\input amstex
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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\borg{1}        % Borg
\def\borgun {2}      % Borg unique
\def\gl {3}       % Gelfand-Levitan
\def\gsac {4}        % Gesztesy-Simon, ac paper
\def\gsmf {5}      % Gesztesy-Simon, m-function paper
\def\gsds {6}        % Gesztesy-Simon, ds paper
\def\haldma {7}    % Hald Mantle
\def\hsturm {8}    % Hochstadt 73
\def\hl {9}         % Hochstadt-Lieberman
\def\levs {10}       % Levinson
\def\lev {11}       % Levitan
\def\levbook {12}  % Levitan book
\def\lg {13}       % Levitan-Gasymov
\def\mal {14}         % Malamud
\def\malsp {15}         % Malamud
\def\mar {16}        % Marchenko
\def\simon {17}       % Simon


\topmatter
\title Inverse Spectral Analysis With Partial Information on
the Potential, III. Updating Boundary Conditions
\endtitle
\rightheadtext{Inverse Spectral Analysis: Updating
Boundary Conditions}
\author Rafael del Rio$^1$, Fritz Gesztesy$^2$, and
Barry Simon$^3$
\endauthor
\leftheadtext{R. del Rio, F.~Gesztesy, and B.~Simon}
\thanks$^1$ IIMAS-UNAM, Apdo.~Postal 20-726, Admon No.~20,
01000~Mexico D.F., Mexico. E-mail: delrio\@servidor.unam.mx
\endthanks
\thanks$^2$ Department of Mathematics, University of Missouri,
Columbia, MO~65211, USA. E-mail: fritz\@math.missouri.edu
\endthanks
\thanks$^3$ 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 CONACYT
Project 05567P-E and the National Science Foundation under
Grant Nos.~DMS-9623121 and DMS-9401491.
\endthanks
\thanks To be submitted to {\it{Intl.~Math.~Research Notes}}
\endthanks
\date June 3, 1997
\enddate
\abstract We discuss results where information on parts of the
discrete spectra of one-dimensional Schr\"odinger operators
$H=-\frac{d^2}{dx^2}+q$ in $L^2((0,1))$ or of a finite Jacobi
matrix together with partial information on $q$ uniquely
determines $q$ a.e.~on $[0,1]$.
These extend classical results of Borg and Hochstadt-Lieberman
as well as results in paper~II of this series.
\endabstract
\endtopmatter

\document

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

This paper is a postscript to two earlier papers [\gsmf, \gsds]
in that it provides a new way of looking at the problems
considered
in those papers that allows the same methods to prove additional
results.

To explain our results, we recall earlier theorems of Borg [\borg]
(see also [\hsturm, \levs--\mal]) and
of Hochstadt-Lieberman [\hl] (see also [\haldma, \malsp]).
Throughout this paper assume
$q\in L^1((0,1))$ to be real-valued and consider the operator
$H=-\frac{d^2}{dx^2} + q$ in $L^2 ((0,1))$ with boundary
conditions
$$
\align
u'(0) + h_0 u(0) &=0, \tag 1.1 \\
u'(1) + h_1 u(1) &= 0, \tag 1.2
\endalign
$$
where $h_j \in \Bbb R\cup \{\infty\}$, $j=0,1$
(with $h_0 = \infty$
shorthand for the boundary condition $u(0)=0$). Fix $h_1 \in
\Bbb R$
but think of $H(h_0)$ as a family of operators depending on $h_0$
as a parameter. Then Borg's and Hochstadt-Lieberman's
results can
be paraphrased as follows:

\example{Borg [\borg]} The spectra of $H(h_0)$ for two values
of $h_0$
determine $q$.
\endexample

\example{Hochstadt-Lieberman [\hl]} The spectra of $H(h_0)$
for one
value of $h_0$ and $q$ on $[0, \frac12]$ determine $q$.
\endexample

In [\gsds], two of us proved a result that can be paraphrased as

\proclaim{Theorem of [\gsds]} Half the spectra of one $H(h_0)$
and $q$ on $[0,\frac34]$ determine $q$.
\endproclaim

