Content-Type: multipart/mixed; boundary="-------------0511301851509" This is a multi-part message in MIME format. ---------------0511301851509 Content-Type: text/plain; name="05-408.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-408.keywords" Inverse spectral theory, Borg-Marchenko theorem ---------------0511301851509 Content-Type: application/x-tex; name="akwe05uabfinal.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="akwe05uabfinal.tex" %October 15, 2005; final version submitted to Ian Knowles %AMS-TeX 2.1 file for UAB 2005 proceedings, Contemporary Mathematics style \input amstex \documentstyle{conm-p} \NoBlackBoxes % The following items provide publication information for the AMS-P logo \issueinfo{00}% volume number {}% issue number {}% month {XXXX}% year \topmatter \title The Borg-Marchenko Theorem with a Continuous Spectrum\endtitle \author Tuncay Aktosun and Ricardo Weder \endauthor \leftheadtext{TUNCAY AKTOSUN AND RICARDO WEDER}% \address Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019, USA \endaddress \email aktosun\@uta.edu\endemail % \thanks will become a 1st page footnote. % Use \endgraf to indicate a new paragraph; a blank line or \par will % be recognized as an error. \thanks The research leading to this article was supported in part by the National Science Foundation under grant DMS-0204437, the Department of Energy under grant DE-FG02-01ER45951, Universidad Nacional Aut\'onoma de M\'exico under Proyecto PAPIIT-DGAPA IN 101902, and by CONACYT under Proyecto P42553-F. \endthanks \address Instituto de Investigaciones en Matem\'aticas Aplicadas y en Sistemas, Universidad Nacional Aut\'onoma de M\'exico, Apartado Postal 20-726, IIMAS-UNAM, M\'exico DF 01000, M\'exico \endaddress \email weder\@servidor.unam.mx\endemail \thanks The second author is Fellow Sistema Nacional de Investigadores. \endthanks % Math Subject Classifications \subjclass Primary 34A55; Secondary 34B24, 34L05, 34L40, 47E05, 81U40 \endsubjclass %\dedicatory This paper is dedicated to our authors.\enddedicatory \keywords Inverse spectral problem, inverse scattering, Borg-Marchenko theorem, Krein's spectral shift function, half-line Schr\"odinger equation\endkeywords \abstract The Schr\"odinger equation is considered on the half line with a selfadjoint boundary condition when the potential is real valued, integrable, and has a finite first moment. It is proved that the potential and the two boundary conditions are uniquely determined by a set of spectral data containing the discrete eigenvalues for a boundary condition at the origin, the continuous part of the spectral measure for that boundary condition, and a subset of the discrete eigenvalues for a different boundary condition. This result provides a generalization of the celebrated uniqueness theorem of Borg and Marchenko using two sets of discrete spectra to the case where there is also a continuous spectrum. The proof employed yields a method to recover the potential and the two boundary conditions, and it also constructs data sets used in various inversion methods. A comparison is made with the uniqueness result of Gesztesy and Simon using Krein's spectral shift function as the inversion data. \endabstract \endtopmatter \document \head 1. Introduction\endhead Consider the Schr\"odinger equation on the half line $$-\psi''+V(x)\,\psi=k^2\psi,\qquad x\in\bold R^+,\tag 1.1$$ where the prime denotes the derivative with respect to $x,$ the potential $V$ is real valued and measurable, and $\int_0^\infty dx\,(1+x)\,|V(x)|$ is finite. Such potentials are said to make up the Faddeev class. Let $H_\alpha$ for a fixed $\alpha\in(0,\pi]$ denote the unique selfadjoint realization [1] of the corresponding Schr\"odinger operator on $L^2(\bold R^+)$ with the boundary condition $$\sin\alpha\cdot \psi'(k,0)+\cos\alpha\cdot\psi(k,0)=0,\tag 1.2$$ which can also be written as $$\cases \psi'(k,0)+\cot\alpha\cdot\psi(k,0)=0,\qquad \alpha\in(0,\pi),\\ %\stretch \psi(k,0)=0,\qquad \alpha=\pi.\endcases$$ Note that $\alpha\mapsto\cot\alpha$ is a monotone decreasing mapping of $(0,\pi)$ onto $\bold R,$ and hence $\alpha$ is uniquely determined by $\cot\alpha.$ It is known [1,2] that $H_\alpha$ has no positive or zero eigenvalues, has no singular-continuous spectrum, has at most a finite number of (simple) negative eigenvalues, and its absolutely-continuous spectrum consists of $k^2\in[0,+\infty).$ Borg [3] and Marchenko [4] independently analyzed (1.1) with the boundary condition (1.2) when there is no continuous spectrum. They showed [3-5] that two sets of discrete spectra associated with two distinct boundary conditions at $x=0$ (with a fixed boundary condition, if any, at $x=+\infty$) uniquely determine the potential and the two boundary conditions. A continuous spectrum often appears in applications, and it usually arises when the potential vanishes at infinity. In our paper we present an extension of the celebrated Borg-Marchenko result in the presence of a continuous spectrum; namely, we show that the potential and the two boundary conditions are uniquely determined by an appropriate data set containing the discrete eigenvalues and the continuous part of the spectral measure corresponding to one boundary condition and a subset of the discrete eigenvalues corresponding to a different boundary condition. Another extension of the Borg-Marchenko result with a continuous spectrum is given by Gesztesy and Simon [6], where a uniqueness result is presented when the corresponding Krein's spectral shift function is used as the data in the class of real-valued potentials that are integrable on $[0,R]$ for all $R>0.