\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 182, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/182\hfil Reconstructing the potential function] {Reconstructing the potential function for indefinite Sturm-Liouville problems using infinite product forms} \author[M. Dehghan, A. Jodayree \hfil EJDE-2013/182\hfilneg] {Mohammad Dehghan, Ali Asghar Jodayree } % in alphabetical order \address{Mohammad Dehghan \newline Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran} \email{m-dehghan@tabrizu.ac.ir, Tel. +989113512619} \address{Ali Asghar Jodayree \newline Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran} \email{akbarfam@yahoo.com} \thanks{Submitted July 15, 2013. Published August 7, 2013.} \subjclass[2000]{34B24, 34A55, 34E20, 34E05} \keywords{Indefinite Sturm-Liouville problem; turning point; dual equations; \hfill\break\indent infinite product form} \begin{abstract} In this article we consider the linear second-order equation of Sturm-Liouville type $$y''+(\lambda\phi^2(t)-q(t))y=0, \quad 0\leq t\leq 1,$$ where $\lambda$ is a real parameter, $q(t)$ is the potential function and $\phi^2(t)$ is the weight function. We use the infinite product representation of the derivative of the solution to the differential equation with Dirichlet-Neumann conditions, and for the system of dual equations which is needed for expressing inverse problem and for retrieving potential. It must be mentioned that the weight function has a zero whose order is an integer called a \emph{turning point}. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction}\label{intro} We consider the indefinite Sturm-Liouville equation $$\label{orig} ly:=-y''+q(t)y=\lambda \phi^{2}(t)y, \quad 0\leq t\leq x,$$ with Dirichlet conditions $$\label{con1} y(0)=y(x)=0,$$ and with Dirichlet-Neumann conditions $$\label{con2} y(0)=y'(x)=0,$$ where $\lambda=\rho^2$ is the spectral parameter, $x$ is a fixed point in the interval $(0,1)$ and also the weight function $\phi^{2}(t)$ and the potential function $q(t)$ satisfies \begin{itemize} \item $\phi^{2}(t)=(t-t_0)^{l_0}\phi_0(t)$ is real and has one zero, $t_0$, so called turning point of odd order $l_0\in \mathbf{N}$ in $[0,1]$ and also $\phi_0(t)$ is positive and twice continuously differentiable. \item $q(t)$ is bounded and integrable on $[0,1]$. \end{itemize} The asymptotic solutions of \eqref{orig} depend on a complex parameter $\rho$ as $|\rho|\to \infty$. We assume $t_0$ to be a turning point of type IV; i.e., $l_0$ is odd. The operator $l$ defined in \eqref{orig} is called the indefinite Sturm-Liouville operator. The differential equation \eqref{orig} with conditions \eqref{con1} and \eqref{con2} are denoted by $L_1(\phi^2(t),q(t),x)$ and $L_2(\phi^2(t),q(t),x)$, respectively. Inverse spectral theory can be considered as the determination of an operator (usually differential) from its spectral data. The literature on such subjects is immense for the definite cases in which the weight function $\phi^2(t)$ is positive throughout the interval such as \cite{levitan,rund,koy,ozk}, while for the indefinite cases is not. The transformation operator method and the Gelfand-Levitan integral equation with respect to the kernel of the transformation operator \cite{kach} in this case is not suitable for the solution of the inverse problems. The eigenvalue problem for the indefinite Sturm-Liouville problem has been discussed in \cite{kong}. The