\input amstex \documentstyle{amsppt} \magnification=1200 \baselineskip=15 pt \loadbold \TagsOnRight \NoBlackBoxes \topmatter \title Uniqueness Theorems in Inverse Spectral Theory for One-Dimensional Schr\"odinger Operators \endtitle \rightheadtext{Uniqueness Theorems in Inverse Spectral Theory} \author F.~Gesztesy$^{1}$ and B.~Simon$^{2}$ \endauthor \leftheadtext{F.~Gesztesy and B.~Simon} \thanks {\it{1991 Mathematics Subject Classification}}: Primary 34B24, 34L05, 81Q10; Secondary 34B20, 47A10. \endthanks \thanks{\it{Key words and phrases}}: Schr\"odinger operators, inverse spectral theory, Krein's spectral shift function \endthanks \thanks $^{1}$ Department of Mathematics, University of Missouri, Columbia, MO 65211. E-mail: mathfg\@mizzou1.\linebreak missouri.edu \endthanks \thanks $^{2}$ Division of Physics, Mathematics, and Astronomy, California Institute of Technology, 253-37,\linebreak Pasadena, CA 91125. This material is based upon work supported by the National Science Foundation under Grant No.~DMS-9101715. The Government has certain rights in this material. \endthanks \thanks To appear in {\it{Trans.~Amer.~Math.~Soc.}} \endthanks \abstract New unique characterization results for the potential $V(x)$ in connection with Schr\"odinger operators on $\Bbb R$ and on the half-line $[0,\infty)$ are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schr\"odinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schr\"odinger operators with confining potentials on the entire real line. \endabstract \endtopmatter \document \flushpar{\bf \S1. Introduction} The purpose of this article is to prove a variety of new uniqueness theorems for potentials $V(x)$ in one-dimensional Schr\"odinger operators $-\frac{d^2}{dx^2}+V$ on $\Bbb R$ and on the half-line $\Bbb R_{+}=[0,\infty)$ in terms of appropriate Krein spectral shift functions recently introduced in a series of papers describing new trace formulas for $V(x)$ on $\Bbb R$ [15],[17],[19],[20] and on $\Bbb R_{+}$ [14]. First we briefly recall these trace formulas for Schr\"odinger operators $H=-\frac{d^2}{dx^2}+V$ on the real line $\Bbb R$ assuming $V$ to be real-valued, continuous, and bounded from below. In addition to $H$, one also considers the family of operators $H^{\beta}_{y}=- \frac{d^2}{dx^2}+V$, $\beta\in\Bbb R\cup\{\infty\}$, $y\in\Bbb R$, with an additional boundary condition of the type $g'(y_{\pm})+\beta g(y_{\pm})=0$ for elements $g$ in the domain of $H^{\beta}_{y}$; see (A.30) and (3.2) for detailed domain descriptions. Here, in obvious notation, $\beta=\infty$ denotes the corresponding operator $H^{\infty}_{y}$ with an additional Dirichlet boundary condition at $y\in\Bbb R$. Denoting by $\xi^{\beta} (\lambda, y)$ Krein's spectral shift function for the pair $(H^{\beta}_{y}, H)$, $\beta\in\Bbb R\cup\{\infty\}$, $y\in\Bbb R$ (see (3.12)--(3.18)), the following trace formulas have been derived in [15] in the Dirichlet case $\beta=\infty$ and in [20] for $\beta\in\Bbb R$: $$\split V(x) = E_{0}+\lim\limits_{z\to i\infty} \int\limits^{\infty}_{E_0} & d\lambda \,\frac{z^2}{(\lambda - z)^{2}}\,[1-2\xi^{\infty} (\lambda, x)], \\ &\qquad E_{0}=\inf\{\sigma(H)\}, \beta=\infty, x\in\Bbb R, \endsplit \tag 1.1$$ $$\split V(x)=2\beta^{2}+E^{\beta}_{0}(x)+ \lim\limits_{z\to i\infty} & \int\limits^{\infty}_{E^{\beta}_{0}(x)} d\lambda \,\frac{z^2}{(\lambda - z)^{2}}\, [1+2\xi^{\beta}(\lambda, x)], \\ &\qquad E^{\beta}_{0}(x)=\inf\{\sigma H^{\beta}_{x})\}, \beta\in\Bbb R, x\in\Bbb R. \endsplit \tag 1.2$$ (Here $\sigma(\,\cdot\,)$ denotes the spectrum.) These trace formulas extend previous results by [7--9],[12],[22],[26],[28],[29],[34],[35],[39],[40] in the short-range, periodic, and certain almost periodic cases. A similar result can be derived for half-line Schr\"odinger operators. Assuming again $V$ to be real-valued, continuous, and bounded from below, denote by $H_{+,\alpha}=-\frac{d^2}{dx^2}+V$, $\alpha\in [0,\pi)$ the family of Schr\"odinger operators on the half-line $\Bbb R_{+}= [0,\infty)$ with the boundary condition $\sin(\alpha)g'(0_{+})+ \cos(\alpha)g(0_{+})=0$ for elements $g$ in the domain of $H_{+,\alpha}$ (cf.~(A.14)). For $\alpha_{1},\alpha_{2} \in (0,\pi)$, $\alpha_{1}\neq \alpha_{2}$, let $\xi_{\alpha_{1},\alpha_{2}}(\lambda)$ be Krein's spectral shift function for the pair $(H_{+,\alpha_{2}}, H_{+,\alpha_{1}})$ (cf.~(2.8)--(2.10)). Then the following trace formula can be inferred from the results in [14]: $$V(0)=\cot^{2}(\alpha)+\lim\limits_{z\to i\infty} \biggl\{-z-i\cot(\alpha) z^{1/2}+2\int\limits_{\Bbb R}d\lambda\frac{z^2}{(\lambda-z)^{2}}\, \xi_{0,\alpha}(\lambda)\biggr\}, \quad \alpha\in (0,\pi). \tag 1.3$$ A quick look at (1.1), (1.2), and (1.3) reveals the fact that $\xi^{\beta}(\lambda, x)$, $\lambda, x\in\Bbb R$ determines $V(x)$, $x\in\Bbb R$ and $\xi_{0,\alpha}(\lambda)$, $\lambda\in\Bbb R$ determines $V(0)$ in the half-line case. However, clearly both of these statements describe a mismatch and hence miss the point: $\xi^{\beta}(\lambda, x)$ depends on two real variables as opposed to one in $V(x)$ and analogously, $\xi_{0,\alpha}(\lambda)$ depends on one real variable while $V(0)$ is just a constant. From the point of view of inverse spectral theory, the problems that need clarification appear to be the following: Does $\xi^{\beta}(\lambda, x_{0})$ for fixed $x_{0}\in\Bbb R$ and all $\lambda\in\Bbb R$ determine $V(x)$ for all $x\in\Bbb R$ and similarly, does $\xi_{\alpha_{1}, \alpha_{2}} (\lambda)$, $\alpha_{1}\neq\alpha_{2}$ for all $\lambda\in\Bbb R$ determine $V(x)$ for all $x\geq 0$ in the half-line case? The present paper provides complete solutions to these problems. In Section 2 we treat the half-line case and provide an affirmative answer to the problem posed: $\xi_{\alpha_{1}, \alpha_{2}}(\lambda)$, $\alpha_{1}\neq\alpha_{2}$ for a.e.~$\lambda\in\Bbb R$ indeed uniquely determines $V(x)$ for a.e.~$x\geq 0$ (cf.~Theorem 2.4) extending a well-known result of Borg [5] and Marchenko [32], obtained independently from each other around 1952 for operators with purely discrete spectrum, to arbitrary spectral types (see Corollary 2.5). We conclude Section 2 with an application of our main Theorem 2.4 to three-dimensional Schr\"odinger operators with spherically symmetric potentials and state a new uniqueness theorem in this context (cf.~Theorem 2.6). Section 3 is devoted to Schr\"odinger operators on the entire real line. While the corresponding question posed concerning $\xi^{\beta}(\lambda, x_{0})$ turns out to have a negative answer, that is, $\xi^{\beta}(\lambda, x_{0})$ for fixed $x_{0}\in\Bbb R$ and a.e.~$\lambda \in\Bbb R$ in general cannot determine $V$ uniquely for a.e.~$x\in\Bbb R$, Theorem 3.2 shows that $\xi^{\beta_1} (\lambda, x_{0})$ and $\xi^{\beta_2} (\lambda, x_{0})$, $\beta_{1} \neq\beta _2$ for a.e.~$\lambda\in\Bbb R$ uniquely determine $V$ a.e.