$ In our generalization of the Borg-Marchenko theorem, we specify the data in terms of a subset of the spectral measure; namely, the amplitude of the Jost function and the eigenvalues. The connection between the data used in [6] and ours is analyzed in Section~5. The problem under study has applications in the acoustical analysis of the human vocal tract. The related inverse problem can be described [7-9] as determining a scaled curvature of the duct of the vocal tract when a constant-frequency sound is uttered. Such an inverse problem has important applications in speech recognition. Our paper is organized as follows. In Section~2 we introduce the preliminary material related to the Jost function, the phase shift, and the spectral measure. In Section~3 we present our generalized Borg-Marchenko theorem. In Section~4 we briefly outline the Gel'fand-Levitan, Marchenko, and the Faddeev-Marchenko procedures to recover the potential. In Section~5 we show how the data used in our theorem uniquely constructs Krein's spectral shift function and vice versa. Finally, in Section~6 we present two examples to illustrate the theory presented in our paper. \head 2. Preliminaries \endhead Recall that the Jost function associated with $H_\alpha$ is defined as [2,10-12] $$F_\alpha(k):=\cases -i[f'(k,0)+\cot\alpha\cdot f(k,0)],\qquad \alpha\in(0,\pi),\\ %\stretch f(k,0),\qquad \alpha=\pi,\endcases\tag 2.1$$ with $f(k,x)$ denoting the Jost solution to (1.1) satisfying the asymptotics $$f(k,x)=e^{ikx}[1+o(1)],\quad f'(k,x)=ik e^{ikx}[1+o(1)],\qquad x\to+\infty.\tag 2.2$$ From (1.1) and (2.2) it follows that $$f(-k,0)=f(k,0)^\ast,\quad f'(-k,0)=f'(k,0)^\ast,\qquad k\in\bold R,\tag 2.3$$ where the asterisk denotes complex conjugation, and hence for $k\in\bold R$ we get $$F_\alpha(-k)=\cases -F_\alpha(k)^\ast,\qquad \alpha\in(0,\pi),\\ F_\pi(k)^\ast,\qquad \alpha=\pi.\endcases\tag 2.4$$ We use $\bold C^+$ for the upper half complex plane, $\overline{\bold C^+}:=\bold C^+\cup\bold R$ for its closure, and $\bold I^+:=i(0,+\infty)$ for the positive imaginary axis in $\bold C^+.$ It is known [2,10-12] that $F_\alpha(k)$ is analytic in $\bold C^+$ and continuous in $\overline{\bold C^+},$ the zeros in $\bold C^+$ of $F_\alpha,$ if any, can only occur on $\bold I^+$ and such zeros are all simple, $F_\alpha(k)\ne 0$ for $k\in\bold R\setminus\{0\},$ and that either $F_\alpha(k)$ is nonzero at $k=0$ (generic case) or it has a simple zero there (exceptional case). Because of (2.4), knowledge of $F_\alpha(k)$ for $k\in\bold R^+$ is equivalent to that for $k\in\bold R.$ Let $k=i\kappa_{\alpha j}$ for $j=1,\dots,N_\alpha$ represent the zeros of $F_\alpha$ on $\bold I^+.$ Thus, the set $\{-\kappa_{\alpha j}^2\}_{j=1}^{N_\alpha}$ corresponds to the discrete eigenvalues of $H_\alpha.$ The negative of the phase of the Jost function $F_\alpha$ is usually known as the phase shift $\phi_\alpha,$ i.e. $$e^{-i\phi_\alpha(k)}:=\displaystyle\frac{F_\alpha(k)}{|F_\alpha(k)|},\qquad k\in\bold R,\tag 2.5$$ where it is understood that $\phi_\alpha(+\infty)=0.$ For $k\in\bold R,$ the phase shift $\phi_\alpha$ satisfies $$\phi_\alpha(-k)=\cases \pi-\phi_\alpha(k),\qquad \alpha\in(0,\pi),\\ -\phi_\pi(k),\qquad \alpha=\pi.\endcases\tag 2.6$$ The scattering matrix $S_\alpha(k)$ for $k\in\bold R$ associated with $H_\alpha$ is defined as $$S_\alpha(k):=e^{2i\phi_\alpha(k)}= \cases -\displaystyle\frac{F_\alpha(-k)}{F_\alpha(k)}, \qquad \alpha\in(0,\pi),\\ \displaystyle\frac{F_\pi(-k)}{F_\pi(k)},\qquad \alpha=\pi. \endcases\tag 2.7$$ The number of discrete eigenvalues $N_\alpha$ is related to the phase shift $\phi_\alpha$ by Levinson's theorem [2,11], i.e. $$\phi_\alpha(0^+)=\cases \left(N_\alpha+\displaystyle\frac{1+d_\alpha}{2}\right)\pi, \qquad \alpha\in(0,\pi),\\ \left(N_\pi+\displaystyle\frac{d_\pi}{2}\right)\pi, \qquad \alpha=\pi,\endcases \tag 2.8$$ where we have defined $$d_\alpha:=\cases 0,\qquad \text { if } F_\alpha(0)\ne 0,\\ 1,\qquad \text { if } F_\alpha(0)=0.\endcases$$ The spectral measure $\rho_\alpha$ corresponding to $H_\alpha$ can be determined via [2,10-12] $$d\rho_\alpha(\lambda)=\cases \displaystyle\frac{\sqrt{\lambda}}{\pi}\,\displaystyle\frac{1} {|F_\alpha(\sqrt{\lambda})|^2}\,d\lambda,\qquad \lambda >0,\\ %\stretch \displaystyle\sum_{j=1}^{N_\alpha} g_{\alpha j}^2\, \delta(\lambda+\kappa_{\alpha j}^2)\,d\lambda, \qquad \lambda <0, \endcases$$ where $\delta(\cdot)$ is the Dirac delta distribution, $\lambda:=k^2,$ and the norming constants $g_{\alpha j}$ are given by $$g_{\alpha j}:=\cases \displaystyle\frac{|f(i\kappa_{\alpha j},0)|} {||f(i\kappa_{\alpha j},\cdot)||},\qquad \alpha\in(0,\pi),\\ %\stretch \displaystyle\frac{|f'(i\kappa_{\pi j},0)|}{||f(i\kappa_{\pi j},\cdot)||},\qquad \alpha=\pi,\endcases\tag 2.9$$ with $||\cdot||$ denoting the standard norm in $L^2(0,+\infty).$ It is known [2,10-12] that $\{V,\alpha\}$ is uniquely determined by the corresponding spectral measure $\rho_\alpha$ and that the reconstruction can be achieved by solving the Gel'fand-Levitan integral equation. \head 3. The Borg-Marchenko Theorem with a Continuous Spectrum \endhead Our generalized Borg-Marchenko theorem consists of identifying the appropriate data set leading to the unique determination of the potential $V$ in (1.1) and the two distinct boundary parameters $\alpha$ and $\beta$ in (1.2) with $0<\beta<\alpha\le \pi.$ Here, we briefly state our theorem and summarize the steps to construct $\{V,\beta,\alpha\}$ and refer the reader to [12] for the proof and further details. Let the set $\{-\kappa_{\beta j}^2\}_{j=1}^{N_\beta}$ correspond to the discrete eigenvalues of $H_\beta.