potential $q(x)$ of an indefinite Sturm-Liouville problem has been determined uniquely by three spectra in {\cite{fu}}. We present a new approach to reconstruct the operator $l$ (indefinite Sturm-Liouville operator); i.e., retrieving potential function $q(t)$ in \eqref{orig} from its spectral data. The question at the core of this paper involves the determination of the infinite product representation for the derivative of the solution of the indefinite Sturm-Liouville problem as well as the reconstruction of the potential function $q(t)$ by means of two spectra, while the weight function $\phi^2(t)$ is given. Differential equations with an indefinite weight function appear in several mathematical physics problems. For instance, turning points correspond to the limit of motion of a wave mechanical particle bounded by a potential field. Turning points arise also in various fields such as optics, elasticity, spectroscopy, stratification and radio engineering problems to design directional couplers for non-uniform electronic lines (see \cite{daho,lit,mchugh,mesh,sve,wasow} for further references). The presence of turning points yields fundamental qualitative changes in the study of this kind of differential equation. In problem $L_1(\phi^2(t),q(t),x)$, for the special case $\phi^2(t)=t$ in the interval $[-1,1]$, Jodayree et al. obtained the infinite product representation of the solution in the closed form \cite{jodayree1}: $U(x,\lambda)=\begin{cases} \frac{p(x)}{(-x)^{1/4}}\prod_{k\geq 1}\frac{\lambda-\lambda_k(x)}{z_k^2(x)}, &-1\leq x<0, \\[4pt] \frac{\pi\sqrt{x}}{6}\prod_{k\geq 1}\frac{(\lambda-r_k(x))p^2(0)}{\tilde{j}^2_k} \prod_{k\geq 1}\frac{f^2(x)(u_k(x)-\lambda)}{\tilde{j}^2_k}, & 00. Finding the solution in the infinite product form led to construct the dual equations which are necessary to retrieve the potential function q(t) in the inverse problem \cite{jodayree2}. Barcilon \cite{barcilon} introduced \{\lambda_n(x)\} and \{\mu_n(x)\} as the eigenvalues of classical Sturm-Liouville equation (vibrating string equation), $$\label{string} y''+\lambda\phi^2(t)y=0, \quad x\leq t\leq L,$$ with conditions y(x)=y(L)=0 and y'(x)=y(L)=0, respectively, in where x is a fixed point in the interval (0,L). In contrast with problem \eqref{orig}, the function \phi^2(t) is positive throughout the interval (0,L). It has been shown that if u(t,\lambda) is the solution of equation \eqref{string} with the initial conditions u(L,\lambda)=0 and \frac{\partial u}{\partial t}(L,\lambda)=1, then by using Hadamard's factorization, for fixed x belonging to [0,L], it can be written \begin{gather*} u(x,\lambda)=-(L-x)\prod_{k=1}^{\infty}(1-\frac{\lambda}{\lambda_n(x)}), \\ u'(x,\lambda)=\prod_{k=1}^{\infty}(1-\frac{\lambda}{\mu_n(x)}), \end{gather*} which are the infinite product form of the solution and its derivative for vibrating string problem \cite{barcilon}. He also derived the dual equations of problem \eqref{string} in the form $$\label{dual:string} \begin{gathered} \frac{d\lambda_n(x)}{dx}=-\frac{\lambda_n(x)}{L-x} \frac{\prod_{k=1}^{\infty}(1-\frac{\lambda_n(x)}{\mu_k(x)})} {\prod_{k\neq n}^{\infty}(1-\frac{\lambda_n(x)}{\lambda_k(x)})}, \\ \frac{d\mu_n(x)}{dx}=\mu_n^2(x)\phi^2(x)(L-x) \frac{\prod_{k=1}^{\infty}(1-\frac{\mu_n(x)}{\lambda_k(x)})} {\prod_{k\neq n}^{\infty}(1-\frac{\mu_n(x)}{\mu_k(x)})}, \end{gathered}$$ with the initial condition \[ \lambda_n(0)=\lambda_n,\quad \mu_n(0)=\mu_n.