~except in the Dirichlet and Neumann cases $\beta_{1}=0$, $\beta_{2}=\infty$ respectively, $\beta_{1}=\infty$, $\beta_{2}=0$. In the latter case, $V$ is uniquely determined up to reflection symmetry with respect to $x_0$. When combining $\xi^{\beta}(\lambda, x_{0})$, $\lambda\in\Bbb R$ with additional Dirichlet data and/or norming constants, further unique characterizations of $V$ can be achieved. This is illustrated in connection with Theorem 3.6 which provides a new uniqueness result for Schr\"odinger operators on $\Bbb R$ with purely discrete spectra. Since our techniques rely heavily on the use of certain properties of Herglotz functions and especially on the Weyl-Titchmarsh theory, we collected a variety of pertinent results in Appendix A. Perhaps we should emphasize at this point that we do not discuss explicit reconstruction procedures for $V(x)$ in this paper (the reader can find standard results on reconstruction techniques, e.g., in [13],[29],[30],[32], and [33]). Here we exclusively focus on deriving new minimal sets of spectral data which uniquely determine the potential $V$ a.e. The basic outline of our philosophy of how to recover $V(x)$ from $\xi^{\infty}(\lambda, x_{0})$, $\lambda\in\Bbb R$ and Dirichlet data is described in [15]. We shall return to this topic elsewhere. Analogous results for second-order finite difference operators are in prepration [18]. \vskip 0.3in \flushpar{\bf \S 2. Schr\"odinger Operators on $\boldkey[\bold 0 \boldkey,\boldsymbol\infty\boldkey)$} In this section we shall describe a uniqueness result for Schr\"odinger operators on the half-line $[0,\infty)$, which extends a well-known theorem of Borg [5] and Marchenko [32] in the special case of purely discrete spectra to arbitrary spectral types. We shall freely exploit the notation introduced in Appendix A and recall $\tau_{+}$, $H_{+,\alpha}$, $\phi_\alpha$, $\theta_\alpha$, $\psi_{+,\alpha}$, $m_{+,\alpha}$, $d\rho_{+,\alpha}$, and $G_{+,\alpha} (z,x,x')$ as introduced in (A.13)--(A.27). In particular, we shall assume hypothesis (A.12), that is, $$V\in L^{1}([0,R]) \text{ for all R>0}, \qquad V\text{ real-valued} \tag 2.1$$ throughout this section and recall that $H_{+,\alpha}$, defined in terms of separated boundary conditions, is a real operator of uniform spectral multiplicity one. The basic uniqueness criterion for Schr\"odinger operators on the half-line $[0,\infty)$ we shall rely on repeatedly in the following can be stated as follows. \proclaim{Theorem 2.1} {\rom{(See, e.g., [32])}} Suppose $\alpha_{1}, \alpha_{2}\in [0,\pi)$, $\alpha_{1}\neq\alpha_{2}$ and define $H_{+, j, \alpha_{j}}$, $m_{+, j, \alpha_{j}}$, $\rho_{+,j,\alpha_{j}}$ associated with the differential expressions $\tau_{j}=-\frac{d^2}{dx^2}+V_{j}(x)$, $x\geq 0$, where $V_{j}, j=1,2$ satisfy hypothesis \rom{(2.1)}. Then the following are equivalent: \roster \item"\rom{(i)}" $m_{+,1,\alpha_{1}}(z)=m_{+,2,\alpha_{2}}(z)$, $z\in\Bbb C_{+}$. \item"\rom{(ii)}" $\rho_{+,1,\alpha_{1}}((-\infty,\lambda])= \rho_{+,2,\alpha_{2}} ((-\infty,\lambda])$, $\lambda\in\Bbb R$. \item"\rom{(iii)}" $\alpha_{1}=\alpha_{2}$ and $V_{1}(x)=V_{2}(x)$ for a.e.~$x\geq 0$. \endroster \endproclaim We begin our analysis with a simple warm-up relating Green's functions for different boundary conditions at $x=0$. (We also recall our convention of Appendix A to fix the boundary condition (if any) at $x=+\infty$.) \proclaim{Lemma 2.2} Let $\alpha_{j}\in [0,\pi)$, $j=1,2$, $x,x'\in \Bbb R_{+}$, and $z\in\Bbb C\backslash\{\sigma(H_{+,\alpha_{1}})\cup \sigma(H_{+,\alpha_{2}})\}$. Then \flushpar{\rom{(i)}} $$G_{+,\alpha_{2}}(z,x,x')-G_{+,\alpha_{1}}(z,x,x')=- \frac{\psi_{+,\alpha_{1}}(z,x)\psi_{+,\alpha_{1}}(z,x')} {\cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(z)}. \tag 2.2$$ \flushpar{\rom{(ii)}} \align \frac{G_{+,\alpha_{2}}(z,0,0)}{G_{+,\alpha_{1}}(z,0,0)} &=\frac{1} {(\beta_{1}-\beta_{2})\sin^{2}(\alpha_{1})[\cot(\alpha_{2}-\alpha_{1}) +m_{+,\alpha_{1}}(z)]} \tag 2.3 \\ &= (\beta_{1}-\beta_{2})\sin^{2}(\alpha_{2})[\cot(\alpha_{2}-\alpha_{1})- m_{+,\alpha_{2}}(z)],\quad \beta_{j}=\cot(\alpha_{j}), j=1,2. \tag 2.4 \endalign \flushpar{\rom{(iii)}} \align \text{\rom{Tr}}[(H_{+,\alpha_{2}}-z)^{-1}-(H_{+,\alpha_{1}}-z)^{-1}] &= -\frac{d}{dz}\,\ln[\cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(z)] \tag 2.5 \\ &=\frac{d}{dz}\,\ln [\cot(\alpha_{2}-\alpha_{1})-m_{+,\alpha_{2}}(z)]. \tag 2.6 \endalign \endproclaim \demo{Proof} (2.2) is a direct consequence of (A.16)--(A.18), (A.23), and (A.38). Similarly, (2.3) and (2.4) follow by combining (A.25) and (A.38). (2.5) follows from (2.2) and (A.44) in the limit $z_{1}\to z_{2}=z$. (2.6) is clear from $$\cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(z)=[\sin(\alpha_{2} -\alpha_{1})]^{2}[\cot(\alpha_{2}-\alpha_{1})-m_{+,\alpha_{2}}(z)]^{-1}, \tag 2.7$$ a simple consequence of (A.38). \qed \enddemo Since $m_{+,\alpha}(z)$ is a Herglotz function, we may now introduce Krein's spectral shift function [27] $\xi_{\alpha_{1}, \alpha_{2}}(\lambda)$ for the pair $(H_{+,\alpha_{2}}, H_{+, \alpha_{1}})$ according to (A.2), (A.4) by $$\multline \cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(z) =\exp\biggl\{\text{Re}[\ln (\cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(i))] \\ + \int\limits_{\Bbb R} \biggl[\frac{1}{\lambda-z}- \frac{\lambda}{1+\lambda^2}\biggr]\xi_{\alpha_{1}, \alpha_{2}} (\lambda)\,d\lambda\biggr\}, \quad 0\leq\alpha_{1} <\alpha_{2} <\pi, z\in\Bbb C\backslash\Bbb R. \endmultline \tag 2.8$$ This is extended to all $\alpha_{1},\alpha_{2}\in [0,\pi)$ by $$\xi_{\alpha,\alpha} (\lambda)=0, \qquad \xi_{\alpha_{2}, \alpha_{1}} (\lambda)=-\xi_{\alpha_{1}, \alpha_{2}}(\lambda) \text{ for a.e.~\lambda\in\Bbb R}. \tag 2.9$$ (2.7) then implies $$\multline \cot(\alpha_{2}-\alpha_{1})-m_{+,\alpha_{2}}(z) =\exp\biggl\{\text{Re} [\ln(\cot(\alpha_{2}-\alpha_{1})-m_{+,\alpha_{2}}(i))] \\ - \int\limits_{\Bbb R} \biggl[\frac{1}{\lambda-z}-\frac{\lambda} {1+\lambda^2}\biggr] \xi_{\alpha_{1},\alpha_{2}}(\lambda)\, d\lambda \biggr\}, \quad 0\leq\alpha_{1} <\alpha_{2} <\pi, z\in\Bbb C\backslash \Bbb R. \endmultline \tag 2.10$$ Next we summarize a few properties of $\xi_{\alpha_{1},\alpha_{2}} (\lambda)$. \proclaim{Lemma 2.3} {\rom{(i)}} Suppose $0\leq\alpha_{1} <\alpha_{2} <\pi$. Then for a.e.~$\lambda\in\Bbb R$, $$\xi_{\alpha_{1},\alpha_{2}}(\lambda) = \cases \lim\limits_{\epsilon\downarrow 0}\,\pi^{-1} \text{\rom{Im}} \{\ln[\cot(\alpha_{2}-\alpha_{1})+m_{+,\alpha_{1}}(\lambda+ i\epsilon)]\} &\quad\hphantom{...........................} (2.11) \\ -\lim\limits_{\epsilon\downarrow 0}\,\pi^{-1}\text{\rom{Im}}\{\ln [\cot(\alpha_{2}-\alpha_{1})-m_{+,\alpha_{2}}(\lambda+ i\epsilon)]\} &\quad\hphantom{...........................} (2.12) \\ \lim\limits_{\epsilon\downarrow 0}\,\pi^{-1}\text{\rom{Im}}\bigl\{\ln \bigl[\tfrac{1}{\sin(\alpha_{1})}\,\tfrac{G_{+,\alpha_{1}}(\lambda + i\epsilon, 0, 0)}{G_{+,\alpha_{2}}(\lambda + i\epsilon, 0,0)}\bigr] \bigr\}. &\quad\hphantom{...........................} (2.13) \endcases$$ \rom(For $\alpha_{1}=0$, $G_{+,\alpha_{1}}(\lambda+i\epsilon,0,0)/\sin (\alpha_{1})$ has to be replaced by $-1$ in {\rom{(2.13)}} according to {\rom{(A.25)}}.\rom) Moreover, $$0\leq \xi_{\alpha_{1},\alpha_{2}}(\lambda)\leq 1 \text{\rom{ a.e.}} \tag 2.14$$ {\rom{(ii)}} Let $\alpha_{j}\in [0,\pi)$, $1\leq j\leq 3$. Then the chain rule'' $$\xi_{\alpha_{1},\alpha_{3}}(\lambda)=\xi_{\alpha_{1},\alpha_{2}}(\lambda) +\xi_{\alpha_{2},\alpha_{3}}(\lambda) \tag 2.15$$ holds for a.e.~$\lambda\in\Bbb R$. {\rom{(iii)}} For all $\alpha_{1},\alpha_{2}\in [0,\pi)$, $$\xi_{\alpha_{1},\alpha_{2}}\in L^{1}(\Bbb R; (1+\lambda^{2})^{-1}\, d\lambda). \tag 2.16$$ {\rom{(iv)}} Assume $\alpha_{1}, \alpha_{2}\in [0,\pi)$, $\alpha_{1}\neq \alpha_{2}$. Then $$\xi_{\alpha_{1},\alpha_{2}}\in L^{1}(\Bbb R; (1+|\lambda|)^{-1}\, d\lambda) \text{\rom{ if and only if }} \alpha_{1},\alpha_{2}\in (0,\pi). \tag 2.17$$ {\rom{(v)}} For all $\alpha_{1},\alpha_{2}\in [0,\pi)$, $$\text{\rom{Tr}} [(H_{+,\alpha_{2}}-z)^{-1} -(H_{+,\alpha_{1}}-z)^{-1}] =-\int\limits_{\Bbb R} (\lambda-z)^{-2} \xi_{\alpha_{1},\alpha_{2}} (\lambda)\, d\lambda. \tag 2.18$$ \endproclaim \demo{Proof} (i) (2.11)--(2.13) follow from (2.3), (2.4) (resp.