$ We use $F_\beta$ to denote the Jost function associated with $H_\beta,$ and it is obtained by replacing $\alpha$ with $\beta$ on the right hand side of (2.1). Our motivation is as follows. Assume that we are given some data set $\Cal D,$ which contains $|F_\alpha(k)|$ for $k\in\bold R,$ the whole set $\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha},$ and a subset of $\{\kappa_{\beta j}\}_{j=1}^{N_\beta}$ consisting of $N_\alpha$ elements. Alternatively, our data $\Cal D$ may include $|F_\beta(k)|$ for $k\in\bold R$ and the sets $\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}$ and $\{\kappa_{\beta j}\}_{j=1}^{N_\beta}.$ Does $\Cal D$ uniquely determine $\{V,\alpha,\beta\}$? If not, what additional information do we need to include in $\Cal D$ for the unique determination? Can we also present a constructive method to recover $\{V,\alpha,\beta\}$ from $\Cal D$ or from a data set obtained by some augmentation of $\Cal D$? Since $0<\beta<\alpha\le \pi,$ from the interlacing properties of eigenvalues, it is known [2,10-12] that either $N_\beta=N_\alpha,$ in which case we have $$0<\kappa_{\alpha 1}<\kappa_{\beta 1} <\kappa_{\alpha 2}<\kappa_{\beta 2}<\dots< \kappa_{\alpha N_\alpha}<\kappa_{\beta N_\alpha},\tag 3.1$$ or else we have $N_\beta=N_\alpha+1,$ in which case we get $$0<\kappa_{\beta 1}<\kappa_{\alpha 1}< \kappa_{\beta 2}<\kappa_{\alpha 2}<\dots< \kappa_{\alpha N_\alpha}<\kappa_{\beta N_\beta}.\tag 3.2$$ There are eight distinct cases to consider depending on whether $\alpha\in(0,\pi)$ or $\alpha=\pi,$ whether $N_\beta=N_\alpha$ or $N_\beta=N_\alpha+1,$ and whether the data set used contains $|F_\alpha|$ or $|F_\beta|.$ So, let us define the data sets $\Cal D_1,\dots,\Cal D_8$ as follows [12]: $$\Cal D_1:=\{h_{\beta\alpha},|F_\alpha(k)| \text{ for } k\in\bold R, \{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}, \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.3$$ $$\Cal D_2:=\{\beta,|F_\pi(k)| \text{ for } k\in\bold R, \{\kappa_{\pi j}\}_{j=1}^{N_\pi}, \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.4$$ $$\Cal D_3:=\{h_{\beta\alpha},|F_\alpha(k)| \text{ for } k\in\bold R, \{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}, {\text{$N_\alpha$-element subset of}} \ \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.5$$ $$\Cal D_4:=\{\beta,|F_\pi(k)| \text{ for } k\in\bold R, \{\kappa_{\pi j}\}_{j=1}^{N_\pi}, {\text{$N_\pi$-element subset of}} \ \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.6$$ $$\Cal D_5:=\{h_{\beta\alpha},|F_\beta(k)| \text{ for } k\in\bold R, \{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}, \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.7$$ $$\Cal D_6:=\{|F_\beta(k)| \text{ for } k\in\bold R, \{\kappa_{\pi j}\}_{j=1}^{N_\pi}, \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.8$$ $$\Cal D_7:=\{\beta,h_{\beta\alpha},|F_\beta(k)| \text{ for } k\in\bold R, \{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}, \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.9$$ $$\Cal D_8:=\{\beta,|F_\beta(k)| \text{ for } k\in\bold R, \{\kappa_{\pi j}\}_{j=1}^{N_\pi}, \{\kappa_{\beta j}\}_{j=1}^{N_\beta}\},\tag 3.10$$ where we let $$h_{\beta\alpha}:=\cot\beta-\cot\alpha.\tag 3.11$$ Note that $h_{\beta\alpha}>0$ when $0<\beta<\alpha<\pi.$ The data sets $\Cal D_1,\Cal D_3,\Cal D_5,\Cal D_7$ correspond to $\alpha\in(0,\pi)$ and the sets $\Cal D_2,\Cal D_4,\Cal D_6,\Cal D_8$ to $\alpha= \pi;$ the sets $\Cal D_1,\Cal D_2,\Cal D_3,\Cal D_4$ contain $|F_\alpha|$ whereas the sets $\Cal D_5,\Cal D_6,\Cal D_7,\Cal D_8$ contain $|F_\beta|;$ the sets $\Cal D_1,\Cal D_2,\Cal D_5,\Cal D_6$ correspond to $N_\beta=N_\alpha$ whereas the sets $\Cal D_3,\Cal D_4,\Cal D_7,\Cal D_8$ to $N_\beta=N_\alpha+1.$ Our generalized Borg-Marchenko theorem can be stated as follows. \proclaim {Theorem 3.1} Let the realizations $H_\alpha$ and $H_\beta$ for some $0<\beta<\alpha\le\pi$ correspond to a potential $V$ in the Faddeev class with the boundary conditions identified by $\alpha$ and $\beta,$ respectively. Then, each of the data sets $\Cal D_j$ with $j=1,\dots,8$ uniquely determines the corresponding $\{V,\alpha,\beta\}.$ \endproclaim The proof of the above theorem provides a method to recover $\alpha$ and $\beta$ as well as $F_\alpha(k)$ and $F_\beta(k)$ for $k\in\overline{\bold C^+},$ thus also allowing us to construct the data sets used as input in various inversion methods to determine $V.$ For the proof and details we refer the reader to [12], and here we only briefly outline the steps involved. Using $\Cal D_j$ for each of $j=1,2,5,6,7,8$ we first construct $\text{Re}[\Lambda_j(k)]$ for $\bold R,$ where $$\text{Re}[\Lambda_1(k)]=\displaystyle\frac{k\,h_{\beta\alpha}}{|F_\alpha(k)|^2} \displaystyle\prod_{j=1}^{N_\alpha}\displaystyle\frac{k^2+\kappa_{\alpha j}^2}{k^2+\kappa_{\beta j}^2}, \quad\text{Re}[\Lambda_2(k)]=-1+\displaystyle\frac{1}{|F_\pi(k)|^2} \displaystyle\prod_{j=1}^{N_\pi}\displaystyle\frac{k^2+\kappa_{\pi j}^2} {k^2+\kappa_{\beta j}^2},$$ $$\text{Re}[\Lambda_3(k)]=\displaystyle\frac{h_{\beta\alpha}\,k^2}{|F_\alpha(k)|^2} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\alpha}(k^2+\kappa_{\alpha j}^2)} {\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)}, \quad\text{Re}[\Lambda_4(k)]=-1+\displaystyle\frac{k^2}{|F_\pi(k)|^2} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\pi}(k^2+\kappa_{\pi j}^2)} {\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)}, $$ $$\text{Re}[\Lambda_5(k)]=-h_{\beta\alpha}+\displaystyle\frac{k^2\,h_{\beta\alpha}}{|F_\beta(k)|^2} \displaystyle\prod_{j=1}^{N_\beta}\displaystyle\frac{k^2+\kappa_{\beta j}^2}{k^2+\kappa_{\alpha j}^2}, $$ $$\text{Re}[\Lambda_6(k)]=-1+\displaystyle\frac{k^2}{|F_\beta(k)|^2} \displaystyle\prod_{j=1}^{N_\beta}\displaystyle\frac{k^2+\kappa_{\beta j}^2}{k^2+\kappa_{\pi j}^2},$$ $$\text{Re}[\Lambda_7(k)]= h_{\beta\alpha}-\displaystyle\frac{h_{\beta\alpha}}{|F_\beta(k)|^2} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)} {\displaystyle\prod_{j=1}^{N_\beta-1}(k^2+\kappa_{\alpha j}^2)}, $$ $$\text{Re}[\Lambda_8(k)]=-1+\displaystyle\frac{1}{|F_\beta(k)|^2} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)} {\displaystyle\prod_{j=1}^{N_\beta-1}(k^2+\kappa_{\alpha j}^2)}.