One of our goals in this note is to prove

\example{New Result} The spectrum of one $H(h_0)$ and half the
spectrum of another $H(h_0)$ and $q$ on $[0,\frac14]$ determine
$q$.
\endexample

We will also show that

\example{New Result} Two-thirds of the spectra of three $H(h_0)$
determine $q$.
\endexample

Our point is as much a new way of looking at the argument in
[\gsds] as these new results. Fundamental to our approach here
and in [\gsmf, \gsds] is the Titchmarsh-Weyl $m$-function
defined by
$$
m_{h_1}(z) = \frac{u'_{h_1}(z,0)}{u_{h_1}(z,0)}\, ,
$$
where $u_{h_1}(z,x)$ solves $-u''(z,x) + q(x) u(z,x)=
zu(z,x)$ with the boundary condition $(1.2)$. $m_{h_1}$ is a
meromorphic function on $\Bbb C$ (in fact, a Herglotz
function) with
all its zeros and poles on the real axis. Since $h_1 \in \Bbb R$
will be fixed throughout this paper, we will delete the subscript
$h_1$ from now on and simply write $m(z)$ instead. Moreover,
due to
the assumption $h_1 \in \Bbb R$, we will index the
eigenvalues of $H(h_0)$ by $\{\lambda_n\}_{n\in \Bbb N_0}$,
$\Bbb N_0 = \Bbb N \cup \{0\}$.

A fundamental result of Marchenko [\mar] (see also
[\borgun, \gl, \simon]) says

\proclaim{Theorem 1.1} $m(z)$ uniquely determines $q$ a.e.~on
$[0,1]$.
\endproclaim

Our fundamental strategy can be described as follows:

(a) Note that $\lambda$ is an eigenvalue of $H(h_0)$ if
and only
if $m(\lambda)= -h_0$.

(b) Prove a general theorem that knowing $m$ at points
$\lambda_0,
\lambda_1, \dots, \lambda_n, \dots$ determines $m$ as long as
$\{\lambda_n\}_{n\in \Bbb N_0}$ has sufficient density.
Given (a), this will allow
one to prove that if $\lambda_0, \lambda_1, \dots, \lambda_n,
\dots$ have
sufficient density, an infinite sequence of pairs
$\{(\lambda_n, \alpha_n)\}_{n\in \Bbb N_0}$ and the knowledge
that $H(h_0 = \alpha_n)$ has an eigenvalue at $\lambda_n$
determines $m$ (and so $q$ a.e.~on $[0,1]$ by Theorem~1.1).

(c) Use scaling covariance to extend the $[0,1]$ result to one
 for $[x, 1]$ for any $x\in (0,1)$.

(d) Note that a knowledge of $q$ a.e.~on $[0,x]$ allows one
to update
boundary conditions. Explicitly, let $H(h_x)$ be the operator in
$L^2 ((x,1))$ with boundary condition $(1.2)$ but (1.1)
replaced by
$$
u'(x) + h_x u(x) =0. \tag 1.3
$$
Then $\lambda_n$ is an eigenvalue of $H(h_0=\alpha_n)$ if
and only
if it is an eigenvalue of $H(h_{x_0} = \beta_n)$, where
$\beta_n$ is obtained by solving $m'_n (x) = q(x) - \lambda_n -
m^2_n$ on $[0, x_0]$ with the boundary condition $m_n (x=0) =
-\alpha_n$ and setting $\beta_n = -m_n (x=x_0)$.

We will present steps (b) and (c) in Sections~2 and 3 and then
step (d) in Section~4.

We will not explicitly derive them, but the results in [\gsds]
that treat operators on $(0,1)$ and that allow one to trade
$C^{2k}$ conditions on $q$ for $k$ eigenvalues can be
extended to
the context we discuss here.

We also note that the ideas in this paper extend to
Jacobi matrices.

Finally, while the present paper and [\gsmf, \gsds]
concentrate on
discrete spectra, we might point out that our $m$-function
strategy also applies in certain cases involving absolutely
continuous
spectra, see [\gsac].