~(2.7)), (2.8), (A.2), and (A.4). (2.14) is clear from (A.4). (ii) is a consequence of (2.13). (iii) is obvious from $0\leq |\xi_{\alpha_{1}, \alpha_{2}}|\leq 1$ a.e. (iv) By (2.9) we may assume $0\leq\alpha_{1} <\alpha_{2} <\pi$. Then (A.39) yields $$\cot(\alpha_{2}-\alpha_{1})-m_{+,\alpha_{2}}(z) \operatornamewithlimits{=}\limits_{z\to i\infty} \cases 0, & \alpha_{1}= 0 \\ \cot(\alpha_{2}-\alpha_{1})-\cot(\alpha_{2})>0, &0<\alpha_{1}<\alpha_{2}<\pi \endcases \tag 2.19$$ and it suffices to apply Theorem A.1(iii) to $\cot(\alpha_{2}- \alpha_{1})-m_{+,\alpha_{2}}(z)$ taking into account (2.10). (v) follows from (2.5) and from applying $-\frac{d}{dz}\ln(\,\cdot\,)$ to (2.8). \qed \enddemo We note that $\xi_{\alpha_{1}, \alpha_{2}}(\lambda)$ (for $\alpha_{1}, \alpha_{2}\in (0,\pi)$) has been introduced by Javrjan [23],[24]. In particular, he proved (2.5) and (2.18) in the non-Dirichlet cases where $0<\alpha_{1},\alpha_{2}<\pi$. We also remark that (2.18) extends to more general situations of the type $$\text{Tr}[F(H_{+,\alpha_{2}})-F(H_{+,\alpha_{1}})]=\int\limits_{\Bbb R} F'(\lambda)\xi_{\alpha_{1},\alpha_{2}}(\lambda)\,d\lambda \tag 2.20$$ for appropriate functions $F$ (see, e.g., [38]). Given these preliminaries, we are now able to state our main uniqueness result for half-line Schr\"odinger operators. \proclaim{Theorem 2.4} Suppose $V_j$ satisfy hypothesis {\rom{(2.1)}} and introduce the differential expressions $\tau_{j}=-\frac{d^2} {dx^2}+V_{j}(x)$, $x\geq 0$, $j=1,2$. Let $\alpha_{j,\ell}\in [0,\pi)$, $\ell=1,2$, suppose $0\leq \alpha_{1,1} <\alpha_{1,2} <\pi$, $0\leq \alpha_{2,1} <\alpha_{2,2} <\pi$, and define $H_{+,j,\alpha_{j,\ell}}$ for $j,\ell=1,2$ associated with $\tau_j$ as in \rom{(A.14)}. In addition, let $\xi_{j,\alpha_{j,1},\alpha_{j,2}}$, $j=1,2$ be Krein's spectral shift function for the pair $(H_{+,j,\alpha_{j,1}}, H_{+,j,\alpha_{j,2}})$. Then the following are equivalent: \roster \item"\rom{(i)}" $\xi_{1,\alpha_{1,1},\alpha_{1,2}}(\lambda) = \xi_{2,\alpha_{2,1},\alpha_{2,2}}(\lambda)$ for a.e.~$\lambda\in\Bbb R$. \item"\rom{(ii)}" $\alpha_{1,1}=\alpha_{2,1}$, $\alpha_{1,2}=\alpha_{2,2}$, and $V_{1}(x)=V_{2}(x)$ for a.e.~$x\geq 0$. \endroster \endproclaim \demo{Proof} We only need to prove that (i) implies (ii). From Lemma 2.3(iv), one infers that $$\alpha_{j,1}\underset (=)\to > 0 \quad\text{if and only if } \int\limits_{\Bbb R}(1+|\lambda|)^{-1} |\xi_{\alpha_{j,1}, \alpha_{j,2}}(\lambda)|\, d\lambda \underset (=)\to < \infty, \quad j=1,2. \tag 2.21$$ Since by hypothesis $\alpha_{1,1}\underset (=)\to > 0$ if and only if $\alpha_{2,1}\underset (=)\to >0$, one is led to the following case distinction. a) $0<\alpha_{1,1}<\alpha_{1,2}<\pi$, $0<\alpha_{2,1}<\alpha_{2,2} <\pi$. \flushpar Then (2.10) and (A.39) imply \align \int\limits^{\infty}_{z}dz'\int\limits_{\Bbb R} (\lambda-z')^{-2} \xi_{j,\alpha_{j,1},\alpha_{j,2}}(\lambda)\, d\lambda &= \ln\biggl[\frac{\cot(\alpha_{j,2}-\alpha_{j,1})-m_{+,j,\alpha_{j,2}} (z)}{\cot(\alpha_{j,2}-\alpha_{j,1})-\cot(\alpha_{j,2})} \biggr] \tag 2.22 \\ &\operatornamewithlimits{=}\limits_{z\to i\infty} (\beta_{j,2}- \beta_{j,1}) iz^{-1/2} + (\beta^{2}_{j,1}-\beta^{2}_{j,2}) 2^{-1} z^{-1} +o(z^{-1}), \\ &\qquad \hphantom{......} \beta_{j,\ell}=\cot(\alpha_{j,\ell}), j,\ell=1,2. \tag 2.23 \endalign Given (i), the asymptotic behavior (2.23) then yields $$\alpha_{1,1}=\alpha_{2,1} \quad \text{and} \quad \alpha_{1,2}= \alpha_{2,2}. \tag 2.24$$ Insertion of (2.24) into (2.22), still assuming (i), then yields $$m_{+,1,\alpha_{1,2}}(z)=m_{+,2,\alpha_{1,2}}(z) \tag 2.25$$ and hence $V_{1}=V_{2}$ a.e.~by Theorem 2.1. b) $0=\alpha_{1,1}<\alpha_{1,2}<\pi$, $0=\alpha_{2,1}<\alpha_{2,2}<\pi$. \flushpar Then (2.10) and (A.39) imply \align \int\limits^{z}_{i} dz'\int\limits_{\Bbb R} &(\lambda-z')^{-2} \xi_{j,0,\alpha_{j,2}}(\lambda)\,d\lambda \\ &= -\ln\biggl[\frac{\cot(\alpha_{j,2})-m_{+,j,\alpha_{j,2}}(z)} {\cot(\alpha_{j,2})-m_{+,j,\alpha_{j,2}}(i)}\biggr] \tag 2.26 \\ &\operatornamewithlimits{=}\limits_{z\to i\infty} \ln(z^{1/2}) + \ln[i\sin^{2}(\alpha_{j,2})] + \ln[\cot(\alpha_{j,2}) -m_{+,j,\alpha_{j,2}}(i)] \\ &\qquad \hphantom{......} -\cot(\alpha_{j,2}) iz^{-1/2} + o(z^{-1/2}), \quad j=1,2. \tag 2.27 \endalign Given (i), the $O(z^{-1/2})$-term in (2.27) then yields $$\alpha_{1,2} =\alpha_{2,2} \tag 2.28$$ and the $O(1)$-term in (2.27) yields $$m_{+,1,\alpha_{1,2}}(i)=m_{+,2,\alpha_{1,2}}(i). \tag 2.29$$ Inserting (2.28) and (2.29) into (2.26), still assuming (i), then yields $$m_{+,1,\alpha_{1,2}}(z)=m_{+,2,\alpha_{1,2}}(z) \tag 2.30$$ and hence again, $V_{1}=V_{2}$ a.e.~by Theorem 2.1. \qed \enddemo As a corollary, we obtain a well-known uniqueness result originally due to Borg [5] and Marchenko [32] obtained independently in 1952. \proclaim{Corollary 2.5} {\rom{(Borg [5], Theorem 1; Marchenko [32], Theorem 2.3.2; see also [30])}} Define $\tau_j$, $H_{+,j,\alpha}$, $\alpha\in [0,\pi)$ as in Theorem {\rom{2.4}. Assume in addition that $H_{+,1,\alpha_{1}}$ and $H_{+,2,\alpha_{2}}$ have purely discrete spectra for some \rom(and hence for all\rom) $\alpha_{j}\in [0,\pi)$, that is, $$\sigma_{\text{\rm{ess}}} (H_{+,j,\alpha_{j}})=\emptyset \quad \text{\rom{for some }} \alpha_{j}\in [0,\pi), j=1,2. \tag 2.31$$ Then the following are equivalent: \roster \item"\rom{(i)}" $\sigma(H_{+,1,\alpha_{1,1}})=\sigma (H_{+,2,\alpha_{2,1}})$, $\sigma(H_{+,1,\alpha_{1,2}})= \sigma (H_{+,2,\alpha_{2,2}})$, \newline \qquad\hphantom{..................................................} $\alpha_{j,\ell}\in [0,\pi)$, $j,\ell=1,2$, $\sin(\alpha_{1,1}- \alpha_{1,2})\neq 0$. \item"\rom{(ii)}" $\alpha_{1,1}=\alpha_{2,1}$, $\alpha_{1,2}=\alpha_{2,2}$, and $V_{1}(x)=V_{2}(x)$ for a.e.~$x\geq 0$. \endroster \endproclaim \demo{Proof} Without loss of generality, we may assume $0\leq \alpha_{1,1}<\alpha_{1,2}<\pi$, $0\leq\alpha_{2,1}<\alpha_{2,2}<\pi$ and hence need to prove that (i) implies $\xi_{1,\alpha_{1,1}, \alpha_{1,2}}=\xi_{2,\alpha_{2,1},\alpha_{2,2}}$ a.e. First we note that $\xi_{j,\alpha_{j,1},\alpha_{j,2}}(\lambda)$, being Krein's spectral shift function for the pair $(H_{+,j,\alpha_{j,2}}, H_{+,j,\alpha_{j,1}})$, $j=1,2$, increases (decreases) by $1$ whenever $\lambda$ passes an eigenvalue of $H_{+,j,\alpha_{j,1}} (H_{+,j,\alpha_{j,2,}})$ as $\lambda$ increases from $-\infty$ to $+\infty$ and stays constant otherwise. (We recall that $\sigma(H_{+,\alpha})$ is simple.) This step-function behavior, together with $0\leq\xi_{j,\alpha_{j,1}, \alpha_{j,2}}\leq 1$ a.e.,~indeed yields $\xi_{1,\alpha_{1,1},\alpha_{1,2}} =\xi_{2,\alpha_{2,1,}\alpha_{2,2}}$ a.e.~and one can apply Theorem 2.4. \qed \enddemo Roughly speaking, Corollary 2.5 says that two sets of purely discrete spectra $\sigma(H_{+,\alpha_{1}}),\mathbreak \sigma(H_{+,\alpha_{2}})$ associated with distinct boundary conditions at $x=0$ (but a fixed boundary condition (if any) at $+\infty$), that is, $\sin(\alpha_{2}- \alpha_{1})\neq 0$, uniquely determine $V$ a.e. Our main result, Theorem 2.4, removes all a priori spectral hypotheses and shows that Krein's spectral shift function $\xi_{\alpha_{1},\alpha_{2}}(\lambda)$ for the pair $(H_{+,\alpha_{2}}, H_{+,\alpha_{1}})$ with distinct boundary conditions at $x=0$, $\sin(\alpha_{2}-\alpha_{1})\neq 0$, uniquely determines $V$ a.e. This illustrates that Theorem 2.4 is the natural generalization of Borg's and Marchenko's theorem from the discrete spectrum case to arbitrary spectral types. Finally, we give a simple application of Theorem 2.4 in the context of three-dimensional Schr\"odinger operators with spherically symmetric potentials. Assuming hypothesis (2.1) for $V$, we introduce the potential $$v(x)=V(|x|), \quad x\in\Bbb R^{3} \tag 2.32$$ and define the self-adjoint Schr\"odinger operator $h$ in $L^{2}(\Bbb R^{3})$ associated with the differential expression $-\Delta +v(x)$ by decomposition with respect to angular momenta, which represents $h$ as an infinite direct sum of half-line operators in $L^{2}(\Bbb R_{+}; r^{2}\,dr)$ associated with differential expressions of the type $$\widehat{\tau}_{+,\ell}=-\frac{d^2}{dr^2}-\frac{2}{r}\,\frac{d}{dr} + \frac{\ell (\ell+1)}{r^2} +V(r), \quad r=|x|>0, \ell\in\Bbb N_{0}= \Bbb N\cup\{0\}. \tag 2.33$$ A simple unitary transformation reduces (2.33) to $$\tau_{+,\ell}=-\frac{d^2}{dr^2}+\frac{\ell(\ell+1)}{r^2}+V(r) \tag 2.34$$ and associated Hilbert space $L^{2}(\Bbb R_{+})$ (see, e.g., [37], Appendix to Sect.