$$ Then, using the Schwarz integral formula $$\Lambda_j(k)=\displaystyle\frac{1}{\pi i}\int_{-\infty}^\infty \displaystyle\frac{dt}{t-k-i0^+}\, \text{Re}[\Lambda_j(t)],\qquad k\in\overline{\bold C^+},$$ we uniquely construct $\Lambda_j,$ where $0^+$ in the integrand indicates that the values of $\Lambda_j(k)$ for $k\in\bold R$ must be obtained via a limit from $\bold C^+.$ We get $$\Lambda_1(k)=-i+i\,\displaystyle\frac{F_\beta(k)}{F_\alpha(k)} \displaystyle\prod_{j=1}^{N_\alpha}\displaystyle\frac{k^2+\kappa_{\alpha j}^2}{k^2+\kappa_{\beta j}^2},$$ $$\Lambda_2(k)=-1-\displaystyle\frac{1}{k}\displaystyle\frac{F_\beta(0)}{F_\pi(0)} \displaystyle\prod_{j=1}^{N_\pi}\displaystyle\frac{\kappa_{\pi j}^2}{\kappa_{\beta j}^2} +\displaystyle\frac{1}{k}\displaystyle\frac{F_\beta(k)}{F_\pi(k)} \displaystyle\prod_{j=1}^{N_\pi}\displaystyle\frac{k^2+\kappa_{\pi j}^2} {k^2+\kappa_{\beta j}^2},$$ $$\Lambda_3(k)=ik\,\displaystyle\frac{F_\beta(k)}{F_\alpha(k)} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\alpha}(k^2+\kappa_{\alpha j}^2)} {\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)},\quad \Lambda_4(k)=-1+k\,\displaystyle\frac{F_\beta(k)}{F_\pi(k)} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\pi}(k^2+\kappa_{\pi j}^2)} {\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)},$$ $$\Lambda_5(k)=ik-h_{\beta\alpha}- ik\,\displaystyle\frac{F_\alpha(k)}{F_\beta(k)} \displaystyle\prod_{j=1}^{N_\beta}\displaystyle\frac{k^2+\kappa_{\beta j}^2}{k^2+\kappa_{\alpha j}^2}, \quad\Lambda_6(k)=-1+k\, \displaystyle\frac{F_\pi(k)}{F_\beta(k)} \displaystyle\prod_{j=1}^{N_\beta}\displaystyle\frac{k^2+\kappa_{\beta j}^2}{k^2+\kappa_{\pi j}^2},$$ $$\Lambda_7(k)=-ik+h_{\beta\alpha}- \displaystyle\frac{i}{k}\,\displaystyle\frac{F_\alpha(0)}{F_\beta(0)} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\beta}\kappa_{\beta j}^2} {\displaystyle\prod_{j=1}^{N_\beta-1}\kappa_{\alpha j}^2}+ \displaystyle\frac{i}{k}\,\displaystyle\frac{F_\alpha(k)}{F_\beta(k)} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)} {\displaystyle\prod_{j=1}^{N_\beta-1}(k^2+\kappa_{\alpha j}^2)},$$ $$\Lambda_8(k)=-1- \displaystyle\frac{1}{k}\,\displaystyle\frac{F_\pi(0)}{F_\beta(0)} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\beta}\kappa_{\beta j}^2} {\displaystyle\prod_{j=1}^{N_\beta-1}\kappa_{\alpha j}^2}+ \displaystyle\frac{1}{k}\,\displaystyle\frac{F_\pi(k)}{F_\beta(k)} \displaystyle\frac{\displaystyle\prod_{j=1}^{N_\beta}(k^2+\kappa_{\beta j}^2)} {\displaystyle\prod_{j=1}^{N_\beta-1}(k^2+\kappa_{\alpha j}^2)}.$$ Each $\Lambda_j(k)$ is analytic in $\bold C^+,$ continuous in $\overline{\bold C^+},$ and $O(1/k)$ as $k\to\infty$ in $\overline{\bold C^+}.$ Next, with the help of $\Cal D_j$ and $\Lambda_j,$ we uniquely construct $\alpha,$ $\beta,$ $F_\alpha,$ and $F_\beta$ by using the procedure given in Section~4 of [12]. Having these four quantities, we can construct $V$ by using one of the available inversion methods [2,10-12], three of which are outlined in Section~4. Note that $f(k,0)$ and $f'(k,0)$ can then also be constructed via $$f(k,0)=\cases\displaystyle\frac{i}{h_{\beta\alpha}}\left[F_\beta(k)-F_\alpha(k)\right], \qquad \alpha,\beta\in(0,\pi),\\ %\stretch F_\pi(k),\endcases$$ $$f'(k,0)=\cases \displaystyle\frac{i}{h_{\beta\alpha}}\left[\cot\beta\cdot F_\alpha(k)-\cot\alpha\cdot F_\beta(k)\right],\qquad \alpha,\beta\in(0,\pi),\\ %\stretch i\,F_\beta(k)-\cot\beta\cdot F_\pi(k),\qquad\beta\in(0,\pi).\endcases$$ The constructions from $\Cal D_j$ with $j=3,4$ involve an extra step because in each of those two cases exactly one of the discrete eigenvalues needed to construct $\text{Re}[\Lambda_j(k)]$ for $k\in\bold R$ is not contained in $\Cal D_j.$ As a result, we first construct a one-parameter family of $\Lambda_j(k)$ for $k\in\overline{\bold C^+}$ and then determine the eigenvalue missing in $\Cal D_j$ by taking a nontangential limit as $k\to\infty$ on $\bold C^+.$ Once the missing discrete eigenvalue is at hand, the corresponding $\Lambda_j(k)$ for $k\in\overline{\bold C^+}$ is uniquely determined as well. We can then proceed as in the case with $j=1,2,5,6,7,8$ in order to determine $\alpha$ and $\beta,$ $F_\alpha(k)$ and $F_\beta(k)$ for $k\in\overline{\bold C^+},$ the potential $V,$ and any other relevant quantities. \head 4. Reconstruction of the Potential \endhead In Section~3 we have outlined the construction of the quantities $\alpha,$ $\beta,$ and $F_\alpha(k)$ and $F_\beta(k)$ for $k\in\overline{\bold C^+}$ by using each of the data sets $\Cal D_j$ with $j=1,\dots,8$ given in (3.3)-(3.10), respectively. In this section we outline three methods to briefly illustrate the use of such quantities to recover the potential $V.$ In the Gel'fand-Levitan method [2,10-12], one forms the input data $\Cal G_\alpha$ given by $$\Cal G_\alpha:=\{|F_\alpha(k)| \text{ for } k\in\bold R,\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha}, \{g_{\alpha j}\}_{j=1}^{N_\alpha}\},$$ where the $g_{\alpha j}$ are the norming constants appearing in (2.9) and they can also be obtained from (cf. (3.25) of [12]) $$g_{\alpha j}=\cases \displaystyle\sqrt{\displaystyle\frac{2i\kappa_{\alpha j}\,F_\beta(i\kappa_{\alpha j})} {h_{\beta\alpha}\,\dot F_\alpha(i\kappa_{\alpha j})}},\qquad 0<\beta<\alpha<\pi,\\ %\stretch \displaystyle\sqrt{\displaystyle\frac{2\,\kappa_{\pi j}\,F_\beta(i\kappa_{\pi j})}{\dot F_\pi(i\kappa_{\pi j})}},\qquad 0<\beta<\alpha=\pi,\endcases$$ with an overdot indicating the $k$-derivative. The corresponding potential $V$ is uniquely recovered via $$V(x)=2\,\displaystyle\frac{d}{dx}A_\alpha(x,x^-),$$ where $A_\alpha(x,y)$ is obtained by solving the Gel'fand-Levitan integral equation $$A_\alpha(x,y)+G_\alpha(x,y)+\int_0^x dz\, G_\alpha(y,z)\,A_\alpha(x,z)=0,\qquad 0\le y0,$$ with the input data $$\Omega_{\text{r}}(y):=\displaystyle\frac{1}{2\pi}\int_{-\infty}^\infty dk\,L(k)\,e^{iky}+\displaystyle\sum_{j=1}^N c_{\text{r}j}^2\, e^{-\tau_j y}.