.~Simon}{Inverse spectral
analysis with partial information on the potential, I. The
case of an a.c.~component in the spectrum}{Helv.~Phys.~Acta}
{70}{1997}{66--71}
\gap
\item{\gsmf.} F.~Gesztesy and B.~Simon, {\it{$m$-functions and
inverse spectral analysis for finite and semi-infinite Jacobi
matrices}}, to appear in J.~d'Anal.~Math.
\gap
\item{\gsds.} F.~Gesztesy and B.~Simon {\it{Inverse spectral
analysis with partial information on the potential, II. The
case of discrete spectrum}}, preprint, 1997.
\gap
\item{\levin.} B.~Ja.~Levin, {\it{Distribution of Zeros of
Entire Functions}}, rev.~ed., Amer.~Math.~Soc., Providence,
RI, 1980.
\gap
\item{\lev.} \ref{B.~M.~Levitan}{On the determination of a
Sturm-Liouville equation by two spectra}
{Amer.~Math.~Soc.~Transl.}{68}{1968}{1--20}
\gap
\item{\levbook.} B.~M.~Levitan, {\it{Inverse Sturm-Liouville
Problems}}, VNU Science Press, Utrecht, 1987.
\gap
\item{\lg.} \ref{B.~M.~Levitan and M.~G.~Gasymov}{Determination
of a differential equation by two of its spectra}
{Russ.~Math.~Surv.}{19:2}{1964}{1--63}
\gap
\item{\ls.} B.~M.~Levitan and I.~S.~Sargsjan, {\it{Introduction
to Spectral Theory}}, Amer. Math. Soc., Providence, RI, 1975.
\gap
\item{\mar.} V.~A.~Marchenko, {\it {Some questions in the theory
of one-dimensional linear differential operators of the second
order, I}}, Trudy Moskov.~Mat.~Ob\v s\v c. {\bf 1} (1952),
327--420 (Russian); English transl.~in Amer.~Math.~Soc.~Transl.
(2) {\bf 101} (1973), 1--104.
\gap
\item{\piv.} V.~N.~Pivovarchik, {\it{An inverse
Sturm-Liouville problem by three spectra}}, unpublished.
\gap
\item{\simon.} B.~Simon, {\it{A new approach to inverse spectral
theory, I. Fundamental formalism}}, in preparation.
\gap
\item{\tit.} E.~C.~Titchmarsh, {\it{The Theory of Functions}},
2nd ed., Oxford University Press, Oxford, 1985.
\gap

\enddocument