~X.1). Next, let $g(z,x,x')$, $x\neq x'$ denote the Green's function of $h$ (i.e., the integral kernel of $(h-z)^{-1}$) and define another self-adjoint operator $h_\beta$ in $L^{2}(\Bbb R^{3})$ by $$(h_{\beta}-z)^{-1}=(h-z)^{-1}+D_{\beta}(z)^{-1} (\overline{g(z,0,\,\cdot\,)}\,,\,\cdot\,)g(z,\,\cdot\,,0), \quad \beta\in\Bbb R, z\in\Bbb C\backslash\{\sigma(h_{\beta}) \cup\sigma(h)\}, \tag 2.35$$ where $$D_{\beta}(z)=\beta-\lim\limits_{|\epsilon|\downarrow 0}\,[g(z,0,\epsilon)-(4\pi |\epsilon|)^{-1}], \qquad z\in\Bbb C\backslash \sigma(h). \tag 2.36$$ As shown, for example, in [1],[41], $h_\beta$ models $h$ plus an additional point (delta) interaction centered at $x=0$ whose strength is parametrized by $\beta\in\Bbb R$. (Clearly, $h_{\infty}=h$.) The function $D_{\beta}(z)$ is Herglotz and one computes (see [14]) $$\text{Tr}[(h_{\beta}-z)^{-1}-(h-z)^{-1}]=- \frac{d}{dz}\ln[D_{\beta}(z)]. \tag 2.37$$ This then allows one to define Krein's spectral shift function $\xi_{\beta}(\lambda)$ for the pair $(h_{\beta}, h)$ by $$\xi_{\beta}(\lambda)=\lim\limits_{\epsilon\downarrow 0}\, \pi^{-1} \text{Im}\{\ln(D_{\beta}(\lambda+i\epsilon)]\}\text{ a.e.} \tag 2.38$$ which yields $$\text{Tr}[(h_{\beta}-z)^{-1}-(h-z)^{-1}]=-\int\limits_{\Bbb R} (\lambda -z)^{-2}\xi_{\beta}(\lambda)\,d\lambda. \tag 2.39$$ Our uniqueness result for three-dimensional Schr\"odinger operators then reads as follows. \proclaim{Theorem 2.6} Define $h_{j}$, $h_{j,\beta_{j}}$, $\beta_{j}\in\Bbb R$ associated with $-\Delta+v_{j}(x)$, $x\in\Bbb R^{3}$, $j=1,2$ and introduce Krein's spectral shift function $\xi_{j,\beta_{j}}(\lambda)$ for the pair $(h_{j,\beta_{j}},h_{j})$, $j=1,2$. Then the following are equivalent: \roster \item"\rom{(i)}" $\xi_{1,\beta_{1}}(\lambda)=\xi_{2,\beta_{2}} (\lambda)$ for a.e.~$\lambda\in\Bbb R$. \item"\rom{(ii)}" $\beta_{1}=\beta_{2}$ and $v_{1}(x)=v_{2}(x)$ for a.e.~$x\in\Bbb R^{3}$. \endroster \endproclaim \demo{Proof} Since $\tau_{+,\ell}$ is l.p.~at $r=0$ for all $\ell=\Bbb N$, the whole problem can be reduced to the angular momentum sector $\ell= 0$. For $\ell=0$, however, $h$ corresponds to $H_{+,\infty}$ and $h_\beta$ to $H_{+,\alpha}$, $\beta=\cot(\alpha)$ in the notation of (A.14). In particular, $\xi_{\beta}(\lambda)$ introduced in (2.38) corresponds to $\xi_{0,\alpha}(\lambda)$ in our notation (2.8). Hence, an application of Theorem 2.4 completes the proof. \qed \enddemo An analogous result could be derived for two-dimensional Schr\"odinger operators with centrally symmetric potentials. Since this requires the replacement of $\tau_{+}=-\frac{d^2}{dx^2}+V(x)$, $x\geq 0$, by $$\tau_{+}=-\frac{d^2}{dx^2}-\frac{1}{4x^2}+V(x), \quad x>0, \tag 2.40$$ a differential expression singular at $x=0$, we omit further details at this point. \vskip 0.3in \flushpar{\bf \S 3. Schr\"odinger Operators on $\Bbb R$} This section explores uniqueness results for Schr\"odinger operators on the whole real line. As in Section 2, we shall rely on the notation introduced in Appendix A and hence recall $\tau$, $H$, $\phi_{\alpha}$, $\theta_\alpha$, $\psi_{\pm,\alpha}$, $m_{\pm,\alpha}$, $d\rho_{\pm,\alpha}$, and $G(z,x,x')$ as introduced in (A.29)--(A.47). In particular, we shall assume hypothesis (A.28), that is, $$V\in L^{1}_{\text{loc}}(\Bbb R), \quad V\text{ real-valued} \tag 3.1$$ throughout this section. Following [20], we introduce, in addition, the following family of self-adjoint operators $H^{\beta}_{y}$ in $L^{2}(\Bbb R)$, $$\gathered H^{\beta}_{y}f=\tau f, \quad \beta\in\Bbb R\cup\{\infty\}, \quad y\in\Bbb R, \\ \Cal D(H^{\beta}_{y})=\{g\in L^{2}(\Bbb R\mid g,g'\in AC([y,\pm R])\text{ for all }R>0;\, g'(y_{\pm})+\beta g(y_{\pm})=0; \\ \lim\limits_{R\to\pm\infty}\,W(f_{\pm}(z_{\pm}),g)(R)=0;\, \tau g\in L^{2}(\Bbb R)\}. \endgathered \tag 3.2$$ Thus $H^{D}_{y}:=H^{\infty}_{y}(H^{N}_{y}:=H^{0}_{y})$ corresponds to the Schr\"odinger operator with an additional Dirichlet (Neumann) boundary condition at $y$. In obvious notation, $H^{\beta}_{y}$ decomposes into the direct sum of half-line operators $$H^{\beta}_{y}=H^{\beta}_{-,y}\oplus H^{\beta}_{+,y} \tag 3.3$$ with respect to $$L^{2}(\Bbb R)=L^{2}((-\infty, y]) \oplus L^{2}([y,\infty)). \tag 3.4$$ In particular, $H^{\beta}_{+,y}$ equals $H_{+,\alpha}$ for $\beta =\cot(\alpha)$ and $y=0$ in our notation (A.14) and, as indicated at the end of Appendix A, our (variable) reference point $x=y$ will be added as a subscript to obtain $\theta_{\alpha, y}(z,x)$, $\phi_{\alpha, y}(z,x)$, $\psi_{\pm,\alpha,y}(z,x)$, $m_{\pm,\alpha, y}(z)$, $M_{\alpha, y}(z)$, etc. $H$ and $H^{\beta}_{y}$, defined in terms of separated boundary conditions are real operators. Moreover, as observed in Appendix A, the point spectrum of $H$ is simple. Next, we recall a few results from [20]. With $G(z,x,x')$ and $G^{\beta}_{y}(z,x,x')$ the Green's functions of $H$ and $H^{\beta}_{y}$, one obtains $$\split G^{\beta}_{y}(z,x,x') &=G(z,x,x') -\frac{(\beta+\partial_{2})G(z,x,y) (\beta+\partial_{1}) G(z,y,x')}{(\beta+\partial_{1}) (\beta+\partial_{2})G(z,y,y)}, \\ &\qquad \hphantom{......} \beta\in\Bbb R, z\in\Bbb C\backslash \{\sigma(H^{\beta}_{y})\cup\sigma(H)\}, \endsplit \tag 3.5$$ $$G^{\infty}_{y}(z,x,x') = G(z,x,x')-G(z,y,y)^{-1} G(z,x,y) G(z,y,x'), \quad z\in\Bbb C\backslash\{\sigma(H^{\infty}_{y})\cup \sigma(H)\}. \tag 3.6$$ Here $$\gathered \left. \partial_{1}G(z,y,x'):=\partial_{x}G(z,x,x')\right|_{x=y}, \quad \left. \partial_{2}(G,z,x,y):=\partial_{x'}G(z,x,x')\right|_{x'=y}, \\ \left. \partial_{1}\partial_{2} G(z,y,y):=\partial_{x}\partial_{x'} G(z,x,x')\right|_{x=y=x'}, \quad \text{etc.} \endgathered\tag 3.7$$ and $$\partial_{1}G(z,y,x)=\partial_{2}G(z,x,y), \quad x\neq y. \tag 3.8$$ As a consequence, \align \text{Tr}[(H^{\beta}_{y}-z)^{-1}-(H-z)^{-1}] &=-\frac{d}{dz}\,\ln [(\beta+\partial_{1})(\beta+\partial_{2})G(z,y,y)], \quad \beta\in\Bbb R, \tag 3.9 \\ \text{Tr}[(H^{\infty}_{y}-z)^{-1}-(H-z)^{-1}] &=-\frac{d}{dz}\,\ln [G(z,y,y)]. \tag 3.10 \endalign In analogy to $G(z,y,y)$ (cf.~(A.47)), also $$(\beta+\partial_{1})(\beta+\partial_{2}) G(z,y,y) \text{ is Herglotz} \tag 3.11$$ for each $y\in\Bbb R$. Hence, both admit exponential representations of the form $$G(z,y,y)=\exp\biggr\{c_{\infty}+\int\limits_{\Bbb R} \biggl[\frac {1}{\lambda-z}-\frac{\lambda}{1+\lambda^2}\biggl] \xi^{\infty} (\lambda, y)\, d\lambda\biggr\}, \tag 3.12$$ $$c_{\infty}\in\Bbb R, \quad 0\leq\xi^{\infty}(\lambda, y)\leq 1 \text{ a.e.}, \tag 3.13$$ $$\xi^{\infty}(\lambda, y)=\lim\limits_{\epsilon\downarrow 0}\, \pi^{-1} \text{Im}\{\ln [G(\lambda+i\epsilon, y, y)]\} \text{ for a.e.~\lambda\in\Bbb R}, \tag 3.14$$ $$(\beta+\partial_{1})(\beta+\partial_{2})G(z,y,y)=\exp\biggl\{c_{\beta} +\int\limits_{\Bbb R}\biggl[\frac{1}{\lambda-z}-\frac{\lambda} {1+\lambda^2}\biggr] [\xi^{\beta}(\lambda, y)+1]\,d\lambda\biggr\}, \quad \beta\in\Bbb R, \tag 3.15$$ $$c_{\beta}\in\Bbb R, \quad -1\leq\xi^{\beta}(\lambda, y)\leq 0 \text{ a.e.}, \quad \beta\in\Bbb R, \tag 3.16$$ $$\xi^{\beta}(\lambda, y)=\lim\limits_{\epsilon\downarrow 0}\, \pi^{-1} \text{Im}\{\ln [(\beta+\partial_{1})(\beta+\partial_{2}) G(\lambda+i\epsilon, y, y)]\}-1, \quad \beta\in\Bbb R \tag 3.17$$ for each $y\in\Bbb R$. Moreover, $$\text{Tr}[(H^{\beta}_{y}-z)^{-1}-(H-z)^{-1}]=-\int\limits_{\Bbb R} (\lambda-z)^{-2}\xi^{\beta}(\lambda, y)\,d\lambda, \quad \beta\in\Bbb R\cup\{\infty\}. \tag 3.18$$ (Strictly speaking, the results (3.5)--(3.18) have been derived in [20] assuming $\tau$ to be in the l.p.~case at $\pm\infty$. However, these results extend to our present setting without effort.) For later purpose, we also note the identities (for each $y\in\Bbb R$), $$G(z,y,y)=M_{0,y,2,2}(z)=[m_{-,0,y}(z)-m_{+,0,y}(z)]^{-1}, \tag 3.19$$ $$\split \sin^{2}(\alpha)(\beta+\partial_{1})(\beta+\partial_{2}) G(z,y,y) &= M_{\alpha, y,2,2}(z) =[m_{-,\alpha, y}(z)-m_{+,\alpha, y}(z)]^{-1}, \\ &\qquad \hphantom{......