$$ \head 5. Krein's Spectral Shift Function \endhead In this section, we indicate the construction of Krein's spectral shift function $\xi_{\beta\alpha}(k)$ for $k\in\bold R^+\cup\bold I^+$ with $0<\beta<\alpha\le \pi,$ and we also show how to extract $\alpha,$ $\beta,$ $F_\alpha,$ $F_\beta,$ and $V$ from $\xi_{\beta\alpha}.$ Krein's spectral shift function $\xi_{\beta\alpha}(k)$ associated with $H_\alpha$ and $H_\beta$ can be defined in various ways [5,13-15]. We find it convenient to introduce $\xi_{\beta\alpha}$ by relating it to the phase of $F_\alpha(k)/F_\beta(k)$ for $k\in\bold R\cup\bold I^+.$ Let us note that $\xi_{\beta\alpha}$ in [6] is considered when $0\le \beta<\alpha<\pi$, but studying it for $0<\beta<\alpha\le \pi$ does not present any inconvenience because we can choose $\xi_{0\theta}(k)=\xi_{\theta\pi}(k)$ for $k\in\bold R^+\cup\bold I^+$ with $\theta\in(0,\pi),$ as done in our paper. Let $$e^{\pi i \xi_{\beta\alpha}(k)}:=\displaystyle\frac{Z_{\beta\alpha}(k)} {|Z_{\beta\alpha}(k)|},\qquad k\in \bold R^+\cup\bold I^+,\tag 5.1$$ with the normalization $$\xi_{\beta\alpha}(+\infty)=\cases 0,\qquad \alpha\in(0,\pi),\\ 1/2,\qquad \alpha=\pi,\endcases\tag 5.2$$ and the range of $\xi_{\beta\alpha}(k)$ on $\bold I^+$ restricted to the interval $[0,1],$ where we have defined $$Z_{\beta\alpha}(k):=\cases \displaystyle\frac{F_\alpha(k)}{F_\beta(k)},\qquad 0<\beta<\alpha<\pi,\\ \displaystyle\frac{i\,F_\beta(k)}{F_\pi(k)} ,\qquad 0<\beta<\alpha=\pi.\endcases\tag 5.3$$ As seen from (2.5) and (5.1)-(5.3), we can express $\xi_{\beta\alpha}(k)$ for $k\in\bold R^+$ in terms of the phase shifts $\phi_\alpha$ and $\phi_\beta$ as $$\xi_{\beta\alpha}(k)=\cases \displaystyle\frac{1}{\pi}\left[ \phi_\beta(k)-\phi_\alpha(k)\right],\qquad 0<\beta<\alpha<\pi,\\ \displaystyle\frac{1}{2}+\displaystyle\frac{1}{\pi}\left[ \phi_\pi(k)-\phi_\beta(k)\right],\qquad 0<\beta<\alpha=\pi.\endcases\tag 5.4$$ With the help of (2.6) and (5.4) we see that $\xi_{\beta\alpha}(k)$ can be extended from $k\in\bold R^+$ to $k\in\bold R$ oddly, i.e. $$\xi_{\beta\alpha}(-k)= -\xi_{\beta\alpha}(k),\qquad 0<\beta<\alpha\le\pi, \quad k\in\bold R.\tag 5.5$$ Using (3.18) of [12], for $k\in\bold R$ we get $$\text{Im}\left[Z_{\beta\alpha}(k)\right]=\cases \displaystyle\frac{k\,h_{\beta\alpha}}{|F_\beta(k)|^2},\qquad 0<\beta<\alpha< \pi,\\ \displaystyle\frac{k}{|F_\pi(k)|^2}, \qquad 0<\beta<\alpha=\pi.\endcases\tag 5.6$$ Since $F_\alpha(k)$ is nonzero for $k\in\bold R^+,$ (5.6) implies that $\text{Im}\left[Z_{\beta\alpha}(k)\right]>0$ for $k\in\bold R^+.$ Thus, from (5.1) we can conclude that $\xi_{\beta\alpha}(k)\in(0,1)$ when $k\in\bold R^+.$ Furthermore, using (2.8) and (5.4) we get $$\xi_{\beta\alpha}(0^+)=\cases N_\beta-N_\alpha+\displaystyle\frac{d_\beta-d_\alpha}{2},\qquad 0<\beta<\alpha<\pi,\\ N_\pi-N_\beta+\displaystyle\frac{d_\pi-d_\beta}{2} ,\qquad 0<\beta<\alpha=\pi.\endcases$$ Having introduced $\xi_{\beta\alpha}(k)$ for $k\in\bold R\cup\bold I^+,$ let us now analyze the problem of recovering $\alpha,$ $\beta,$ $F_\alpha,$ $F_\beta,$ and $V$ from $\xi_{\beta\alpha}.$ First, given $\xi_{\beta\alpha}(k)$ for $k\in\bold R^+\cup\bold I^+,$ from (5.2) we determine whether $\alpha\in(0,\pi)$ or $\alpha=\pi.$ Next, we can recover $N_\alpha,$ $N_\beta,$ $\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha},$ and $\{\kappa_{\beta j} \}_{j=1}^{N_\beta}$ by using the values of $\xi_{\beta\alpha}(k)$ for $k\in\bold I^+.$ In fact, with the help of (2.1), (2.3), and (5.1)-(5.3), we see that $Z_{\beta\alpha}(k)$ is real valued for $k\in\bold I^+$ and that $\xi_{\beta\alpha}(k)$ on $\bold I^+$ is equal to either $0$ or $1,$ with jump discontinuities at $k=i\kappa_{\alpha j}$ and $k=i\kappa_{\beta j}.$ In other words, in consonant with the interlacing properties given in (3.1) and (3.2), as a result of the simplicity of zeros of $F_\alpha$ and $F_\beta$ on $\bold I^+,$ we have, when $N_\alpha=N_\beta$ and $0<\beta<\alpha<\pi$ $$\xi_{\beta\alpha}(i\omega)=\cases 0,\qquad \omega\in (0,\kappa_{\alpha 1}) \cup (\kappa_{\beta N_\beta},+\infty) \cup_{j=2}^{N_\alpha} (\kappa_{\beta (j-1)}, \kappa_{\alpha j}),\\ 1,\qquad \omega\in \cup_{j=1}^{N_\alpha} (\kappa_{\alpha j}, \kappa_{\beta j}),\endcases$$ and we have, when $N_\alpha=N_\beta-1$ and $0<\beta<\alpha<\pi$ $$\xi_{\beta\alpha}(i\omega)=\cases 0,\qquad \omega\in (\kappa_{\beta N_\beta},+\infty) \cup_{j=1}^{N_\alpha} (\kappa_{\beta j}, \kappa_{\alpha j}),\\ 1,\qquad \omega\in (0,\kappa_{\beta 1}) \cup_{j=1}^{N_\alpha} (\kappa_{\alpha j}, \kappa_{\beta (j+1)}) .\endcases$$ On the other hand, we have, when $0<\beta<\alpha=\pi$ and $N_\beta=N_\pi$ $$\xi_{\beta\pi}(i\omega)=\cases 1,\qquad \omega\in (0,\kappa_{\pi 1}) \cup (\kappa_{\beta N_\beta},+\infty) \cup_{j=2}^{N_\pi} (\kappa_{\beta (j-1)}, \kappa_{\pi j}),\\ 0,\qquad \omega\in \cup_{j=1}^{N_\pi} (\kappa_{\pi j}, \kappa_{\beta j}),\endcases$$ and we have, when $0<\beta<\alpha=\pi$ and $N_\beta=N_\pi+1$ $$\xi_{\beta\pi}(i\omega)=\cases 1,\qquad \omega\in (\kappa_{\beta N_\beta},+\infty) \cup_{j=1}^{N_\pi} (\kappa_{\beta j}, \kappa_{\pi j}),\\ 0,\qquad \omega\in (0,\kappa_{\beta 1}) \cup_{j=1}^{N_\pi} (\kappa_{\pi j}, \kappa_{\beta (j+1)}) .