\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 $c0$, $k=1,\dots,N-1$. 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 $00 \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