} \beta=\cot(\alpha), \alpha\in (0,\pi), \endsplit \tag 3.20$$ and especially, $$\split m_{+,\alpha_{2}, y}(z)^{2} &+\{[m_{-,\alpha_{2}, y}(z)-m_{+,\alpha_{2}, y}(z)] + 2\cot(\alpha_{1}-\alpha_{2})\}m_{+,\alpha_{2}, y}(z) \\ &+ \cot^{2}(\alpha_{1}-\alpha_{2}) +[m_{-,\alpha_{2}, y}(z)- m_{+,\alpha_{2}, y}(z)]\cot(\alpha_{1}-\alpha_{2}) \\ &-[\sin(\alpha_{1}-\alpha_{2})]^{-2}[m_{-,\alpha_{2},y}(z)- m_{+,\alpha_{2},y}(z)][m_{-,\alpha_{1},y}(z)-m_{+,\alpha_{1}, y}(z)]^{-1}=0, \\ & \qquad \hphantom{..........................................} \qquad \alpha_{1}\neq\alpha_{2}, z\in\Bbb C\backslash\Bbb R, \endsplit \tag 3.21$$ following directly from (A.38). As a consequence of Theorem 2.1, the basic uniqueness criterion for Schr\"odinger operators on $\Bbb R$ reads as follows. \proclaim{Theorem 3.1} Suppose $\alpha_{1},\alpha_{2}\in [0,\pi)$, $\alpha_{1}\neq\alpha_{2}$ and assume $V_j$, $j=1,2$ satisfy hypothesis \rom{(3.1)}. Define $H_j$, $m_{\pm,j,\alpha_{j}, y}(z), M_{j, \alpha_{j}, y}(z)$ associated with $\tau_{j}=-\frac{d^2}{dx^2}+V_{j}(x)$, $x\in\Bbb R$, $j=1,2$. Then the following are equivalent: \roster \item"\rom{(i)}" $m_{+,1,\alpha_{1},y}(z)=m_{+,2,\alpha_{2},y}(z)$, $m_{-,1,\alpha_{1},y}(z)=m_{-,2,\alpha_{2},y}(z)$, $z\in\Bbb C_{+}$. \item"\rom{(ii)}" $M_{1,\alpha_{1},y}(z)=M_{2,\alpha_{2},y}(z)$, $z\in\Bbb C_{+}$. \item"\rom{(iii)}" $\alpha_{1}=\alpha_{2}$ and $V_{1}(x)=V_{2}(x)$ for a.e.~$x\in\Bbb R$. \endroster \endproclaim The following is our principal characterization result for Schr\"odinger operators on $\Bbb R$. \proclaim{Theorem 3.2} Let $\beta_{1},\beta_{2}\in\Bbb R\cup \{\infty\}$, $\beta_{1}\neq\beta_{2}$, and $x_{0}\in\Bbb R$. \roster \item"\rom{(i)}" $\xi^{\beta_1}(\lambda, x_{0})$ and $\xi^{\beta_2} (\lambda, x_{0})$ for a.e.~$\lambda\in\Bbb R$ uniquely determine $V(x)$ for a.e.~$x\in\Bbb R$ if the pair $(\beta_{1},\beta_{2})$ differs from $(0,\infty)$, $(\infty,0)$. \item"\rom{(ii)}" If $(\beta_{1},\beta_{2})=(0,\infty)$ or $(\infty, 0)$, assume in addition that $\tau$ is in the limit point case at $+\infty$ and $-\infty$. Then $\xi^{\infty}(\lambda, x_{0})$ and $\xi^{0}(\lambda, x_{0})$ for a.e.~$\lambda\in\Bbb R$ uniquely determine $V$ a.e.~up to reflection symmetry with respect to $x_0$; that is, both $V(x)$, $V(2x_{0}-x)$ for a.e.~$x\in\Bbb R$ correspond to $\xi^{\infty}(\lambda, x_{0})$ and $\xi^{0}(\lambda, x_{0})$ for a.e.~$\lambda\in\Bbb R$. \endroster \endproclaim \demo{Proof} (i) Identifying $x_0$ and $y$ in (3.21), one can solve for $m_{+,\alpha_{2},y}(z)$ to obtain $$\split m&_{+,\alpha_{2}, x_{0}} (z) =-\frac{1}{2} [m_{-,\alpha_{2},x_{0}}(z) -m_{+,\alpha_{2},x_{0}}(z)]-\cot(\alpha_{1}-\alpha_{2}) \\ &\pm\biggl\{\frac{1}{4}[m_{-,\alpha_{2},x_{0}}(z)- m_{+,\alpha_{2},x_{0}}(z)]^{2}+\frac{1}{\sin^{2}(\alpha_{1}- \alpha_{2})}\,\frac{[m_{-,\alpha_{2},x_{0}}(z)- m_{+,\alpha_{2},x_{0}}(z)]}{[m_{-,\alpha_{1},x_{0}}(z)- m_{+,\alpha_{1},x_{0}}(z)]}\biggr\}^{1/2}, \\ & \qquad \hphantom{................................................} \qquad \hphantom{...................................} z\in\Bbb C\backslash\Bbb R. \endsplit \tag 3.22$$ By (3.12), (3.15), (3.19), and (3.20), $[m_{-,\alpha_{j},x_{0}}(z)- m_{+,\alpha_{j},x_{0}}(z)]$ are both determined by $\xi^{\beta_j} (\lambda, x_{0})$, $\beta_{j}=\cot(\alpha_{j})$, $j=1,2$, respectively and hence the right-hand side of (3.22) is determined up to the $+/-$ ambiguity. In order to resolve that ambiguity, we now consider the following case distinction: a) $\alpha_{j}\in (0,\pi)$ (i.e., $\beta_{j}\in\Bbb R$), $j=1,2$. \flushpar Then by (A.39), $$m_{\pm,\alpha_{2},x_{0}}(z)\operatornamewithlimits{=}\limits_{z\to i\infty}\cot(\alpha_{2})+o(z^{-1/2}), \tag 3.23$$ which inserted into (3.22) results in $$m_{+,\alpha_{2},x_{0}}(z)\operatornamewithlimits{=}\limits_{z\to i\infty} \cot(\alpha_{2}-\alpha_{1})+o(z^{-1/2}) \pm \biggl\{ \frac{\sin^{2}(\alpha_{1})}{\sin^{2}(\alpha_{1}-\alpha_{2})\sin^{2} (\alpha_{2})} +O(z^{-1})\biggr\}^{1/2}. \tag 3.24$$ A comparison of (3.23) and (3.24) reveals that only one choice of the sign (the $+$ sign, choosing the branch of $\sqrt{\,\cdot\,}$, such that $\sqrt{x}>0$ for $x>0$) in (3.24) can be compatible with the leading behavior $\cot(\alpha_{2})$ in (3.23). This resolves the sign ambiguity in (3.24) and hence in (3.22) and thus determines $m_{+,\alpha_{2},x_{0}}(z)$. Since $\xi^{\beta_2}(\lambda, x_{0})$ determines $[m_{-,\alpha_{2}, x_{0}}(z)-m_{+,\alpha_{2},x_{0}}(z)]$, $m_{-,\alpha_{2},x_{0}}(z)$ is also determined. Thus, both Weyl $m$-functions $m_{\pm,\alpha_{2},x_{0}}(z)$ are known which in turn determines $V$ a.e.~by Theorem 3.1. b) $\alpha_{2}=0$ (i.e., $\beta_{2}=\infty$), $\alpha_{1}\neq\pi/2$ (i.e., $\beta_{1}\neq 0$). \flushpar Then by (A.40), $$m_{\pm,0,x_{0}}(z)\operatornamewithlimits{=}\limits_{z\to i\infty} \pm iz^{1/2}+o(1), \tag 3.25$$ which inserted into (3.22) yields $$m_{+,0,x_{0}}(z)\operatornamewithlimits{=}\limits_{z\to i\infty} iz^{1/2}-\cot(\alpha_{1})+o(1)\pm\{O(1)\}^{1/2}. \tag 3.26$$ Since by (3.25) the $\{O(1)\}^{1/2}$-term must cancel $-\cot(\alpha_{1})$, this again resolves the sign ambiguity in (3.26) (once more the $+$ sign turns out to be the right one) and hence in (3.22). Thus, $m_{+,0,x_{0}}(z)$ is determined. Since $\xi^{\infty}(\lambda, x_{0})$ determines $[m_{-,0,x_{0}}(z)-m_{+,0,x_{0}}(z)]$ also $m_{-,0,x_{0}}(z)$ and hence $V$ is determined a.e.~as in part a). (ii) In the exceptional case where $(\beta_{1},\beta_{2})=(0,\infty)$, $(\infty, 0)$, the exchange $$V(x)\to V(2x_{0}-x) \text{ implies } m_{\pm,0,x_{0}}(z)\to -m_{\mp,0,x_{0}}(z) \tag 3.27$$ since we assumed the l.p.~case at $\pm\infty$. This substitution leaves $$[m_{-,0,x_{0}}(z)-m_{+,0,x_{0}}(z)]^{-1}=G(z,x_{0},x_{0}) \tag 3.28$$ and $$\split m_{-,0,x_{0}}(z) &m_{+,0,x_{0}}(z) [m_{-,0,x_{0}}(z)-m_{+,0,x_{0}}(z)] ^{-1} \\ & =[m_{-,\pi/2, x_{0}}(z)-m_{+,\pi/2,x_{0}}(z)]^{-1} = \partial_{1}\partial_{2} G(z,x_{0}, x_{0}) \endsplit \tag 3.29$$ and hence $\xi^{\infty}(\lambda,x_{0})$ and $\xi^{0}(\lambda, x_{0})$ invariant (cf.~(3.19) and 3.20)). (Here we used that $m_{\pm,\pi/2, x_{0}}(z)=-[m_{\pm,0,x_{0}}(z)]^{-1}$, see (A.38).) \qed \enddemo \proclaim{Corollary 3.3} Suppose $\tau$ is in the limit point case at $+\infty$ and $-\infty$ and let $\beta\in\Bbb R\cup\{\infty\}$ and $x_{0}\in\Bbb R$. Then $\xi^{\beta}(\lambda, x_{0})$ for a.e.~$\lambda\in\Bbb R$ uniquely determines $V(x)$ for a.e.~$x\in\Bbb R$ if and only if $V$ is reflection symmetric with respect to $x_0$, that is, $V(2x_{0}-x)=V(x)$ a.e. \endproclaim \demo{Proof} First suppose that $V(2x_{0}-x)=V(x)$ a.e. Then (A.38) yields $$m_{-,\alpha,x_{0}}(z)=-m_{+,\pi-\alpha, x_{0}}(z), \quad \alpha\in [0,\pi). \tag 3.30$$ If $\beta\in\Bbb R\backslash\{0\}$ (i.e., $\alpha\in (0,\pi)\backslash \{\pi/2\}$, $\beta=\cot(\alpha)$), then (3.30) implies $$[m_{-,\alpha,x_{0}}(z)-m_{+,\alpha,x_{0}}(z)]^{-1} = [m_{-,\pi-\alpha, x_{0}}(z)-m_{+,\pi-\alpha,x_{0}}(z)]^{-1}. \tag 3.31$$ By (3.15), this yields $\xi^{\beta}(\lambda, x_{0})=\xi^{-\beta} (\lambda, x_{0})$ a.e.~ and hence $V$ is uniquely determined a.e.~by Theorem 3.2. On the other hand, if $\beta=\infty$ or $0$ (i.e., $\alpha=0$ or $\pi/2$) then (3.30) yields $$m_{-,0,x_{0}}(z)=-m_{+,0,x_{0}}(z) \text { or } m_{-,\pi/2,x_{0}}(z)=-m_{+,\pi/2,x_{0}}(z). \tag 3.32$$ This determines $m_{\pm,0,x_{0}}(z)$ or $m_{\pm,\pi/2,x_{0}}(z)$ and hence $V$ a.e.~by Theorem 3.1. Conversely, suppose $V$ is not reflection symmetric with respect to $x_0$. Define $\widehat{V}(x)=V(2x_{0}-x)$ a.e.~and denote by $\widehat{m}_{\pm,\alpha,x_{0}}(z_{0})$, $\widehat{M}_{\alpha,x_{0}} (z)$, and $\widehat{\xi}^{\beta}(\lambda, x_{0})$ the corresponding quantities associated with $\widehat V$. Then $$\widehat{m}_{\pm,\pi-\alpha,x_{0}}(z)=-m_{\mp,\alpha, x_{0}}(z), \quad \alpha\in [0,\pi) \tag 3.33$$ (identifying $\alpha=0$ and $\pi$) and hence $$\widehat{M}_{\pi-\alpha, x_{0}}(z) =\pmatrix M_{\alpha, x_{0}, 1,1}(z) & -M_{\alpha, x_{0}, 1,2}(z) \\ -M_{\alpha, x_{0}, 2,1}(z) & M_{\alpha, x_{0}, 2,2}(z) \endpmatrix \neq M_{\alpha, x_{0}}(z) \tag 3.34$$ since $m_{-,\alpha, x_{0}}(z)\neq -m_{+,\alpha, x_{0}}(z)$ for all $\alpha\in [0,\pi)$. (The latter fact is obvious from the asymptotic behavior (A.39) for $\alpha\in (0,\pi)\backslash\{\pi/2\}$ and also follows from our hypothesis that $V$ is not reflection symmetric w.r.t.