$ In fact, the pair of sequences $(\lambda_n(0),\mu_n(0))$ suffices as data to guarantee the existence and uniqueness of function $\phi^2(t)$ in \eqref{string} {\cite{barcilon}}. Hence, by using the solution $(\lambda_n(x),\mu_n(x))$ of \eqref{dual:string}, one can construct the original equation \eqref{string}. For this reason, the equation \eqref{dual:string} is referred to as dual equation of \eqref{string} in the classical literature. Pranger \cite{pranger} studied the recovery of the function $\phi^2(t)$ from the eigenvalues in equation \eqref{string} with the Dirichlet boundary condition on the interval $[0,1]$, replacing $\{\mu_n\}$ by $\{\lambda'_n\}$ and introducing the infinite product form of the solution to construct the dual equation $\lambda''_n+\frac{2}{x}\lambda_n+2\lambda_n\lambda'_n\sum_{j\neq n}(\frac{\lambda'_j}{\lambda^2_j})(1-\frac{\lambda_n}{\lambda_j})^{-1} -2\frac{(\lambda'_n)^2}{\lambda_n}=0,$ where $\{\lambda_n\}$ are eigenvalues of equation \eqref{string} on the interval $[0,x]$, $00$ so that $\phi^2(t)\geq c$ for all $t$ and $\phi^2(t)\in C^2(0,L)$, then Equation \eqref{string} can be transformed into the canonical Sturm-Liouville equation \cite{hille} $y''+(\lambda-q)y=0.$ In section 2 we introduce some notation which we use throughout this article. In section 3 we find the infinite product form for the derivative of the solution of the indefinite Sturm-Liouville equation \eqref{orig} before and after the turning point at the interval $(0,1)$. The main results of the paper are expressed by theorems \ref{qphi} and \ref{theo}. The infinite product representation for the solution of problem \eqref{orig} in \cite{kheiri} and its derivative given here, enable us to construct the dual equations of this problem, in section 4, which this system of equations identifies the two spectra of eigenvalues for an arbitrary fixed point in the whole interval. Using these two spectra, one can retrieve the potential function $q(t)$ by the algorithm stated in the end of section 4. \section{Preliminaries} Let $\epsilon>0$ be fixed and sufficiently small, and let $D_\epsilon =[0,t_0-\epsilon]\cup [t_0+\epsilon,1]$. Further, we set $\mu=\frac{1}{2+l_0}$ ($l_0$ is order of turning point), $\lambda=\rho^2$ ($\rho$ is a complex parameter) and $\theta=4\mu$. We also denote \begin{gather*} I_+ =\{t: \phi^{2}(t)>0\},\quad I_- =\{t: \phi^{2}(t)<0\},\\ \xi(t)=\begin{cases} 0 & \text{for } t\in I_+(t), \\ 1 & \text{for } t\in I_-(t), \end{cases} \\ \phi^{2}_+(t)=\max (0,\phi^{2}(t)), \quad \phi^{2}_-(t)=\max (0,-\phi^{2}(t)), \\ K_\pm (t)=\begin{cases} 1 & \text{for } t\in I_-(t), \\ \frac{1}{2}\csc(\frac{\pi\mu}{2})\exp(\mp i\frac{\pi}{4}) & \text{for } t\in I_+(t), \end{cases}\\ K_\pm ^*(t)=\begin{cases} \pm i & \text{for } t\in I_-(t), \\ 2\sin(\frac{\pi\mu}{2})\exp(\pm i\frac{\pi}{4}) & \text{for } t\in I_+(t), \end{cases} \end{gather*} Let $S_k=\{\rho: \arg \rho \in [\frac{k\pi}{4},\frac{(k+1)\pi}{4}] \}, \quad k=0,1.$ Here the choice of the root $\phi$ of $\phi^2$ depends on the interval and the sector under consideration and has to be determined carefully. Due to the type of turning point $t_0$, we have \phi(t)=\begin{cases} |\phi(t)| & \text{for } t>t_0, \\ |\phi(t)| e^{ i\frac{\pi}{2}l_0} & \text{for } tt_0 is of the form \begin{gather} S(x,\lambda)=C_{1,1}(x)\prod_{n=1}^\infty(1-\frac{\lambda}{\lambda_n^-(x)}) \prod_{n=1}^\infty(1-\frac{\lambda}{\lambda_n^+(x)}),\label{prods01} \\ S'(x,\lambda)=C_{2,1}(x)\prod_{n=1}^\infty(1-\frac{\lambda}{\mu_n^-(x)}) \prod_{n=1}^\infty(1-\frac{\lambda}{\mu_n^+(x)}).