\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