\endcases\tag 5.7$$ Thus, we are able to recover $N_\alpha,$ $N_\beta,$ $\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha},$ and $\{\kappa_{\beta j} \}_{j=1}^{N_\beta}$ by analyzing the location of the jumps of $\xi_{\beta\alpha}(k)$ for $k\in\bold I^+.$ In order to continue with the recovery, let us define the `reduced' quantities identified with the superscript $[0]$ as follows: $$F_{\alpha}^{[0]}(k):=F_{\alpha}(k) \displaystyle\prod_{j=1}^{N_\alpha}\displaystyle\frac{k+i\kappa_{\alpha j}} {k-i\kappa_{\alpha j}},\qquad \alpha\in (0,\pi].$$ Note that $|F_{\alpha}^{[0]}(k)|=|F_{\alpha}(k)|$ for $k\in\bold R,$ and hence as seen from (5.3) it is natural to let $$Z_{\beta\alpha}^{[0]}(k):=\cases Z_{\beta\alpha}(k) \displaystyle\prod_{j=1}^{N_\alpha}\displaystyle\frac{k+i\kappa_{\alpha j}} {k-i\kappa_{\alpha j}} \displaystyle\prod_{p=1}^{N_\beta}\displaystyle\frac{k-i\kappa_{\beta p}} {k+i\kappa_{\beta p}},\qquad 0<\beta<\alpha< \pi,\\ Z_{\beta\pi}(k) \displaystyle\prod_{j=1}^{N_\pi}\displaystyle\frac{k-i\kappa_{\pi j}} {k+i\kappa_{\pi j}} \displaystyle\prod_{p=1}^{N_\beta}\displaystyle\frac{k+i\kappa_{\beta p}} {k-i\kappa_{\beta p}}, \qquad 0<\beta<\alpha=\pi,\endcases \tag 5.8$$ so that $Z_{\beta\alpha}^{[0]}$ has no zeros or poles in $\overline{\bold C^+}\setminus\{0\}.$ Let $\Cal W^{[0]}$ denote the class of functions $Y(k)$ that are analytic in $\bold C^+$ and continuous in $\overline{\bold C^+}\setminus\{0\}$ satisfying $Y(-k)=Y(k)^\ast$ for $k\in\bold R$ and $1+O(1/k)$ as $k\to\infty$ in $\overline{\bold C^+},$ and that $Y$ is either continuous at $k=0$ or has a finite-order zero or pole at $k=0.$ Let $\Cal W$ denote the extended class of functions that differ from $Y(k)$ in $\Cal W^{[0]}$ by a finite number of multiplicative factors of the form $\displaystyle\frac{k+ia_j} {k+ib_j}$ with real constants $a_j$ and $b_j.$ For such functions $Y(k)$ in $\Cal W,$ we let $\log \left(Y(k)\right)$ denote the branch of the logarithm normalized as $\text{Im}\left[\log \left(Y(\infty)\right)\right]=0.$ \proclaim {Theorem 5.1} The quantities $Z_{\beta\alpha}^{[0]}(k)$ for $0<\beta<\alpha< \pi$ and $Z_{\beta\pi}^{[0]}(k)/[ik]$ for $0<\beta<\pi$ each belong to $\Cal W^{[0]},$ and for $k\in\overline{\bold C^+}$ we have $$Z_{\beta\alpha}^{[0]}(k)=\exp\left(\displaystyle\frac{1}{\pi} \int_{-\infty}^\infty dt\, \displaystyle\frac{\text{\rm Im}\left[\log \left(Z_{\beta\alpha}^{[0]}(t)\right)\right]} {t-k-i0^+}\right),\qquad 0<\beta<\alpha< \pi,\tag 5.9$$ $$Z_{\beta\pi}^{[0]}(k)=ik\cdot\exp\left(\displaystyle\frac{1}{\pi} \int_{-\infty}^\infty dt\, \displaystyle\frac{\text{\rm Im}\left[\log \left(Z_{\beta\pi}^{[0]}(t)/[it]\right)\right]} {t-k-i0^+}\right),\qquad 0<\beta<\pi.\tag 5.10$$ \endproclaim \demo{Proof} For any $\alpha\in(0,\pi]$ it is known [2,10-12] that $F_\alpha(k)$ is analytic in $\bold C^+$ and continuous in $\overline{\bold C^+},$ it is nonzero in $\overline{\bold C^+}$ except for the simple zeros at $k=i\kappa_{\alpha j}$ with $j=1,\dots,N_\alpha$ and perhaps a simple zero at $k=0.$ By using (3.12) and (3.13) of [12], as $k\to\infty$ in $\overline{\bold C^+}$ we obtain $$Z_{\beta\alpha}(k) =\cases 1+\displaystyle\frac{i\,h_{\beta \alpha}}{k} -\displaystyle\frac{h_{\beta\alpha}\,\cot\beta}{k^2}+o(1/k^2), \qquad 0<\beta<\alpha< \pi,\\ ik+\cot\beta +o(1),\qquad 0<\beta<\alpha=\pi,\endcases\tag 5.11$$ and hence from (5.8) and (5.11), as $k\to\infty$ in $\overline{\bold C^+},$ we get $$Z_{\beta\alpha}^{[0]}(k) = 1-\displaystyle\frac{i}{k}\left[ h_{\beta \alpha}+2\sum_{j=1}^{N_\pi}\kappa_{\pi j}-2 \sum_{j=1}^{N_\beta}\kappa_{\beta j}\right]+O(1/k^2), \qquad 0<\beta<\alpha< \pi,$$ $$\displaystyle\frac{Z_{\beta\pi}^{[0]}(k)}{ik}= 1+\displaystyle\frac{1}{ik}\left[ \cot\beta+2\sum_{j=1}^{N_\pi}\kappa_{\pi j}-2 \sum_{j=1}^{N_\beta}\kappa_{\beta j}\right]+o(1/k), \qquad 0<\beta<\pi.$$ Using also (2.4), (5.3), and (5.8), it follows that $Z_{\beta\alpha}^{[0]}(k)$ for $0<\beta<\alpha< \pi$ and $Z_{\beta\pi}^{[0]}(k)/[ik]$ for $0<\beta<\pi$ each belong to $\Cal W^{[0]}.$ Moreover, the logarithms of both these quantities are analytic in $\bold C^+$ and continuous in $\overline{\bold C^+}\setminus\{0\},$ have at most an integrable singularity at $k=0,$ and are $O(1/k)$ as $k\to\infty$ in $\overline{\bold C^+}.$ Thus, the Schwarz integral formula can be used to obtain $$\log \left(Z_{\beta\alpha}^{[0]}(k)\right)=\displaystyle\frac{1}{\pi} \int_{-\infty}^\infty dt\, \displaystyle\frac{\text{Im}\left[\log \left(Z_{\beta\alpha}^{[0]}(t)\right)\right]} {t-k-i0^+},\qquad k\in\overline{\bold C^+},$$ $$\log \left(\displaystyle\frac{Z_{\beta\pi}^{[0]}(k)}{ik}\right)=\displaystyle\frac{1}{\pi} \int_{-\infty}^\infty dt\, \displaystyle\frac{\text{Im}\left[\log \left(Z_{\beta\pi}^{[0]}(t)/[it]\right)\right]} {t-k-i0^+},\qquad k\in\overline{\bold C^+},$$ which give us (5.9) and (5.10), respectively. \quad\qed \enddemo From (5.8) we get $|Z_{\beta\alpha}^{[0]}(k)|=|Z_{\beta\alpha}(k)|$ for $k\in\bold R,$ and hence a comparison with (5.1) and (5.2) leads us to let $$e^{\pi i \xi_{\beta\alpha}^{[0]}(k)}:=\displaystyle\frac{Z_{\beta\alpha}^{[0]}(k)} {|Z_{\beta\alpha}^{[0]}(k)|},\qquad k\in \bold R^+\cup\bold I^+,\tag 5.12$$ with the normalization $$\xi_{\beta\alpha}^{[0]}(+\infty)=\cases 0,\qquad \alpha\in(0,\pi),\\ 1/2,\qquad \alpha=\pi,\endcases$$ so that for $0<\beta<\alpha< \pi$ we have $$\xi_{\beta\alpha}^{[0]}(k)=\xi_{\beta\alpha}(k)+ \displaystyle\frac{1}{\pi i} \log\left(\displaystyle\prod_{j=1}^{N_\alpha}\displaystyle\frac{k+i\kappa_{\alpha j}} {k-i\kappa_{\alpha j}} \displaystyle\prod_{p=1}^{N_\beta}\displaystyle\frac{k-i\kappa_{\beta p}} {k+i\kappa_{\beta p}}\right),\tag 5.13$$ and for $0<\beta<\pi$ we get $$\xi_{\beta\pi}^{[0]}(k)=\xi_{\beta\pi}(k)+ \displaystyle\frac{1}{\pi i} \log\left(\displaystyle\prod_{j=1}^{N_\pi}\displaystyle\frac{k-i\kappa_{\pi j}} {k+i\kappa_{\pi j}} \displaystyle\prod_{p=1}^{N_\beta}\displaystyle\frac{k+i\kappa_{\beta p}} {k-i\kappa_{\beta p}}\right).