\input amstex \loadbold \documentstyle{amsppt} %\magnification=1100 \magnification=\magstephalf \baselineskip=15 pt %\NoBlackBoxes \TagsOnRight \def\gap{\vskip 0.1in\noindent} \def\ref#1#2#3#4#5#6{#1, {\it #2,} #3 {\bf #4} (#5), #6.} %References \def\borg{1} % Borg \def\borgun {2} % Borg unique \def\gl {3} % Gelfand-Levitan \def\gsac {4} % Gesztesy-Simon, ac paper \def\gsmf {5} % Gesztesy-Simon, m-function paper \def\gsds {6} % Gesztesy-Simon, ds paper \def\haldma {7} % Hald Mantle \def\hsturm {8} % Hochstadt 73 \def\hl {9} % Hochstadt-Lieberman \def\levs {10} % Levinson \def\lev {11} % Levitan \def\levbook {12} % Levitan book \def\lg {13} % Levitan-Gasymov \def\mal {14} % Malamud \def\malsp {15} % Malamud \def\mar {16} % Marchenko \def\simon {17} % Simon \topmatter \title Inverse Spectral Analysis With Partial Information on the Potential, III. Updating Boundary Conditions \endtitle \rightheadtext{Inverse Spectral Analysis: Updating Boundary Conditions} \author Rafael del Rio$^1$, Fritz Gesztesy$^2$, and Barry Simon$^3$ \endauthor \leftheadtext{R. del Rio, F.~Gesztesy, and B.~Simon} \thanks$^1$ IIMAS-UNAM, Apdo.~Postal 20-726, Admon No.~20, 01000~Mexico D.F., Mexico. E-mail: delrio\@servidor.unam.mx \endthanks \thanks$^2$ Department of Mathematics, University of Missouri, Columbia, MO~65211, USA. E-mail: fritz\@math.missouri.edu \endthanks \thanks$^3$ Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasade-na, CA~91125, USA. E-mail: bsimon\@caltech.edu \endthanks \thanks This material is based upon work supported by CONACYT Project 05567P-E and the National Science Foundation under Grant Nos.~DMS-9623121 and DMS-9401491. \endthanks \thanks To be submitted to {\it{Intl.~Math.~Research Notes}} \endthanks \date June 3, 1997 \enddate \abstract We discuss results where information on parts of the discrete spectra of one-dimensional Schr\"odinger operators $H=-\frac{d^2}{dx^2}+q$ in $L^2((0,1))$ or of a finite Jacobi matrix together with partial information on $q$ uniquely determines $q$ a.e.~on $[0,1]$. These extend classical results of Borg and Hochstadt-Lieberman as well as results in paper~II of this series. \endabstract \endtopmatter \document \vskip 0.1in \flushpar{\bf \S 1. Introduction} \vskip 0.1in This paper is a postscript to two earlier papers [\gsmf, \gsds] in that it provides a new way of looking at the problems considered in those papers that allows the same methods to prove additional results. To explain our results, we recall earlier theorems of Borg [\borg] (see also [\hsturm, \levs--\mal]) and of Hochstadt-Lieberman [\hl] (see also [\haldma, \malsp]). Throughout this paper assume $q\in L^1((0,1))$ to be real-valued and consider the operator $H=-\frac{d^2}{dx^2} + q$ in $L^2 ((0,1))$ with boundary conditions \align u'(0) + h_0 u(0) &=0, \tag 1.1 \\ u'(1) + h_1 u(1) &= 0, \tag 1.2 \endalign where $h_j \in \Bbb R\cup \{\infty\}$, $j=0,1$ (with $h_0 = \infty$ shorthand for the boundary condition $u(0)=0$). Fix $h_1 \in \Bbb R$ but think of $H(h_0)$ as a family of operators depending on $h_0$ as a parameter. Then Borg's and Hochstadt-Lieberman's results can be paraphrased as follows: \example{Borg [\borg]} The spectra of $H(h_0)$ for two values of $h_0$ determine $q$. \endexample \example{Hochstadt-Lieberman [\hl]} The spectra of $H(h_0)$ for one value of $h_0$ and $q$ on $[0, \frac12]$ determine $q$. \endexample In [\gsds], two of us proved a result that can be paraphrased as \proclaim{Theorem of [\gsds]} Half the spectra of one $H(h_0)$ and $q$ on $[0,\frac34]$ determine $q$. \endproclaim One of our goals in this note is to prove \example{New Result} The spectrum of one $H(h_0)$ and half the spectrum of another $H(h_0)$ and $q$ on $[0,\frac14]$ determine $q$. \endexample We will also show that \example{New Result} Two-thirds of the spectra of three $H(h_0)$ determine $q$. \endexample Our point is as much a new way of looking at the argument in [\gsds] as these new results. Fundamental to our approach here and in [\gsmf, \gsds] is the Titchmarsh-Weyl $m$-function defined by $$m_{h_1}(z) = \frac{u'_{h_1}(z,0)}{u_{h_1}(z,0)}\, ,$$ where $u_{h_1}(z,x)$ solves $-u''(z,x) + q(x) u(z,x)= zu(z,x)$ with the boundary condition $(1.2)$. $m_{h_1}$ is a meromorphic function on $\Bbb C$ (in fact, a Herglotz function) with all its zeros and poles on the real axis. Since $h_1 \in \Bbb R$ will be fixed throughout this paper, we will delete the subscript $h_1$ from now on and simply write $m(z)$ instead. Moreover, due to the assumption $h_1 \in \Bbb R$, we will index the eigenvalues of $H(h_0)$ by $\{\lambda_n\}_{n\in \Bbb N_0}$, $\Bbb N_0 = \Bbb N \cup \{0\}$. A fundamental result of Marchenko [\mar] (see also [\borgun, \gl, \simon]) says \proclaim{Theorem 1.1} $m(z)$ uniquely determines $q$ a.e.~on $[0,1]$. \endproclaim Our fundamental strategy can be described as follows: (a) Note that $\lambda$ is an eigenvalue of $H(h_0)$ if and only if $m(\lambda)= -h_0$. (b) Prove a general theorem that knowing $m$ at points $\lambda_0, \lambda_1, \dots, \lambda_n, \dots$ determines $m$ as long as $\{\lambda_n\}_{n\in \Bbb N_0}$ has sufficient density. Given (a), this will allow one to prove that if $\lambda_0, \lambda_1, \dots, \lambda_n, \dots$ have sufficient density, an infinite sequence of pairs $\{(\lambda_n, \alpha_n)\}_{n\in \Bbb N_0}$ and the knowledge that $H(h_0 = \alpha_n)$ has an eigenvalue at $\lambda_n$ determines $m$ (and so $q$ a.e.~on $[0,1]$ by Theorem~1.1). (c) Use scaling covariance to extend the $[0,1]$ result to one for $[x, 1]$ for any $x\in (0,1)$. (d) Note that a knowledge of $q$ a.e.~on $[0,x]$ allows one to update boundary conditions. Explicitly, let $H(h_x)$ be the operator in $L^2 ((x,1))$ with boundary condition $(1.2)$ but (1.1) replaced by $$u'(x) + h_x u(x) =0. \tag 1.3$$ Then $\lambda_n$ is an eigenvalue of $H(h_0=\alpha_n)$ if and only if it is an eigenvalue of $H(h_{x_0} = \beta_n)$, where $\beta_n$ is obtained by solving $m'_n (x) = q(x) - \lambda_n - m^2_n$ on $[0, x_0]$ with the boundary condition $m_n (x=0) = -\alpha_n$ and setting $\beta_n = -m_n (x=x_0)$. We will present steps (b) and (c) in Sections~2 and 3 and then step (d) in Section~4. We will not explicitly derive them, but the results in [\gsds] that treat operators on $(0,1)$ and that allow one to trade $C^{2k}$ conditions on $q$ for $k$ eigenvalues can be extended to the context we discuss here. We also note that the ideas in this paper extend to Jacobi matrices. Finally, while the present paper and [\gsmf, \gsds] concentrate on discrete spectra, we might