\label{prods2} \end{gather} Index '1' in C_{r,1} (r=1,2) means that the fixed point x lies after turning point (x>t_0). The function C_{1,1}(x) has been estimated in \cite{kheiri}: \label{c11} \begin{aligned} C_{1,1}(x)&=\frac{1}{16}\pi|\phi(0)\phi(x)|^{-1/2}\csc(\frac{\pi \mu}{2})p(t_0)^{1/2}f(x)^{1/2} \\ &\quad \times\prod_{n=1}^{\infty}-\frac{\lambda_n^-(x)p^2(t_0)}{\tilde{j}_n^2} \prod_{n=1}^{\infty}\frac{\lambda_n^+(x)f^2(x)}{\tilde{j}_n^2}, \end{aligned} where p(x) and f(x) are defined in \eqref{p} and \eqref{f} respectively and \tilde{j}_n(n=1,2,\dots) are the positive zeros of the derivative of the Bessel function of first kind (J_1'(z)). Let J_\nu(z) and J'_\nu(z) be the Bessel function of order \nu and its derivative, respectively. From \cite{abram} we have \[ J_\nu(z)=\frac{(z/2)^\nu}{\Gamma(\nu+1)}\prod_{m=1}^\infty (1-\frac{z^2}{j_{\nu,m}^2}), where \begin{gather*} j_{\nu,m}\sim \beta-\frac{\alpha-1}{8\beta}-\frac{4(\alpha-1)(7\alpha-31)}{3(8\beta)^3} -\dots, \\ \beta=(m+\frac{\nu}{2}-\frac{1}{4})\pi,\quad \alpha=4\nu^2. \end{gather*} By inserting $\nu=0$, we can write $J_0(z)=\prod_{m=1}^\infty(1-\frac{z^2}{j_{0,m}^2}),$ where $j_{0,m}^2=m^2\pi^2-\frac{m\pi^2}{2}+O(1),\quad m=1,2,\dots,$ are the positive zeros of $J_0(z)$. Also, from \cite{abram}, We have $J'_\nu(z)=\frac{(z/2)^{\nu-1}}{2\Gamma(\nu)} \prod_{m=1}^\infty(1-\frac{z^2}{\tilde{j}_{\nu,m}^2}),\quad \nu>0,$ where \begin{gather*} \tilde{j}_{\nu,m}\sim\beta'-\frac{\alpha+3}{8\beta'} -\frac{4(7\alpha^2+82\alpha-9)}{3(8\beta')^3}-\dots, \\ \beta'=(m+\frac{\nu}{2}-\frac{3}{4})\pi,\quad \alpha=4\nu^2. \end{gather*} In reference to \cite{abram}, as a result of $J'_0(z)=-J_1(z)$ the zeros of $J_1(z)$ and $J'_0(z)$ are the same, namely $\tilde{j}_{0,m}=j_{1,m}$ for $m=1,2,\dots$. Therefore, we can write $J'_0(z)=-J_1(z)=-\frac{z}{2}\prod_{m=1}^\infty(1-\frac{z^2}{j_{1,m}^2}),$ where $j_{1,m}=(m+\frac{1}{4})\pi+\dots,\quad m=1,2,\dots.$ Replacing $m$ by $m-1$ in the previous relation we obtain \begin{gather*} j_{1,m-1}=(m-\frac{3}{4})\pi+\dots,\quad m=2,3,\dots, \\ j_{1,m-1}^2=m^2\pi^2-\frac{3}{2}m\pi^2+O(1),\quad m=2,3,\dots. \end{gather*} Consequently, $\frac{-j_{0,n}^2}{p^2(t_0)\mu_n^-(x)}=1+O(\frac{1}{n^2}), \quad \frac{j_{1,n-1}^2}{f^2(x)\mu_n^+(x)}=1+O(\frac{1}{n^2}).$ Therefore, the infinite products $\prod_{n=1}^\infty\frac{-j_{0,n}^2}{p^2(t_0)\mu_n^-(x)}$ and $\prod_{n=2}^\infty\frac{j_{1,n-1}^2}{f^2(x)\mu_n^+(x)}$ are absolutely convergent for each $x>t_0$. Then, from \eqref{prods2}, we may write $$S'(x,\lambda)=B_{2,1}(x)(1-\frac{\lambda}{\mu_1^+})\prod_{n=1}^\infty \frac{(\lambda-\mu_n^-(x))p^2(t_0)}{j_{0,n}^2} \prod_{n=2}^\infty\frac{(\mu_n^+(x)-\lambda)f^2(x)}{j_{1,n-1}^2},\label{prods1}$$ where $$\label{bc1} B_{2,1}(x)=C_{2,1}(x)\prod_{n=1}^\infty \frac{-j_{0,n}^2}{p^2(t_0)\mu_n^-(x)}\prod_{n=2}^\infty \frac{j_{1,n-1}^2}{f^2(x)\mu_n^+(x)}.