\vskip 0.3in
\flushpar{\bf \S 2. Zeros of the $\boldkey m$-function}
\vskip 0.1in

If $a\in\Bbb R$, let $a_+ = \max (a,0)$. Then

\proclaim{Theorem 2.1} Let $\{\lambda_n\}_{n\in \Bbb N_0}$ be
a sequence
of distinct positive real numbers satisfying
$$
\sum^\infty_{n=0} \frac{(\lambda_n -
\frac{1}{4} \pi^2 n^2)_+}{n^2}
<\infty. \tag 2.1
$$
Let $m_1, m_2$ be the $m$-functions for two operators
$H_j=-\frac{d^2}{dx^2}+ q_j$ in $L^2((0,1))$ with boundary
conditions
$$
u'(1) + h^{(j)}_1 u(1) = 0
$$
and $h^{(j)}_1 \in \Bbb R$, $j=1,2$. Suppose that
$m_1 (\lambda_n) = m_2 (\lambda_n)$
for all $n \in \Bbb N_0$. Then $m_1 = m_2$ \rom(and hence
$q_1=q_2$
a.e.~on $[0,1]$ and $h_1^{(1)} = h_1^{(2)}$\rom).
\endproclaim

\remark{Remarks} 1. In our examples, $\lambda_n \sim
\pi^2 n^2 + C$
as $n \to \infty$ (cf.~(3.1)), so (2.1) is satisfied,
for instance,
by considering two distinct spectra of $H(h_0)$.

2. We allow the case $m_1 (\lambda_n)=m_2(\lambda_n) =\infty$.
\endremark

\vskip 0.1in

As a preliminary result we note the following

\proclaim{Lemma 2.2} Suppose $\{\lambda_n\}_{n\in \Bbb N_0}$
is a
sequence of positive real numbers satisfying {\rom{(2.1)}} and
$$
\sum^{\infty}_{n=0} \lambda^{-1}_n < \infty. \tag 2.2
$$
Define $f(z) := \prod^\infty_{n=0} (1- \frac{z}{\lambda_n})$,
then
$$
\varlimsup\limits\Sb |y|\to \infty \\ y\in \Bbb R \endSb
\frac {|y|^{1/2} \text{\rom{ sinh}}(2|y|^{1/2})}{|f(iy)|} <
\infty. \tag 2.3
$$
\endproclaim

\demo{Proof} Let $y\in \Bbb R$. Then
$\text{sinh}(2|y|^{1/2}\,) / |y|^{1/2} = |\sin
(2i|y|^{1/2}) / |y|^{1/2}|$ and
$$
\frac{\sin (2\sqrt z\,)}{2\sqrt z} = \prod^\infty_{n=1} \biggl(
1-\frac{4z}{\pi^2 n^2}\biggr),
$$
so (2.3) becomes
$$
\varlimsup_{|y|\to\infty} \frac{|y|}{1+ \frac{|y|}{\lambda_0}} \,
\prod^\infty_{n=1} \biggl[ \frac{( 1 + \frac{4|y|}{\pi^2 n^2})}
{(1+ \frac{|y|}{\lambda_n})} \biggr] <\infty \tag 2.4
$$
using $2^{-1/2}(1+|x|) \leq (1+x^2)^{1/2} \leq (1+|x|)$.

If $0\leq a\leq b$, then $(\frac{1+a|y|}{1+b|y|}) \leq 1$,
and if $a >b>0$,
then
$$\gather
\frac{(1+a|y|)}{1+b|y|} = 1 + \frac{(a-b)|y|}{1+b|y|} \leq 1 +
\frac{a-b}{b} = \frac{a}{b}, \\
\prod^\infty_{n=1} \frac{(1 + \frac{4|y|}{\pi^2 n^2})}
{(1+\frac{|y|}{\lambda_n})} \leq \prod_{n : \lambda_n >
\frac{1}{4} \pi^2 n^2}
\frac{4\lambda_n}{\pi^2 n^2} = \prod_{n=1}^{\infty}
\biggl[ 1 +
\frac{(\lambda_n - \frac14 \pi^2 n^2)_+}
{\frac14 \pi^2 n^2}\biggr]
<\infty
\endgather
$$
if (2.1) holds. \qed
\enddemo