~$x_0$ for $\alpha=0, \pi/2$. Alternatively, it also follows from our hypothesis and Theorem 3.1.) (3.34) however, shows that $\xi^{\beta}(\lambda, x_{0})=\widehat{\xi}^{-\beta}(\lambda, x_{0})$ is common to $V$ and $\widehat{V}\neq V$. \qed \enddemo In view of Corollary 2.5, it seems appropriate to formulate Theorem 3.2 in the special case of purely discrete spectra. \proclaim{Corollary 3.4} Suppose $H$ \rom(and hence $H^{\beta}_{y}$ for all $y\in\Bbb R$, $\beta\in\Bbb R\cup\{\infty\}$\rom) has purely discrete spectrum, that is, $\sigma_{\text{\rm{ess}}}(H)=\emptyset$ and let $\beta_{1},\beta_{2}\in\Bbb R\cup\{\infty\}$, $\beta_{1}\neq \beta_{2}$, and $x_{0}\in\Bbb R$. \roster \item"\rom{(i)}" $\sigma(H)$, $\sigma(H^{\beta_j}_{x_0})$, $j=1,2$ uniquely determine $V$ a.e.~if the pair $(\beta_{1},\beta_{2})$ differs from $(0,\infty)$ and $(\infty, 0)$. \item"\rom{(ii)}" If $(\beta_{1},\beta_{2})=(0,\infty)$ or $(\infty, 0)$, assume in addition that $\tau$ is in the limit point case at $+\infty$ and $-\infty$. Then $\sigma(H)$, $\sigma(H^{\infty}_{x_0})$, and $\sigma(H^{0}_{x_0})$ uniquely determine $V$ a.e.~up to reflection symmetry with respect to $x_0$, that is, both $V(x)$ and $\widehat{V} (x)=V(2x_{0}-x)$ for a.e.~$x\in\Bbb R$ correspond to $\sigma(H)=\sigma (\widehat{H})$, $\sigma(H^{\infty}_{x_0})=\sigma(\widehat{H}^{\infty} _{x_0})$, and $\sigma(H^{0}_{x_0})=\sigma(\widehat{H}^{0}_{x_0})$. Here, in obvious notation, $\widehat H$, $\widehat{H}^{\infty}_{x_0}$, $\widehat{H}^{0}_{x_0}$ correspond to $\widehat{\tau}=-\frac{d^2} {dx^2}+\widehat{V}(x)$, $x\in\Bbb R$. \item"\rom{(iii)}" Suppose $\tau$ is in the limit point case at $+\infty$ and $-\infty$ and let $\beta\in\Bbb R\cup\{\infty\}$. Then $\sigma(H)$ and $\sigma(H^{\beta}_{x_0})$ uniquely determine $V$ a.e.~if and only if $V$ is reflection symmetric with respect to $x_0$. \item"\rom{(iv)}" Suppose that $V$ is reflection symmetric with respect to $x_0$ and $\tau$ is non-oscillatory at $+\infty$ and $- \infty$. Then $V$ is uniquely determined a.e.~by $\sigma(H)$ in the sense that $V$ is the only potential symmetric with respect to $x_0$ with spectrum $\sigma(H)$. \endroster \endproclaim \demo{Proof} (i) We denote $\sigma(H)=\{e_{n}\}_{n\in J_{0}}$, $\sigma(H^{\beta}_{x_0})=\{\lambda^{\beta}_{n}(x_{0})\}_{n\in I^{\beta}}$, where $I^{\beta}=J_{0}$, $\beta\in\Bbb R$, and $I^{\infty} =J$, with $J_{0}=\Bbb N_{0}$ or $\Bbb Z$ and $J=\Bbb N$ or $\Bbb Z$ depending on whether or not $H$ is bounded from below. Moreover, we use the ordering $e_{n} 0$. Then Floquet theory yields that the spectra of $H$ and $H^{\infty}_{x_0}$ are of the type \alignat2 \sigma(H)&=\bigcup\limits_{n\in\Bbb N} [E_{2(n-1)}, E_{2n-1}], &&\quad E_{0}0. Then \xi^{\infty}(\lambda, x_{0}) for a.e.~\lambda\in\Bbb R and \sigma_{j}(x_{0}), j\in I(x_{0}) uniquely determine V for a.e.~x\in\Bbb R. \endproclaim For the proof, it suffices to note that (cf., e.g., [15],[20],[26]) \xi^{\infty}(\lambda, x_{0})=\cases \tfrac{1}{2}, &\lambda\in (E_{2(n-1)}, E_{2n-1}), n\in\Bbb N \\ 1, &\lambda\in (E_{2n-1},\mu_{n}(x_{0}), n\in\Bbb N \\ 0, &\lambda\in (-\infty, E_{0}), (\mu_{n}(x_{0}), E_{2n}), n\in\Bbb N \endcases \tag 3.44 $$in connection with the periodic case (3.41), (3.42). This result extends to algebro-geometric quasi-periodic finite-gap potentials and certain classes of almost-periodic potentials; we omit further details at this point. After this warm-up we turn to a new uniqueness result for operators with purely discrete spectra. Assume$$ \sigma_{\text{\rm{ess}}}(H)=\emptyset \quad \text{and denote $\sigma(H)=\{e_{n}\}_{n\in J_{0}}$} \tag 3.45 $$such that$$ \sigma(H^{\infty}_{x_0})=\{\mu_{n}(x_{0})\}_{n\in J}, \quad e_{n-1}\leq \mu_{n}(x_{0})\leq e_{n}, n\in J, \tag 3.46 $$where J_{0}=\Bbb N_0 or \Bbb Z and J=\Bbb N or \Bbb Z are appropriate index sets depending on whether or not H is bounded from below. Next we divide the spectrum of H^{\infty}_{x_{0}} into simple and (twice) degenerate Dirichlet eigenvalues, that is, those which are disjoint from \sigma(H) and those which coincide with an element of \sigma(H),$$\gathered J= I(x_{0})\cup I'(x_{0}), \quad I(x_{0})\cap I'(x_{0})=\emptyset, \\ \{\mu_{j}(x_{0})\}_{j\in I(x_{0})} \cap \sigma (H)=\emptyset, \quad \{\mu_{j'}(x_{0})\}_{j'\in I'(x_{0})}\subset \sigma(H) \endgathered \tag 3.47 $$(i.e., \mu_{j'}(x_{0})\in\{e_{j'-1}, e_{j'}\} for j'\in I'(x_{0})). As a last ingredient we need the norming constants associated with the (twice) degenerate Dirichlet eigenvalues \{\mu_{j'}(x_{0})\}_{j'\in I'(x_{0})} denoted by$$ c_{\pm, j'}(x_{0})>0, \quad j'\in I'(x_{0}). \tag 3.48 $$Quite generally, the norming constant c_{+,n}(x_{0})>0 (respectively c_{-,n}(x_{0})>0) associated with \mu_{n}(x_{0})\in\sigma(H^{\infty} _{+,x_{0}}) (respectively \mu_{n}(x_{0})\in\sigma(H^{\infty}_{- ,x_{0}})) is given by minus (respectively plus) the residue of the corresponding Weyl m-function m_{+,0,x_{0}}(z) (respectively m_{- ,0,x_{0}}(z)) at z=\mu_{n}(x_{0}). Equivalently, one has$$ c_{\pm, n}(x_{0})=\|\phi_{0,x_{0}}(\mu_{n}(x_{0}), \,\cdot\,)\|^{-2} _{L^{2}(\Bbb R_{\pm})} \tag 3.49 $$(cf.~(A.37)). Given these preparations we can state the following result. \proclaim{Theorem 3.6} Let x_{0}\in\Bbb R and suppose H has purely discrete spectrum, that is, \sigma_{\text{\rm{ess}}}(H)=\emptyset, \sigma(H)=\{e_{n}\}_{n\in J_{0}}. Then \xi^{\infty}(\lambda, x_{0}) for a.e.~\lambda\in\Bbb R, \sigma_{j}(x_{0}), j\in I(x_{0}), and c_{+,j'}(x_{0}), c_{-,j'}(x_{0}), j'\in I'(x_{0}) uniquely determine V for a.e.~x\in\Bbb R. \endproclaim \demo{Proof} The step function \xi^{\infty}(\lambda, x_{0}) determines the Green's function G(z,x_{0}, x_{0}) of H by (3.12) and hence$$ [m_{-,0,x_{0}}(z)-m_{+,0,x_{0}}(z)]=G(z,x_{0}, x_{0})^{-1} \tag 3.50 $$is determined. Since \sigma_{\text{\rm{ess}}}(H)=\emptyset, both m_{\pm, 0, x_{0}}(z) are meromorphic (on \Bbb C) with first-order poles (and zeros) on \Bbb R. Since by hypothesis we know the left/right distribution of all simple Dirichlet eigenvalues \{\mu_{j}(x_{0})\}_{j\in I(x_{0})}, we can infer the corresponding residue of m_{-,0,x_{0}}(z) (respectively m_{+,0,x_{0}}(z)) from the knowledge of G(z,x_{0}, x_{0})^{-1}=[m_{-,0,x_{0}}(z)- m_{+,0,x_{0}}(z)]. But for the remaining (twice) degenerate Dirichlet eigenvalues \{\mu_{j'}(x_{0})\}_{j'\in I'(x_{0})} of H^{\infty}_{x_0}, the residues of m_{\pm, -,x_{0}}(z) at z=\mu_{j'}(x_{0}), j'\in I'(x_{0}) equals \mp c_{\pm, j'}(x_{0}) and hence is known as well. Thus, the principal parts of m_{\pm, 0, x_{0}}(z) are determined. Since the corresponding half-line spectral measures d\rho_{\pm, 0, x_{0}}(\lambda) associated with H^{\infty}_{\pm, x_{0}}=H_{\pm, 0, x_{0}} are pure point measures supported on \sigma(H_{\pm,0,x_{0}}) of corresponding mass c_{\pm, n}(x_{0}), they are completely determined under our hypothesis. But d\rho_{\pm, 0,x_{0}}(\lambda) uniquely determines V a.e.~on [x_{0}, \pm\infty) by Theorem 2.1. \qed \enddemo If in addition V is symmetric with respect to x_0 and \tau is in the limit point case at +\infty and -\infty, then I(x_{0})=\emptyset, I'(x_{0})=J, m_{+,0,x_{0}}(z)=-m_{- ,0,x_{0}}(z) and hence \xi^{\infty}(\lambda, x_{0}) alone uniquely determines V a.e., recovering again the result of Borg [5] and Marchenko [32] recorded in Corollary 3.4(iii). The reader might want to compare our method of proof of Theorem 3.6 with the inverse spectral approach to confining potentials on the half-line \Bbb R_+ as presented in [21]. \vskip 0.2in \example{Acknowledgments} F.G.