point out that our $m$-function strategy also applies in certain cases involving absolutely continuous spectra, see [\gsac]. \vskip 0.3in \flushpar{\bf \S 2. Zeros of the $\boldkey m$-function} \vskip 0.1in If $a\in\Bbb R$, let $a_+ = \max (a,0)$. Then \proclaim{Theorem 2.1} Let $\{\lambda_n\}_{n\in \Bbb N_0}$ be a sequence of distinct positive real numbers satisfying $$\sum^\infty_{n=0} \frac{(\lambda_n - \frac{1}{4} \pi^2 n^2)_+}{n^2} <\infty. \tag 2.1$$ Let $m_1, m_2$ be the $m$-functions for two operators $H_j=-\frac{d^2}{dx^2}+ q_j$ in $L^2((0,1))$ with boundary conditions $$u'(1) + h^{(j)}_1 u(1) = 0$$ and $h^{(j)}_1 \in \Bbb R$, $j=1,2$. Suppose that $m_1 (\lambda_n) = m_2 (\lambda_n)$ for all $n \in \Bbb N_0$. Then $m_1 = m_2$ \rom(and hence $q_1=q_2$ a.e.~on $[0,1]$ and $h_1^{(1)} = h_1^{(2)}$\rom). \endproclaim \remark{Remarks} 1. In our examples, $\lambda_n \sim \pi^2 n^2 + C$ as $n \to \infty$ (cf.~(3.1)), so (2.1) is satisfied, for instance, by considering two distinct spectra of $H(h_0)$. 2. We allow the case $m_1 (\lambda_n)=m_2(\lambda_n) =\infty$. \endremark \vskip 0.1in As a preliminary result we note the following \proclaim{Lemma 2.2} Suppose $\{\lambda_n\}_{n\in \Bbb N_0}$ is a sequence of positive real numbers satisfying {\rom{(2.1)}} and $$\sum^{\infty}_{n=0} \lambda^{-1}_n < \infty. \tag 2.2$$ Define $f(z) := \prod^\infty_{n=0} (1- \frac{z}{\lambda_n})$, then $$\varlimsup\limits\Sb |y|\to \infty \\ y\in \Bbb R \endSb \frac {|y|^{1/2} \text{\rom{ sinh}}(2|y|^{1/2})}{|f(iy)|} < \infty. \tag 2.3$$ \endproclaim \demo{Proof} Let $y\in \Bbb R$. Then $\text{sinh}(2|y|^{1/2}\,) / |y|^{1/2} = |\sin (2i|y|^{1/2}) / |y|^{1/2}|$ and $$\frac{\sin (2\sqrt z\,)}{2\sqrt z} = \prod^\infty_{n=1} \biggl( 1-\frac{4z}{\pi^2 n^2}\biggr),$$ so (2.3) becomes $$\varlimsup_{|y|\to\infty} \frac{|y|}{1+ \frac{|y|}{\lambda_0}} \, \prod^\infty_{n=1} \biggl[ \frac{( 1 + \frac{4|y|}{\pi^2 n^2})} {(1+ \frac{|y|}{\lambda_n})} \biggr] <\infty \tag 2.4$$ using $2^{-1/2}(1+|x|) \leq (1+x^2)^{1/2} \leq (1+|x|)$. If $0\leq a\leq b$, then $(\frac{1+a|y|}{1+b|y|}) \leq 1$, and if $a >b>0$, then $$\gather \frac{(1+a|y|)}{1+b|y|} = 1 + \frac{(a-b)|y|}{1+b|y|} \leq 1 + \frac{a-b}{b} = \frac{a}{b}, \\ \prod^\infty_{n=1} \frac{(1 + \frac{4|y|}{\pi^2 n^2})} {(1+\frac{|y|}{\lambda_n})} \leq \prod_{n : \lambda_n > \frac{1}{4} \pi^2 n^2} \frac{4\lambda_n}{\pi^2 n^2} = \prod_{n=1}^{\infty} \biggl[ 1 + \frac{(\lambda_n - \frac14 \pi^2 n^2)_+} {\frac14 \pi^2 n^2}\biggr] <\infty \endgather$$ if (2.1) holds. \qed \enddemo \demo{Proof of Theorem 2.1} We follow the arguments in [\gsmf, \gsds] fairly closely. One can write $m_j (z) = \frac{Q_j(z)}{P_j(z)}$, $j=1,2$, where \roster \item"\rom{(1)}" $P_j, Q_j$ are entire functions satisfying \align |P_j(z)| &\leq C \exp(\sqrt{|z|}\,), \tag 2.5a \\ |Q_j(z)| &\leq C (1+\sqrt{|z|}\,) \exp(\sqrt{|z|}\,). \tag 2.5b \endalign \item"\rom{(2)}" $$m_j (z) = \pm i \sqrt{z} + o(1) \text{ as } z \to \pm i\infty. \tag 2.6$$ \endroster (We use the square root branch with $\text{Im}\, (\sqrt{z}) \geq 0$.) Suppose $m_1 \neq m_2$. Then $P_2(z) Q_1(z) - P_1(z) Q_2(z) := H(z)$ is an entire function of order at most $\frac12$ and not identically zero. Since $H(\lambda_n)=0$, we conclude that $\sum_{n\in\Bbb N_0} \lambda^{-a}_n <\infty$ if $a>\frac12$. In particular, (2.2) holds, and we can define $f(z) = \prod^\infty_{n=0} (1-\frac{z}{\lambda_n})$. Next, define $$G(z) := \frac{H(z)}{f(z)} = \frac{P_1(z) P_2(z)}{f(z)}\, (m_1(z) - m_2(z)). \tag 2.7$$ Since $H(\lambda_n)=0$, $G(z)$ is an entire function. By (2.3), $$\varlimsup_{|y|\to\infty} \frac{|y|^{1/2} \exp{(2 |y|^{1/2})}}{|f(iy)|} < \infty,$$ so by (2.5) and (2.6), $$|G(iy)| \leq \frac{\exp{(2 |y|^{1/2})}}{f(iy)} \, |m_1(iy) - m_2(iy)| =o(|y|^{-1/2})$$ goes to zero as $|y|\to\infty$. The Phragm\'en-Lindel\"of argument of [\gsds] then yields the contradiction $G(z)\equiv 0$, that is, $m_1=m_2$. \qed \enddemo \remark{Remark} The above yields $o(|y|^{-1/2})$ even though $o(1)$ would have been sufficient. We have thrown away half a zero. That means one can prove the following result. \endremark \proclaim{Theorem 2.2} Let $\{\lambda_n\}_{n\in\Bbb N_0}$ and $\{\mu_n\}_{n\in\Bbb N_0}$ be two sequences of real numbers satisfying $$\sum^\infty_{n=0} \frac{(\lambda_n - \pi^2 n^2)_+}{n^2} < \infty \qquad \text{and} \qquad \sum^\infty_{n=0} \frac{(\mu_n - \pi^2 n^2)_+}{n^2} < \infty, \tag 2.8$$ with $\mu_m \neq \lambda_n$ for all $m,n \in \Bbb N_0$. Let $m_1, m_2$ be the $m$-functions for two operators $H_j=-\frac{d^2}{dx^2}+q_j$, $j=1,2$ in $L^2 ((0,1))$ with boundary conditions $$u'(1) + h^{(j)}_1 u(1)=0$$ and $h^{(j)}_1\in\Bbb R$, $j=1,2$. Suppose that $m_1(z) = m_2(z)$ for all $z$ in $\{\lambda_n\}^\infty_{n=0} \cup \{\mu_n\}^\infty_{n=0}$ except perhaps for one. Then $m_1 = m_2$ \rom(and hence $q_1=q_2$ a.e.~on $[0,1]$ and $h_1^{(1)} = h_1^{(2)}$\rom). \endproclaim By scaling, one sees the following analog of Theorem~2.1 holds (there is also an analog of Theorem~2.2): \proclaim{Theorem 2.3} Let $a0$ we have $$\#\{\lambda \in \{S_1 \cup S_2 \cup S_3\} \text{\rom{ with }} \lambda\leq \lambda_0 \} \geq \tfrac23 \#\{\lambda \in \{\sigma_1 \cup \sigma_2 \cup \sigma_3 \} \text{\rom{ with }} \lambda\leq \lambda_0 \} -1.$$ Then $q$ is uniquely determined a.e.~on $[0,1]$. \endproclaim In particular, two-thirds of three spectra determine $q$. \vskip 0.3in \flushpar {\bf \S 4. Updating $\boldkey m$} \vskip 0.1in We are now able to understand why partial information on $q$ --- knowing it on $[0,a]$ --- lets us get away with less information on eigenvalues, a phenomenon originally discovered by Hochstadt-Lieberman [\hl] in the special case where $a=\frac12$. We note that $m(z,x)$ satisfies the Ricatti-type equation $$m'(z,x) = q(x)-z-m^2(z,x). \tag 4.1$$ If we know that $\lambda$ is an eigenvalue of $H(h_0)$, then $m(\lambda, 0)=-h_0$. If we know $q$ on $[0,a]$, we can use (4.1) to compute $m(\lambda, a) := -h_a$ and so infer that $\lambda$ is an eigenvalue of $H(h_a)$, the operator in $L^2((a,1))$. By Theorem~2.3, that means we only need a lower density of eigenvalues of the various $H(h_a)$. A typical result is the following \proclaim{Theorem 4.1} Let $\sigma_N$ and $\sigma_D$ be the eigenvalues of $H(h_0 =0)$ and $H(h_0 =\infty)$, respectively. Let $S_N \subseteq \sigma_N$, $S_D \subseteq \sigma_D$. Fix $a\in (0,1)$. Suppose for $\lambda_0 >0$ sufficiently large that $$\#\{\lambda \in \{S_N \cup S_D \} \text{\rom{ with }} \lambda\leq \lambda_0 \} \geq (1-a)\# \{\lambda\in \{\sigma_N \cup \sigma_D \} \text{\rom{ with }} \lambda\leq \lambda_0 \}.