\demo{Proof of Theorem 2.1} We follow the arguments in
[\gsmf, \gsds] fairly closely. One can write $m_j (z) =
\frac{Q_j(z)}{P_j(z)}$, $j=1,2$, where
\roster
\item"\rom{(1)}" $P_j, Q_j$ are entire functions satisfying
$$\align
|P_j(z)| &\leq C \exp(\sqrt{|z|}\,), \tag 2.5a \\
|Q_j(z)| &\leq C (1+\sqrt{|z|}\,) \exp(\sqrt{|z|}\,). \tag 2.5b
\endalign
$$
\item"\rom{(2)}"
$$
m_j (z) = \pm i \sqrt{z} + o(1)  \text{ as } z \to \pm i\infty.
\tag 2.6
$$
\endroster
(We use the square root branch with $\text{Im}\,
(\sqrt{z}) \geq 0$.)

Suppose $m_1 \neq m_2$. Then $P_2(z) Q_1(z) - P_1(z) Q_2(z) :=
H(z)$ is an entire function of order at most $\frac12$ and not
identically zero. Since $H(\lambda_n)=0$, we conclude
that $\sum_{n\in\Bbb N_0}
\lambda^{-a}_n <\infty$ if $a>\frac12$. In particular,
(2.2) holds,
and we can define $f(z) = \prod^\infty_{n=0}
(1-\frac{z}{\lambda_n})$.
Next, define
$$
G(z) := \frac{H(z)}{f(z)} = \frac{P_1(z) P_2(z)}{f(z)}\,
(m_1(z) - m_2(z)). \tag 2.7
$$
Since $H(\lambda_n)=0$, $G(z)$ is an entire function. By (2.3),
$$
\varlimsup_{|y|\to\infty} \frac{|y|^{1/2}
\exp{(2 |y|^{1/2})}}{|f(iy)|}
< \infty,
$$
so by (2.5) and (2.6),
$$
|G(iy)| \leq \frac{\exp{(2 |y|^{1/2})}}{f(iy)} \,
|m_1(iy) - m_2(iy)|
=o(|y|^{-1/2})
$$
goes to zero as $|y|\to\infty$. The Phragm\'en-Lindel\"of
argument
of [\gsds] then yields the contradiction $G(z)\equiv 0$, that is,
 $m_1=m_2$. \qed
\enddemo

\remark{Remark} The above yields $o(|y|^{-1/2})$ even though
$o(1)$ would have been sufficient. We have thrown away half
a zero.
That means one can prove the following result.
\endremark

\proclaim{Theorem 2.2} Let $\{\lambda_n\}_{n\in\Bbb N_0}$
and $\{\mu_n\}_{n\in\Bbb N_0}$ be two sequences of real numbers
satisfying
$$
\sum^\infty_{n=0} \frac{(\lambda_n - \pi^2 n^2)_+}{n^2} <
\infty \qquad \text{and} \qquad \sum^\infty_{n=0}
\frac{(\mu_n - \pi^2 n^2)_+}{n^2} < \infty, \tag 2.8
$$
with $\mu_m \neq \lambda_n$ for all $m,n \in \Bbb N_0$. Let
$m_1, m_2$ be the $m$-functions for two operators
$H_j=-\frac{d^2}{dx^2}+q_j$, $j=1,2$ in $L^2 ((0,1))$
with boundary conditions
$$
u'(1) + h^{(j)}_1 u(1)=0
$$
and $h^{(j)}_1\in\Bbb R$, $j=1,2$. Suppose that $m_1(z) = m_2(z)$
for all $z$ in
$\{\lambda_n\}^\infty_{n=0} \cup \{\mu_n\}^\infty_{n=0}$ except
perhaps for one. Then $m_1 = m_2$ \rom(and hence $q_1=q_2$
a.e.~on $[0,1]$ and $h_1^{(1)} = h_1^{(2)}$\rom).
\endproclaim

By scaling, one sees the following analog of Theorem~2.1 holds
(there
is also an analog of Theorem~2.2):