\tag 5.14$$ Using (5.5), (5.13), and (5.14) we can extend $\xi_{\beta\alpha}^{[0]}(k)$ oddly from $k\in\bold R^+$ to $k\in\bold R,$ i.e. $$\xi_{\beta\alpha}^{[0]}(-k)= -\xi_{\beta\alpha}^{[0]}(k),\qquad 0<\beta<\alpha\le \pi, \quad k\in\bold R.\tag 5.15$$ From (5.12) and (5.15), for $k\in\bold R$ we get $$\xi_{\beta\alpha}^{[0]}(k)=\cases\displaystyle\frac{1}{\pi}\,\text{Im}\left[ \log \left(Z_{\beta\alpha}^{[0]}(k)\right)\right],\qquad 0<\beta<\alpha< \pi,\\ \displaystyle\frac{1}{\pi}\,\text{Im}\left[ \log \left(\displaystyle\frac{Z_{\beta\pi}^{[0]}(k)}{ik} \right)\right]+\displaystyle\frac12\,\text{sign}(k),\qquad 0<\beta<\alpha=\pi,\endcases\tag 5.16$$ and similarly, for $k\in\bold R$ we have $$\xi_{\beta\alpha}(k)=\cases\displaystyle\frac{1}{\pi}\,\text{Im}\left[ \log \left(Z_{\beta\alpha}(k)\right)\right],\qquad 0<\beta<\alpha< \pi,\\ \displaystyle\frac{1}{\pi}\,\text{Im}\left[ \log \left(\displaystyle\frac{Z_{\beta\pi}(k)}{ik}\right)\right]+\displaystyle\frac12\,\text{sign}(k),\qquad 0<\beta<\alpha=\pi.\endcases\tag 5.17$$ Since $Z_{\beta\alpha}^{[0]}(k)>0$ for $k\in\bold I^+,$ we have $$\xi_{\beta\alpha}^{[0]}(k)=0,\qquad 0<\beta<\alpha\le\pi, \quad k\in\bold I^+,\tag 5.18$$ and hence, from (5.13) and (5.14), for $k\in\bold I^+$ we get $$\xi_{\beta\alpha}(k)=\cases \displaystyle\frac{1}{\pi i} \log\left(\displaystyle\prod_{j=1}^{N_\alpha}\displaystyle\frac{k-i\kappa_{\alpha j}} {k+i\kappa_{\alpha j}} \displaystyle\prod_{p=1}^{N_\beta}\displaystyle\frac{k+i\kappa_{\beta p}} {k-i\kappa_{\beta p}}\right),\qquad 0<\beta<\alpha< \pi,\\ \displaystyle\frac{1}{\pi i} \log\left(\displaystyle\prod_{j=1}^{N_\pi}\displaystyle\frac{k+i\kappa_{\pi j}} {k-i\kappa_{\pi j}} \displaystyle\prod_{p=1}^{N_\beta}\displaystyle\frac{k-i\kappa_{\beta p}} {k+i\kappa_{\beta p}}\right),\qquad 0<\beta<\alpha=\pi.\endcases\tag 5.19$$ Using (5.16) in (5.9) and (5.10), we obtain the following. \proclaim {Corollary 5.3} For $0<\beta<\alpha\le \pi,$ the quantity $Z_{\beta\alpha}^{[0]}(k)$ for $k\in\overline{\bold C^+}$ can be obtained from $\xi_{\beta\alpha}^{[0]}(k)$ given for $k\in\bold R^+$ via (5.15) and $$Z_{\beta\alpha}^{[0]}(k)=\cases \exp\left( \displaystyle\int_{-\infty}^\infty dt\, \displaystyle\frac{\xi_{\beta\alpha}^{[0]}(t)} {t-k-i0^+}\right),\qquad 0<\beta<\alpha< \pi,\\ ik\cdot\exp\left( \displaystyle\int_{-\infty}^\infty dt\, \displaystyle\frac{\xi_{\beta\pi}^{[0]}(t)-(1/2)\,\text{\rm sign}(t) } {t-k-i0^+}\right),\qquad 0<\beta<\alpha=\pi.\endcases$$ \endproclaim Now let us continue with the recovery of $\alpha$ and $\beta$ from $\xi_{\beta\alpha}.$ We have already constructed $N_\alpha,$ $N_\beta,$ $\{\kappa_{\alpha j}\}_{j=1}^{N_\alpha},$ and $\{\kappa_{\beta j} \}_{j=1}^{N_\beta}.$ Next, with the help of (5.13), (5.14), and (5.15) we obtain $\xi_{\beta\alpha}^{[0]}(k)$ for $k\in\bold R.$ Then, using Corollary~5.3 we get $Z_{\beta\alpha}^{[0]}(k)$ and via (5.8) we obtain $Z_{\beta\alpha}(k)$ for $k\in\overline{\bold C^+}.$ Having $Z_{\beta\alpha}(k)$ in hand, we can recover $\alpha$ and $\beta$ by using (3.11) and (5.11). Next, we continue with the construction of $F_\alpha$ and $F_\beta.$ When $\alpha\ne\pi,$ we proceed as follows. Having $h_{\beta\alpha}$ and $Z_{\beta\alpha}(k)$ for $k\in\bold R,$ via (5.6) we construct $|F_\beta(k)|^2$ as $$|F_\beta(k)|^2= \displaystyle\frac{k\,h_{\beta\alpha}}{\text{Im}\left[Z_{\beta\alpha}(k)\right]}, \qquad k\in\bold R.$$ Knowing $|F_\beta(k)|$ for $k\in\bold R,$ we can then construct $F_\beta$ via (cf. (3.6) of [12]) $$F_\beta(k)=k\left(\displaystyle\prod_{j=1}^{N_\beta}\displaystyle\frac{k-i\kappa_{\beta j}}{k+i\kappa_{\beta j}}\right) \exp\left(\displaystyle\frac{-1}{\pi i}\int_{-\infty}^\infty dt\, \displaystyle\frac{\log |t/F_\beta(t)|}{t-k-i0^+}\right), \qquad k\in\overline{\bold C^+}.\tag 5.20$$ On the other hand, if $\alpha=\pi$ we proceed as follows. From (5.6) we get $|F_\pi(k)|^2$ as $$|F_\pi(k)|^2= \displaystyle\frac{k}{\text{Im}\left[Z_{\beta\pi}(k)\right]}, \qquad k\in\bold R. $$ Having $|F_\pi(k)|$ for $k\in\bold R$ at hand, we can then construct $F_\pi$ via (cf. (3.7) of [12]) $$F_\pi(k)=\left(\displaystyle\prod_{j=1}^{N_\pi} \displaystyle\frac{k-i\kappa_{\pi j}}{k+i\kappa_{\pi j}}\right) \exp\left(\displaystyle\frac{1}{\pi i}\int_{-\infty}^\infty dt\, \displaystyle\frac{\log |F_\pi(t)|}{t-k-i0^+}\right),\qquad k\in\overline{\bold C^+}.\tag 5.21$$ Finally, if $\alpha\ne \pi$ then via the first line of (5.3) we can recover $F_\alpha(k)$ from the already constructed quantities $F_\beta(k)$ and $Z_{\beta\alpha}(k)$ for $k\in\overline{\bold C^+}.$ Similarly, if $\alpha=\pi$ then via the second line of (5.3) we can recover $F_\beta(k)$ from the already constructed quantities $F_\pi(k)$ and $Z_{\beta\pi}(k)$ for $k\in\overline{\bold C^+}.$ A comparison with (3.3)-(3.10) reveals that we have now constructed the data sets $\Cal D_1,\Cal D_3,\Cal D_5,\Cal D_7$ when $\alpha\in(0,\pi)$ and $\Cal D_2,\Cal D_4,\Cal D_6,\Cal D_8$ when $\alpha= \pi.$ Finally, we can recover $V$ via one of the methods described in Section~4. Let us also note that, by using any one of the eight data sets $\Cal D_j$ given in (3.3)-(3.10), we can construct the quantities $F_\alpha(k)$ and $F_\beta(k)$ for $k\in\overline{\bold C^+}$ as outlined in Section~3, then obtain $Z_{\beta\alpha}$ for $k\in\overline{\bold C^+}$ given in (5.3), and also recover $\xi_{\beta\alpha}(k)$ for $k\in\bold R^+\cup\bold I^+$ via (5.17) and (5.19). \head 6. Examples \endhead In this section we present two examples to illustrate the recovery of the potential and the boundary conditions from the data set $\Cal D_3$ given in (3.5) and also from Krein's spectral shift function $\xi_{\beta\alpha}.