~is indebted to the Department of Mathematics at Caltech for its hospitality and support during the summers of 1993 and 1994 where some of this work was done. \endexample \vskip 0.3in \flushpar {\bf Appendix A: Herglotz Functions and Weyl-Titchmarsh Theory} We briefly summarize a few basic facts on Herglotz functions and then recall some of the essential elements of the Weyl-Titchmarsh theory for Schr\"odinger operators on the half-line [0,\infty) as well as on \Bbb R relevant in Sections 2 and 3. We start with Herglotz functions (also called Pick or Nevanlinna-Pick functions). Denoting \Bbb C_{\pm} :=\{z\in\Bbb C\mid \pm\text{Im}(z)>0\}, any analytic map m:\Bbb C_{+}\to\Bbb C_{+} is called Herglotz. One conveniently defines m on \Bbb C_- by m(\bar z)=\overline{m(z)} for z\in\Bbb C_+. Herglotz functions admit particular representations (Borel transforms) in terms of certain measures on \Bbb R. Since this aspect is of fundamental importance in the context of inverse spectral theory of Schr\"odinger operators, we recall the following classical results of Aronszajn and Donoghue [2]. \proclaim{Theorem A.1 [2]} Let m be a Herglotz function. Then, \roster \item"\rom{(i)}" There exists a measure d\rho on \Bbb R and a \xi\in L^{1}_{\text{\rm{loc}}}(\Bbb R) real-valued such that$$\align m(z) &= a+bz+\int\limits_{\Bbb R} \biggl[\frac{1}{\lambda-z} - \frac{\lambda}{1+\lambda^2}\biggr]\,d\rho(\lambda) \tag A.1 \\ &=\exp\biggl\{c+\int\limits_{R}\biggl[\frac{1}{\lambda-z}-\frac {\lambda}{1+\lambda^2}\biggr]\biggr\} \xi(\lambda)\,d\lambda, \tag A.2 \endalign $$where$$ \int\limits_{\Bbb R}\frac{d\rho(\lambda)}{1+\lambda^2}<\infty, \quad a=\text{\rm{Re}}[m(i)], b\geq 0 \tag A.3 $$and$$ 0\leq\xi\leq 1 \text{\rm{ a.e.}}, \quad c=\text{\rm{Re}}\{\ln[m(i)]\}. \tag A.4 $$\item"\rom{(ii)}" \rom(Fatou's lemma\rom)$$\gather \rho((\lambda, \mu])=\lim\limits_{\delta\downarrow 0}\, \lim\limits _{\epsilon\downarrow 0}\,\pi^{-1}\int\limits^{\mu+\delta}_{\lambda+\delta} d\nu \,\text{\rm{Im}}[m(\nu+i\epsilon)], \tag A.5 \\ \xi(\lambda)=\lim\limits_{\epsilon\downarrow 0}\,\pi^{-1}\text{\rm{Im}} \{\ln[m(\lambda+i\epsilon)]\}\text{\rm{ a.e.}} \tag A.6 \endgather $$\item"\rom{(iii)}" Let m,n\in\Bbb N and b=0. Then$$ \int\limits^{0}_{-\infty} (1+\lambda^{2})^{-1}|\lambda|^{m} |\xi(\lambda)|\,d\lambda +\int\limits^{\infty}_{0} (1+\lambda^{2})^{-1} |\lambda|^{n} |\xi(\lambda)|\,d\lambda <\infty \tag A.7 $$if and only if$$\gathered \int\limits^{0}_{-\infty} (1+\lambda^{2})^{-1} |\lambda|^{m}\, d\rho(\lambda) +\int\limits^{\infty}_{0} (1+\lambda^{2})^{-1} |\lambda|^{n}\,d\rho(\lambda)<\infty \\ \text{\rm{and}} \quad \lim\limits_{z\to i\infty} \, m(z) = a- \int\limits_{\Bbb R} (1+\lambda^{2})^{-1}\lambda \,d\rho(\lambda) >0. \endgathered \tag A.8 $$\item"\rom{(iv)}"$$ m(z)=1+\int\limits_{\Bbb R}(\lambda-z)^{-1}\,d\rho(\lambda) \quad \text{\rm{with }} \int\limits_{\Bbb R} d\rho(\lambda) <\infty \tag A.9 $$if and only if$$ m(z)=\exp\biggl[\int\limits_{\Bbb R}(\lambda-z)^{-1}\xi(\lambda)\, d\lambda\biggr] \quad\text{\rm{with }} 0\leq\xi\leq 1 \text{\rm { a.e. and }} \xi\in L^{1}(\Bbb R). \tag A.10 $$In this case$$ \int\limits_{\Bbb R} d\rho(\lambda) =\int\limits_{\Bbb R} \xi(\lambda) \, d\lambda. \tag A.11 $$\item"\rom{(v)}" Any poles and zeros of m are simple and located on the real axis, the residues at poles being negative. \endroster \endproclaim The link between Herglotz functions and rank-one perturbations of self-adjoint operators is developed in detail in [38]. In particular, its universal applicability and unifying aspects in connection with the spectral theory of ordinary differential operators and finite-difference operators are amply illustrated in [16],[25],[38]. Next we turn to Schr\"odinger operators on the half-line \Bbb R_{+} :=[0,\infty). The following material can be found, for example, in [6],[31], and [36]. Suppose$$ V\in L^{1}([0, R]) \text{ for all } R>0, \quad V \text{ real-valued} \tag A.12 $$and introduce the differential expression$$ \tau_{+}=-\frac{d^2}{dx^2}+V(x), \quad x\geq 0. \tag A.13 $$Associated with \tau_+ we introduce the following self-adjoint operator H_{+,\alpha} in L^{2}(\Bbb R_{+}). Pick a z_{+}\in \Bbb C\backslash\Bbb R and a solution f_{+}(z_{+},\,\cdot\,)\in L^{2}(\Bbb R_{+}) of \tau_{+}\psi=z_{+}\psi (the existence of such an f_{+}(z_{+}, x) is a fundamental result of Weyl's theory) and define$$\gathered H_{+,\alpha}f=\tau_{+}f, \quad \alpha\in [0,\pi), \\ f\in\Cal D(H_{+,\alpha})=\{g\in L^{2}(\Bbb R_{+}) \mid g,g'\in AC([0,R])\text{ for all } R>0; \\ \sin(\alpha)g'(0_{+})+\cos(\alpha)g(0_{+})=0; \, \lim\limits _{R\to\infty}\, W(f_{+}(z_{+}), g)(R)=0; \, \tau_{+}g\in L^{2}(\Bbb R_{+})\}. \endgathered \tag A.14 $$Here W(f,g)(x)=f(x)g'(x)-f'(x)g(x) denotes the Wronskian of f and g and the boundary condition \lim\limits_{R\to\infty} W(f_{+}(z_{+}), g)=0 at x=+\infty can be omitted if and only if \tau_+ is in the limit point (l.p.) case at +\infty, that is, if and only if f_{+}(z_{+},x) is unique (up to constant multiples). If \tau_+ is in the limit circle (l.c.) case at +\infty, H_{+,\alpha} depends on the choice of f_{+}(z_{+},x) and for definiteness we shall fix the boundary condition at +\infty,'' that is, always employ the same f_{+}(z_{+}, \,\cdot\,) in the definition (A.14) of H_{+,\alpha} for all values of \alpha\in [0,\pi). Due to our choice of (symmetric) separated boundary conditions in (A.14), H_{+,\alpha} is a real operator (i.e., g\in\Cal D(H_{+,\alpha}) implies \bar{g}\in\Cal D(H_{+,\alpha}) and H_{+,\alpha}\bar{g}=\overline{(H_{+,\alpha}g)}), see, for example, [36], Section 6.4, with uniform spectral multiplicity one, cf.~[10], Corollary XIII.5.5. Next we introduce the fundamental system \phi_{\alpha}(z,x), \theta_{\alpha}(z,x), z\in\Bbb C of solutions of$$ \tau_{+}\psi(z,x)=z\psi(z,x), \quad x\geq 0 \tag A.15 $$satisfying$$ \phi_{\alpha}(z,0)=-\theta'_{\alpha}(z,0)=-\sin(\alpha), \quad \phi'_{\alpha}(x,0)=\theta_{\alpha}(z,0)=\cos(\alpha) \tag A.16 $$such that W(\theta_{\alpha}(z), \phi_{\alpha}(z))=1. Furthermore, let \psi_{+,\alpha}(z,x), z\in\Bbb C\backslash\Bbb R be the unique solution of (A.15) which satisfies$$\gathered \psi_{+,\alpha}(z,\,\cdot\,)\in L^{2}(\Bbb R_{+}), \quad \sin(\alpha)\psi'_{+,\alpha}(z,0_{+}) + \cos(\alpha) \psi_{+,\alpha} (z,0_{+})=1, \\ \lim\limits_{R\to\infty}\, W(f_{+}(z_{+}), \psi_{+,\alpha}(z)) (R)=0, \quad z\in\Bbb C\backslash\Bbb R \endgathered \tag A.17 $$(the latter condition being superfluous, i.e., automatically fulfilled, if \tau_+ is l.p.~at +\infty). Uniqueness of \psi_{+,\alpha}(z,x) is a consequence of Weyl's theory and the fact that we are imposing conditions separately at 0 and \infty in (A.17); see, for example, [10], Theorem XIII.2.32. \psi_{+,\alpha}(z,x) is of the form$$ \psi_{+,\alpha}(z,x)=\theta_{\alpha}(z,x)+m_{+,\alpha}(z) \phi_{\alpha}(z,x) \tag A.18 $$with m_{+,\alpha}(z) being Weyl's m-function. m_{+,\alpha}(z) is well known to be a Herglotz function (cf.