\proclaim{Theorem 2.3} Let $a<b$ and
$\{\lambda_n\}_{n\in\Bbb N_0}$ be a sequence of distinct
positive real numbers satisfying
$$
\sum^\infty_{n=0} \frac{(\lambda_n -
\frac{\pi^2 n^2}{4(b-a)^2})_+}
{n^2} <\infty. \tag 2.9
$$
Let $m_1, m_2$ be the $m$-functions for two operators
$H_j=-\frac{d^2}{dx^2} + q_j$, $j=1,2$ in $L^2 ((a,b))$ with
boundary
conditions {\rom{(1.3)}} at $x=a$ and
$$
u'(b) + h^{(j)}_b u(b) =0,
$$
where $h^{(j)}_b \in \Bbb R$, $j=1,2$. Suppose that
$m_1 (\lambda_n)= m_2 (\lambda_n)$
for all $n\in \Bbb N_0$. Then $m_1 = m_2$ \rom(and hence
$q_1=q_2$
a.e.~on $[a,b]$ and $h_b^{(1)} = h_b^{(2)}$\rom).
\endproclaim

\vskip 0.3in
\flushpar{\bf \S 3. Whole Interval Results}
\vskip 0.1in

Fix $h_1\in \Bbb R$, let $H(h_0)$ be the operator on
$L^2 ((0,1))$
with $u'(1) + h_1 u(1) =0$ and $u'(0) + h_0 u(0)=0$ boundary
conditions, and denote by $\lambda_n (h_0)$ the corresponding
eigenvalues of $H(h_0)$.
Then, for $h_0\in \Bbb R$, it is known (see, e.g., the
references in [\gsds]) that
$$
\lambda_n = (n\pi)^2 + 2(h_1 - h_0) + \int^1_0 q(x)\, dx
+ o(1) \text{ as } n\to \infty \tag 3.1
$$
and for $h_0 =\infty$,
$$
\lambda_n = [(n+\tfrac12) \pi]^2 + 2h_1 + \int^1_0 q(x)\, dx
+ o(1) \text{ as } n\to \infty. \tag 3.2
$$

To say that $H(h_0)$ has eigenvalue $\lambda$ is equivalent to
$m(\lambda) = -h_0$. Thus, Theorem~2.1 implies

\proclaim{Theorem 3.1} Let $H_1 (h_0), H_2(h_0)$ be
associated with
two potentials $q_1, q_2$ on $[0,1]$ and two potentially
distinct
boundary conditions $h^{(1)}_1, h^{(2)}_1 \in \Bbb R$ at
$x=1$.
Suppose that
$\{(\lambda_n, h^{(n)}_0)\}_{n\in\Bbb N_0}$ is a sequence
of pairs
with $\lambda_0 <
\lambda_1 <\cdots \to \infty$ and $h^{(n)}_0 \in\Bbb R \cup
\{\infty\}$ so that both $H_1 (h^{(n)}_0)$ and
$H_2 (h^{(n)}_0)$
have eigenvalues at $\lambda_n$. Suppose that {\rom{(2.1)}}
holds. Then $q_1 = q_2$ a.e.~on $[0,1]$ and
$h^{(1)}_1 = h^{(2)}_1$.
\endproclaim

Given (3.1), (3.2) we immediately have Borg's theorem [\borg]
as a corollary
(this is essentially the usual proof), but more is true. For
example,
by using Theorem~2.2 one infers:

\proclaim{Corollary 3.2 [\borg]} Fix
$h^{(1)}_0, h^{(2)}_0 \in \Bbb R$.
Then all the eigenvalues of $H(h^{(1)}_0)$ and all the
eigenvalues
of $H(h^{(2)}_0)$, save one, uniquely determine
$q$ a.e.~on $[0,1]$.
\endproclaim

\proclaim{Corollary 3.3} Let $h^{(1)}_0, h^{(2)}_0, h^{(3)}_0
\in \Bbb R$ and denote by $\sigma_j=\sigma(H(h^{(j)}_0))$
the spectra
of $H(h^{(j)}_0)$, $j=1,2,3$. Assume $S_j
\subseteq \sigma_j$, $j=1,2,3$ and suppose that for all
sufficiently large $\lambda_0 >0$ we have
$$
\#\{\lambda \in \{S_1 \cup S_2 \cup S_3\} \text{\rom{ with }}
\lambda\leq \lambda_0 \} \geq \tfrac23 \#\{\lambda \in
\{\sigma_1 \cup \sigma_2 \cup \sigma_3 \}
\text{\rom{ with }} \lambda\leq \lambda_0 \} -1.
$$
Then $q$ is uniquely determined a.e.~on $[0,1]$.
\endproclaim

In particular, two-thirds of three spectra determine $q$.