$ \example{Example 6.1} In our first example, assume that we are given $\Cal D_3$ with $h_{\beta\alpha}=5,$ $N_\alpha=1,$ $N_\beta=2,$ $\kappa_{\alpha 1}=2,$ $\kappa_{\beta 2}=4,$ and $|F_\alpha(k)|^2=k^2+4$ for $k\in\bold R,$ and we are interested in constructing all the relevant quantities such as $\cot\alpha,$ $\cot\beta,$ $F_\alpha,$ $F_\beta,$ and $V.$ By using the method outlined in Section~3, with the details given in the proof of Theorem~2.3 of [12], we get $$\kappa_{\beta 1}=1,\quad \cot\alpha=-8/5,\quad \cot\beta=17/5,$$ $$F_\alpha(k)=k-2i,\quad F_\beta(k)=\displaystyle\frac{(k-i)(k-4i)}{k+2i}.$$ As indicated in Sections~3-5, we can then obtain other relevant quantities such as $$f(k,0)=\displaystyle\frac{5k-8i}{5(k+2i)},\quad f'(k,0)=\displaystyle\frac{25ik^2+40k+36i}{25(k+2i)}, $$ the quantities used in the Gel'fand-Levitan and Marchenko methods, e.g. $$g_{\alpha 1}=\sqrt{\displaystyle\frac 25},\quad m_{\alpha 1}=\sqrt{40},\quad S_\alpha(k)=\displaystyle\frac{k+2i}{k-2i},$$ the quantities used in the Faddeev-Marchenko method, e.g. $$N=1,\quad \tau_1=\displaystyle\frac{(\sqrt{34}+4)i}{5}, \quad c_{\text{r}1}=\displaystyle\frac{3}{\sqrt{5\sqrt{34}}}, \quad L(k)=\displaystyle\frac{-18}{25k^2-40ik+18},$$ the quantities relevant to Krein's spectral shift function, e.g. $$Z_{\beta\alpha}^{[0]}(k)=\displaystyle\frac{(k+2i)^2} {(k+i)(k+4i)},\quad \xi_{\beta\alpha}^{[0]}(k)=\displaystyle\frac{1}{\pi}\, \text{Im}\left[\log \left(\displaystyle\frac{(k+2i)^2} {(k+i)(k+4i)}\right)\right],$$ $$Z_{\beta\alpha}(k)=\displaystyle\frac{(k+2i)(k-2i)} {(k-i)(k-4i)},\quad \xi_{\beta\alpha}(k)=\displaystyle\frac{1}{\pi}\, \text{Im}\left[\log \left(\displaystyle\frac{(k+2i)(k-2i)} {(k-i)(k-4i)}\right)\right],$$ and finally the potential and the Jost solution $$V(x)=-\displaystyle\frac{288e^{4x}} {\left(9+e^{4x}\right)^2},\quad f(k,x)=e^{ikx}\left[1-\displaystyle\frac{36i}{(k+2i)\left(9+e^{4x}\right)}\right].$$ \endexample \example{Example 6.2} As our second example, let us assume that we have as our data Krein's spectral shift function given by $$\xi_{\beta\alpha}(k)=\cases \displaystyle\frac12+ \displaystyle\frac{1}{\pi}\, \text{Im}\left[\log \left(\displaystyle\frac{(k-i)(k+i)(k-3i)} {k(k-2i)(k+2i)}\right)\right],\qquad k\in\bold R^+,\\ 0,\qquad k\in i(0,1)\cup i(2,3),\\ 1,\qquad k\in i(1,2)\cup i(3,+\infty),\endcases \tag 6.1$$ and we would like to construct all the relevant quantities such as $\cot\alpha,$ $\cot\beta,$ $F_\alpha,$ $F_\beta,$ and $V.$ By letting $k\to+\infty$ in (6.1) we get $\xi_{\beta\alpha}(+\infty)=1/2,$ and hence we see from (5.2) that $\alpha=\pi.$ Next, we analyze our $\xi_{\beta\alpha}(k)$ on $\bold I^+$ and see the jump discontinuities at $k=i,2i,3i.$ A comparison with (5.7) indicates that $N_\alpha=1,$ $N_\beta=2,$ $\kappa_{\alpha 1}=2,$ $\kappa_{\beta 1}=1,$ and $\kappa_{\beta 2}=3.$ Next, using (5.14) and (5.18) we obtain $$\xi_{\beta\alpha}^{[0]}(k)=\cases \displaystyle\frac12 +\displaystyle\frac{1}{\pi}\, \text{Im}\left[\log \left(\displaystyle\frac{(k+i)^2(k+3i)} {k(k+2i)^2}\right)\right],\qquad k\in\bold R,\\ 0,\qquad k\in\bold I^+,\endcases$$ and then, via Corollary~5.3, we get $$Z_{\beta\alpha}^{[0]}(k)=\displaystyle\frac {i(k+i)^2(k+3i)}{(k+2i)^2}.\tag 6.2$$ Next, using (6.2) in (5.8) we obtain $$Z_{\beta\alpha}(k)=\displaystyle\frac {i(k+i)(k-i)(k-3i)}{(k+2i)(k-2i)}.\tag 6.3$$ Using (6.3) in (5.11) we get $\cot\beta=3,$ and then with the help of (5.20), (5.21), and then (5.3) we obtain $$F_\pi(k)=\displaystyle\frac {k-2i}{k+i},\quad F_\beta(k)=\displaystyle\frac {(k-i)(k-3i)}{k+2i}.$$ Proceeding as in the first example, we can then construct all the relevant quantities. For example, we have $$S_\pi(k)=\displaystyle\frac{(k+i)(k+2i)}{(k-i)(k-2i)},\quad m_{\pi 1}=4\sqrt{3},\quad g_{\pi 1}=\sqrt{3},$$ $$L(k)=\displaystyle\frac{-9i}{2k^3+5k+9i},\quad N=1,\quad \tau_1=2.14444\overline{1}\,i,\quad c_{{\text r}1}=0.63118\overline{2},$$ where the overline indicates a roundoff at the digit. Finally, we obtain the potential and the Jost solution $$V(x)=\displaystyle\frac{24e^{-2x}-480e^{-4x}+720e^{-6x}-480e^{-8x}+600e^{-10x}} {\left(1-3e^{-2x}+15e^{-4x}-5e^{-6x}\right)^2},$$ $$ f(k,x)=e^{ikx}\left[1+\displaystyle\frac{\displaystyle\frac{6i(e^{-2x}-5e^{-6x})}{k+i}+ \displaystyle\frac{60i(-e^{-4x}+e^{-6x})}{k+2i}} {1-3e^{-2x}+15e^{-4x}-5e^{-6x}} \right].$$ \endexample \refstyle{N} \widestnumber\key{15} \Refs \ref\key{1} \by J. Weidmann \book Spectral theory of ordinary differential operators \publ Lecture Notes in Math. {\bf 1258}, Springer \publaddr Berlin \yr 1987 \endref \ref\key{2} \by B. M. 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Weder \paper Inverse spectral-scattering problem with two sets of discrete spectra for the radial Schr\"odinger equation \jour IMA preprint \#1960 (2004); available at the url http://arxiv.org/PS\_cache/math-ph/pdf/0402/0402019.pdf %\vol 21 %\yr 2005 %\pages 899-914 \endref \ref\key{13} \by M. G. Krein \paper Perturbation determinants and a formula for the traces of unitary and self-adjoint operators \jour Soviet Math. Dokl. \vol 3 \yr 1962 \pages 707--710 \endref \ref\key{14} \by M. S. Birman and D. R. Yafaev \paper The spectral shift function. The papers of M. G. Krein and their further development \jour St. Petersburg Math. J. \vol 4 \yr 1993 \pages 833--870 \endref \ref\key{15} \by D. R. Yafaev \book Mathematical Scattering Theory \publ Am. Math. Soc. \publaddr Providence, RI \yr 1992 \endref \endRefs \enddocument ---------------0511301851509--