$$ Then $S_N, S_D$ and $q$ on $[0,a]$ uniquely determine $q$ a.e.~on $[0,1]$. \endproclaim This follows immediately from the updating idea. For example, if $a=\frac34$, we can recover Theorem~1.3 of [\gsds] (it is essentially a reworking of the proof in [\gsds]); but for $a\in (0,\frac12)$, the result is new and implies, for example, that $q$ on $[0,\frac14]$, all the Neumann eigenvalues, and half the Dirichlet eigenvalues determine $q$ a.e.~on $[0,1]$. \vskip 0.3in \Refs \endRefs \vskip 0.1in \item{\borg.}\ref{G.~Borg}{Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe}{Acta Math.}{78}{1946}{1--96} \gap \item{\borgun.} G.~Borg, {\it{Uniqueness theorems in the spectral theory of $y''+(\lambda -q(x))y=0$}}, Proc.~11th Scandinavian Congress of Mathematicians, Johan Grundt Tanums Forlag, Oslo, 1952, 276--287. \gap \item{\gl.} I.M.~Gel'fand and B.M.~Levitan, {\it{On the determination of a differential equation from its special function}}, Izv.~Akad.~Nauk SSR. Ser.~Mat. {\bf 15} (1951), 309--360 (Russian); English transl.~in Amer.~Math. Soc.~Transl.~Ser. (2) {\bf 1} (1955), 253--304. \gap \item{\gsac.} \ref{F.~Gesztesy and B.~Simon}{Inverse spectral analysis with partial information on the potential, I. The case of an a.c.~component in the spectrum}{Helv.~Phys.~Acta}{70}{1997}{66--71} \gap \item{\gsmf.} F.~Gesztesy and B.~Simon, {\it{$m$-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices}}, preprint, 1996. \gap \item{\gsds.} F.~Gesztesy and B.~Simon, {\it{Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum}}, preprint, 1997. \gap \item{\haldma.}\ref{O.H.~Hald}{Inverse eigenvalue problem for the mantle} {Geophys.~J.~R.~Astr.~Soc.}{62}{1980}{41--48} \gap \item{\hsturm.}\ref{H.~Hochstadt}{The inverse Sturm-Liouville problem} {Commun.~Pure Appl.~Math.}{26}{1973}{715--729} \gap \item{\hl.}\ref{H.~Hochstadt and B.~Lieberman}{An inverse Sturm-Liouville problem with mixed given data}{SIAM J.~Appl.~Math.} {34}{1978}{676--680} \gap \item{\levs.}\ref{N.~Levinson}{The inverse Sturm-Liouville problem} {Mat.~Tidskr.}{B}{1949}{25--30} \gap \item{\lev.}\ref{B.~Levitan}{On the determination of a Sturm-Liouville equation by two spectra}{Amer. Math. Soc.~Transl.} {68}{1968}{1--20} \gap \item{\levbook.} B.~Levitan, {\it{Inverse Sturm-Liouville Problems}}, VNU Science Press, Utrecht, 1987. \gap \item{\lg.} B.M.~Levitan and M.G.~Gasymov, {\it {Determination of a differential equation by two of its spectra}}, Russ.~Math.~Surv. {\bf 19:2} (1964), 1--63. \gap \item{\mal.}\ref{M.M.~Malamud}{Similarity of Volterra operators and related questions of the theory of differential equations of fractional order}{Trans. Moscow Math. Soc.}{55}{1994}{57--122} \gap \item{\malsp.} M.M.~Malamud, {\it{Spectral analysis of Volterra operators and inverse problems for systems of differential equations}}, preprint, 1997. \gap \item{\mar.} V.A.~Marchenko, {\it {Some questions in the theory of one-dimensional linear differential operators of the second order, I}}, Trudy Moskov.~Mat.~Ob\v s\v c. {\bf 1} (1952), 327--420 (Russian); English transl.~in Amer.~Math.~Soc.~Transl. (2) {\bf 101} (1973), 1--104. \gap \item{\simon.} B.~Simon, {\it{A new approach to inverse spectral theory, I. The basic formalism}}, in preparation. \enddocument