~also the comment following (A.27)). To avoid repetitions, we list properties of m_{+,\alpha}(z) a bit later (together with those of m_{-,\alpha}(z)). Here we just note that the Herglotz property of m_{+,\alpha}(z) together with the asymptotic behavior (A.39), (A.40) yields the existence of a measure d\rho_{+,\alpha} such that$$\alignat2 m_{+,\alpha} &= a_{+,\alpha} +\int\limits_{\Bbb R} \biggl[\frac{1} {\lambda-z}-\frac{\lambda}{1+\lambda^2}\biggr]\,d\rho_{+,\alpha} (\lambda), &&\quad \alpha\in [0,\pi) \tag A.19 \\ &=\cot(\alpha)+\int\limits_{\Bbb R}(\lambda-z)^{-1} d\rho_{+,\alpha} (\lambda), &&\quad \alpha\in (0,\pi), \tag A.20 \endalignat $$with$$ \int\limits_{\Bbb R}\frac{d\rho_{+,\alpha}(\lambda)} {1+|\lambda|} \ \cases <\infty, & \alpha\in (0,\pi) \\ =\infty, & \alpha=0. \endcases \tag A.21 $$The Green's function G_{+,\alpha}(z,x,x') of H_{+,\alpha} finally reads$$ ((H_{+,\alpha}-z)^{-1}f)(x) =\int\limits^{\infty}_{0} dx' G_{+,\alpha} (z,x,x')f(x'), \quad z\in\Bbb C\backslash\sigma(H_{+,\alpha}), f\in L^{2}(\Bbb R_{+}), \tag A.22 \align G_{+,\alpha}(z,x,x') &= \cases \phi_{\alpha}(z,x)\psi_{+,\alpha}(z,x'), & 0\leq x\leq x' \\ \phi_{\alpha}(z,x')\psi_{+,\alpha}(z,x), & 0\leq x' \leq x \endcases \tag A.23 \\ &= \int\limits_{\Bbb R} (\lambda-z)^{-1} \phi_{\alpha}(\lambda,x) \phi_{\alpha}(\lambda, x')\,d\rho_{+,\alpha}(\lambda), \tag A.24 \endalign $$where \sigma(\,\cdot\,) denotes the spectrum. In particular, (A.18), (A.23), and (A.24) yield$$\alignat2 G_{+,\alpha}(z,0,0) &= -\sin(\alpha)[\cos(\alpha)-m_{+,\alpha}(z) \sin(\alpha)], &&\quad \alpha\in [0,\pi) \tag A.25 \\ &= \sin^{2}(\alpha)\int\limits_{\Bbb R}(\lambda-z)^{-1}\, d\rho_{+,\alpha}(\lambda), &&\quad \alpha\in (0,\pi) \tag A.26 \endalignat $$and for each x\geq 0,$$ G_{+,\alpha}(z,x,x) \text{ is Herglotz}. \tag A.27 $$While the latter result is obvious from (A.24) (note we have \phi_{\alpha}(\lambda, x)\operatornamewithlimits{=}\limits_{|\lambda| \to\infty} O(1) for \alpha\in (0,\pi) and \phi_{0}(\lambda, x) \operatornamewithlimits{=}\limits_{|\lambda|\to\infty} O(|\lambda|^{-1/2}) for fixed x\in\Bbb R), the fact (A.27) is easily proved directly using the first resolvent equation and self-adjointness of H_{+,\alpha}. (This statement holds quite generally for the diagonal integral kernel of resolvents of self-adjoint operators in connection with general measure spaces as long as the diagonal kernel is well-defined. In particular, it holds for the diagonal Green's function of finite difference operators.) Together with (A.25) this yields a direct proof that m_{+,\alpha}(z) is Herglotz too. Finally, we recall a few facts in connection with Schr\"odinger operators on \Bbb R. Assuming$$ V\in L^{1}_{\text{\rm{loc}}}(\Bbb R), \quad V \text{ real-valued}, \tag A.28 $$one introduces the differential expression$$ \tau=-\frac{d^2}{dx^2}+V(x), \quad x\in\Bbb R \tag A.29 $$and picks z_{\pm}\in\Bbb C\backslash\Bbb R and solutions f_{\pm} (z_{\pm}, \,\cdot\,)\in L^{2}(\Bbb R_{\pm}) (\Bbb R_{-}:= (-\infty, 0]) of \tau\psi(z)=z\psi(z) for z=z_{+}, respectively z_-. One then defines a self-adjoint operator H in L^{2}(\Bbb R) by$$\align & Hf =\tau f, \\ & f\in\Cal D(H) =\{g\in L^{2}(\Bbb R)\mid g,g'\in AC_{\text{\rm{loc}}} (\Bbb R);\,\lim\limits_{R\to\pm\infty}\,W(f_{\pm}(z_{\pm}), g)(R)=0; \, \tau g\in L^{2}(\Bbb R)\}, \tag A.30 \endalign $$where again, the boundary condition at +\infty (or -\infty) can be omitted if and only if \tau is l.p.~at +\infty (or -\infty), that is, if and only if f_{+}(z_{+}, \,\cdot\,) (or f_{-}(z_{-}, \,\cdot\,)) is unique up to constant multiples. Again, when considering restrictions of \tau to \Bbb R_{\pm}, we shall fix the boundary condition at +\infty and/or -\infty if \tau is l.c.~at +\infty and/or -\infty. As in the half-line case (A.14), the separated boundary conditions in (A.30) imply that H is a real operator (see, e.g., [36], Section 6.4). Moreover, the point spectrum \sigma_{p}(H) of H (the set of eigenvalues of H) is simple (this follows, e.g., from [10], Theorem XIII.2.32). Next we define \phi_{\alpha}(z,x), \theta_{\alpha}(z,x) as in (A.15), (A.16) (replacing \tau_+ by \tau) and introduce the uniquely determined solutions \psi_{\pm, \alpha}(z,x) of$$ \tau\psi(z,x) =z\psi(z,x), \quad x\in\Bbb R \tag A.31 $$satisfying$$\gathered \psi_{\pm,\alpha}(z,\,\cdot\,)\in L^{2}(\Bbb R_{\pm}), \quad \sin(\alpha)\psi'_{\pm,\alpha}(z,0)+\cos(\alpha)\psi_{\pm,\alpha} (z,0)=1, \\ \lim\limits_{R\to\pm\infty}\, W(f_{\pm}(z_{\pm}), \psi_{\pm,\alpha}(z)) (R)=0, \quad z\in\Bbb C\backslash\Bbb R \endgathered \tag A.32 $$(the latter condition being superfluous at +\infty and/or -\infty, i.e., automatically fulfilled if \tau is l.p.~at +\infty and/or -\infty). Existence and uniqueness of \psi_{\pm,\alpha}(z,x) follows from Theorem XIII.2.32 in [10]; they admit the following representation$$ \psi_{\pm,\alpha}(z,x)=\theta_{\alpha}(z,x)+m_{\pm,\alpha}(z) \phi_{\alpha}(z,x) \tag A.33 $$in terms of the Weyl m-functions m_{\pm,\alpha}(z). With our conventions$$\gather \pm m_{\pm,\alpha}(z)\text{ is Herglotz}, \quad \pm\text{Im}[m_{\pm,\alpha}(z)]>0, \quad \pm z\in\Bbb C_{+}, \tag A.34 \\ \overline{m_{\pm,\alpha}(z)} =m_{\pm,\alpha}(\bar z), \quad z\in\Bbb C\backslash\Bbb R, \tag A.35 \\ W(\psi_{+,\alpha}(z), \psi_{-,\alpha}(z)) = m_{-,\alpha}(z) - m_{+,\alpha}(z). \tag A.36 \endgather $$Moreover, we recall the following facts$$ \pm\lim\limits_{\epsilon\downarrow 0}\, i\epsilon\,m_{\pm,\alpha} (\lambda+i\epsilon) =\cases 0, &\phi_{\alpha}(\lambda,\,\cdot\,)\notin L^{2}(\Bbb R_{\pm}) \\ -\|\phi_{\alpha}(\lambda, \,\cdot\,)\|^{-2}_{2}, &\phi_{\alpha} (\lambda, \,\cdot\,)\in L^{2}(\Bbb R_{\pm}), \lambda\in\Bbb R, \endcases \tag A.37 \align m_{\pm,\alpha_{1}}(z) &=\frac{-\sin(\alpha_{1}-\alpha_{2}) + \cos(\alpha_{1}-\alpha_{2}) m_{\pm, \alpha_{2}}(z)} {\cos(\alpha_{1}-\alpha_{2}) +\sin(\alpha_{1}-\alpha_{2}) m_{\pm,\alpha_{2}}(z)}, \tag A.38 \\ m_{\pm,\alpha}(z) &\operatornamewithlimits{=}\limits_{z\to i\infty} \cot(\alpha)\pm\frac{i}{\sin^{2}(\alpha)}\, z^{-1/2} - \frac {\cos(\alpha)}{\sin^{3}(\alpha)}\,z^{-1} + o(z^{-1}), \quad \alpha\in (0,\pi), \tag A.39 \\ m_{\pm, 0}(z) &\operatornamewithlimits{=}\limits_{z\to i\infty} \pm iz^{1/2} + o(1), \tag A.40 \\ m_{\pm,\alpha}(z) &= a_{\pm,\alpha} \pm\int\limits_{\Bbb R} \biggl[ \frac{1}{\lambda-z} -\frac{\lambda}{1+\lambda^2}\biggr]\, d\rho_{\pm,\alpha}(\lambda), \quad \alpha\in [0,\pi) \tag A.41 \\ &=\cot(\alpha) \pm\int\limits_{\Bbb R} (\lambda-z)^{-1} \, d\rho_{\pm,\alpha}(\lambda), \quad \alpha\in (0,\pi), \tag A.42 \endalign $$with$$ \int\limits_{\Bbb R} \frac{d\rho_{\pm,\alpha}(\lambda)} {1+|\lambda|} \ \cases < \infty, & \alpha\in (0,\pi) \\ =\infty, & \alpha =0, \endcases \tag A.43 \aligned \pm\int\limits^{\pm\infty}_{0} dx\,\psi_{\pm,\alpha}(z_{1},x) \psi_{\pm,\alpha}(z_{2},x) & = \pm\frac{m_{\pm,\alpha}(z_{1})- m_{\pm, \alpha}(z_{2})}{z_{1}-z_{2}} \\ &= \int\limits_{\Bbb R} (\lambda-z_{1})^{-1}(\lambda-z_{2})^{-1}\, d\rho_{\pm,\alpha}(\lambda). \endaligned \tag A.44 $$While the meaning of (A.38) is clear whenever \tau is l.p.~at \pm\infty, its interpretation in the l.c.~case is as follows: Pick an m_{+,\alpha_{2}}(z) (respectively m_{-,\alpha_{2}}(z)) on the corresponding limit circle of \tau at +\infty (respectively -\infty) for \alpha_2. Then the left-hand-side of (A.38) defines a point m_{+,\alpha_{1}}(z) (respectively m_{-,\alpha_{1}}(z)) on the corresponding limit circle of \tau at +\infty (respectively -\infty) for \alpha_1. As a consequence, a more sophisticated notation for \psi_{\pm,\alpha}(z,x), m_{\pm,\alpha}(z), d\rho_{\pm,\alpha} (\lambda), etc.~would have to include an additional subscript \varphi_{\pm}(\alpha)\in [0,\pi) parametrizing points on the limit circle at \pm\infty for \alpha. For simplicity, we decided to omit this additional subscript in the limit circle case. Perhaps the asymptotic expansions (A.39) and (A.40) also warrant a comment. Under our general hypothesis (A.12), the standard literature usually provides somewhat weaker asymptotic formulas. The actual results (A.39), (A.40) appear to be due to Everitt [11] (see also [3]). The Green's function G(z,x,x') of H is then characterized by$$\align ((H-z)^{-1}f)(x) &= \int\limits_{\Bbb R} dx'\,G(z,x,x')f(x'), \quad z\in\Bbb C\backslash\sigma(H), f\in L^{2}(\Bbb R), \tag A.45 \\ G(z,x,x') &=\frac{1}{m_{-,\alpha}(z)-m_{+,\alpha}(z)} \ \cases \psi_{-,\alpha}(z,x)\psi_{+,\alpha}(z,x'), &x\leq x' \\ \psi_{-,\alpha}(z,x')\psi_{+,\alpha}(z,x), &x'\leq x. \endcases \tag A.46 \endalign $$Again (cf.~the paragraph following (A.27)), for each x\in \Bbb R, the diagonal Green's function$$ G(z,x,x)\text{ is Herglotz}. \tag A.47 $$We emphasize that our choice of reference point x=0 in (A.16) was purely a matter of convenience. In Section 3 it turns out to be advantageous to introduce a (variable) reference point x=y instead. 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