\vskip 0.3in
\flushpar {\bf \S 4. Updating $\boldkey m$}
\vskip 0.1in

We are now able to understand why partial information on $q$
--- knowing it on $[0,a]$ --- lets us get away with less
information on eigenvalues, a phenomenon originally discovered by
Hochstadt-Lieberman [\hl] in the special case where
$a=\frac12$. We note that $m(z,x)$ satisfies the Ricatti-type
equation
$$
m'(z,x) = q(x)-z-m^2(z,x). \tag 4.1
$$
If we know that $\lambda$ is an eigenvalue of $H(h_0)$, then
$m(\lambda, 0)=-h_0$. If we know $q$ on $[0,a]$, we can use (4.1)
to compute $m(\lambda, a) := -h_a$ and so infer that $\lambda$
is an eigenvalue of $H(h_a)$, the operator in $L^2((a,1))$.
By Theorem~2.3, that means we only need a lower density of
eigenvalues of the various $H(h_a)$. A typical result is the
following

\proclaim{Theorem 4.1} Let $\sigma_N$ and $\sigma_D$ be the
eigenvalues of $H(h_0 =0)$
and $H(h_0 =\infty)$, respectively. Let $S_N \subseteq \sigma_N$,
$S_D \subseteq \sigma_D$. Fix $a\in (0,1)$. Suppose
for $\lambda_0 >0$
sufficiently large that
$$
\#\{\lambda \in \{S_N \cup S_D \} \text{\rom{ with }}
\lambda\leq \lambda_0 \}
\geq (1-a)\# \{\lambda\in \{\sigma_N \cup \sigma_D \}
\text{\rom{ with }}
\lambda\leq \lambda_0 \}.
$$
Then $S_N, S_D$ and $q$ on $[0,a]$ uniquely determine
$q$ a.e.~on $[0,1]$.
\endproclaim

This follows immediately from the updating idea. For example, if
$a=\frac34$, we can recover Theorem~1.3 of [\gsds] (it is
essentially a reworking of the proof in [\gsds]); but for
$a\in (0,\frac12)$, the result is new and implies, for example,
that $q$ on $[0,\frac14]$, all the Neumann eigenvalues, and half
the Dirichlet eigenvalues determine $q$ a.e.~on $[0,1]$.


\vskip 0.3in
\Refs
\endRefs

\vskip 0.1in

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Sturm-Liouvilleschen
Eigenwertaufgabe}{Acta Math.}{78}{1946}{1--96}
\gap
\item{\borgun.} 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{\gl.} I.M.~Gel'fand and B.M.~Levitan, {\it{On the
determination
of a differential equation from its special function}},
Izv.~Akad.~Nauk SSR. Ser.~Mat. {\bf 15} (1951),
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English transl.~in Amer.~Math. Soc.~Transl.~Ser. (2)
{\bf 1} (1955),
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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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\gap
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\gap
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{Mat.~Tidskr.}{B}{1949}{25--30}
\gap
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{68}{1968}{1--20}
\gap
\item{\levbook.} B.~Levitan, {\it{Inverse
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VNU Science Press, Utrecht, 1987.
\gap
\item{\lg.} B.M.~Levitan and M.G.~Gasymov, {\it
{Determination of a
differential equation by two of its spectra}},
Russ.~Math.~Surv. {\bf 19:2}
(1964), 1--63.
\gap
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\gap
\item{\malsp.} M.M.~Malamud, {\it{Spectral analysis of
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\gap
\item{\mar.} V.A.~Marchenko, {\it {Some questions in the
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\gap
\item{\simon.} B.~Simon, {\it{A new approach to inverse
spectral
theory, I. The basic formalism}}, in preparation.


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