$$ \begin{lemma}\label{lem2} Let $j_{0,m}$ be the positive zeros of $J_0(z)$ and for fixed $x$ in $(t_0,1)$ $\mu_m^-(x)=-\frac{m^2\pi^2}{p^2(t_0)}+\frac{3}{2}\frac{m\pi^2}{p^2(t_0)}+O(1), m\geq1,$ be a negative sequence of continuous functions. The infinite product $\prod_{m=1}^\infty\frac{(\lambda-\mu_m^-(x))p^2(t_0)}{j_{0,m}^2}$ is an entire function of $\lambda$ for fixed $x$, whose roots are precisely $\mu_m^-(x)$, $m\geq1$. Moreover, $\prod_{m=1}^\infty\frac{(\lambda-\mu_m^-(x))p^2(t_0)}{j_{0,m}^2} =J_0(i\sqrt{\lambda}p(t_0))(1+O(\frac{\log n}{n})),$ uniformly on the circles $|\lambda|=\frac{n^2\pi^2}{p^2(t_0)}$. \end{lemma} \begin{proof} This follows from using the method of the proof of lemma \ref{lem1}. For more details, see \cite{jodayree1}. \end{proof} \begin{lemma}\label{lem3} Let $j_{1,m}$ be the positive zeros of $J_1(z)$ and for fixed $x$ in $(t_0,1)$ $\mu_m^+(x)=\frac{m^2\pi^2}{f^2(x)}-\frac{m\pi^2}{2f^2(x)}+O(1),\quad m\geq1,$ be a positive sequence of continuous functions. Then, the infinite product $\prod_{m=2}^\infty\frac{(\mu_m^+(x)-\lambda)f^2(x)}{j_{1,m-1}^2}$ is an entire function of $\lambda$ for fixed $x$, whose roots are precisely $\mu_m^+(x)$, $m\geq1$. Moreover, $\prod_{m=2}^\infty\frac{(\mu_m^+(x)-\lambda)f^2(x)}{j_{1,m-1}^2} =-\frac{2}{\sqrt{\lambda}f(x)}J'_0(\sqrt{\lambda}f(x)) (1+O(\frac{\log n}{n})),$ uniformly on the circles $|\lambda|=\frac{n^2\pi^2}{p^2(t_0)}$. \end{lemma} \begin{proof} This follows from using the method of the proof of lemma $\ref{lem1}$. For more details, see \cite{jodayree1}. \end{proof} \begin{theorem}\label{teo2} Let $S'(t,\lambda)$ be the derivative of the solution of problem \eqref{orig} in association with initial condition \eqref{sinitial}. Then, for each fixed $x>t_0$, \begin{align*} S'(x,\lambda) &=-\frac{1}{4}|\phi(0)|^{-1/2}|\phi(x)|^{1/2}i^{1/2}\pi \mu_1^+(x) e^{i\frac{\pi}{4}}\csc(\frac{\pi\mu}{2})f^{3/2}(x)p^{1/2}(t_0)\\ &\quad \times\prod_{n=1}^\infty -\frac{\mu_n^-(x)p^2(t_0)}{j^2_{0,n}}\prod_{n=2}^\infty \frac{\mu^+_n(x)f^2(x)}{j^2_{1,n-1}} \prod_{n=1}^\infty (1-\frac{\lambda}{\mu_n^-(x)})\prod_{n=2}^\infty (1-\frac{\lambda}{\mu_n^+(x)}), \end{align*} where $p(x)$ and $f(x)$ is defined in \eqref{p} and \eqref{f}. Sequences $\{\mu_n^+(x)\}$ and $\{\mu_n^-(x)\}$ represent the positive and negative eigenvalues of $L_2(\phi^2(t),q(t),x)$, respectively and $j_{\nu,n} (\nu=0,1)$ are the positive zeros of $J_{\nu}(z)$. \end{theorem} \begin{proof} Using \eqref{asym2} for $t_00$ in $[0,t_0)\cup(t_0,1]$; i.e., $t_0$ is a turning point of type IV and the sequences $\{\lambda_n^-\}$, $\{\lambda_n^+\}$, $\{\mu_n^-\}$ and $\{\mu_n^+\}$ satisfy the following relations: \begin{gather*} \sqrt{\lambda_n^+}=\frac{n\pi-\frac{\pi}{4}}{\int_{t_0}^1|\phi(\tau)|d\tau} +O(\frac{1}{n}),\quad \sqrt{-\lambda_n^-}=\frac{n\pi-\frac{\pi}{4}}{\int^{t_0}_0|\phi(\tau)|d\tau} +O(\frac{1}{n}), \\ \sqrt{\mu_n^+}=\frac{n\pi-3\frac{\pi}{4}}{\int_{t_0}^1|\phi(\tau)|d\tau} +O(\frac{1}{n}),\quad \sqrt{-\mu_n^-}=\frac{n\pi-\frac{\pi}{4}}{\int^{t_0}_0|\phi(\tau)|d\tau)} +O(\frac{1}{n}). \end{gather*} \begin{enumerate} \item By solving the dual equation \eqref{dual1} with initial conditions $$\label{initial1} \lambda^-_n(1)=\lambda^-_n,\quad \lambda^+_n(1)=\lambda^+_n,\quad \mu^-_n(1)=\mu^-_n,\quad \mu^+_n(1)=\mu^+_n,$$ we find $\lambda_n^-(x)$, $\lambda_n^+(x)$, $\mu_n^-(x)$ and $\mu_n^+(x)$ for $x\in (t_0,1)$. \item Calculate $q(x)=\frac{C'_{2,1}(x)}{C_{1,1}(x)}$ where $C_{1,1}(x)$ and $C_{2,1}(x)$ are defined in \eqref{c11} and \eqref{c21}, respectively. \item By solving the dual equation \eqref{dual0} with initial conditions $$\label{initial2} \lambda_n(t_0)=\lim_{x\to t^+_0}\lambda_n^-(x),\quad \mu_n(t_0)=\lim_{x\to t^+_0}\mu_n^-(x),$$ we find $\lambda_n(x)$ and $\mu_n(x)$ for $x\in (0,t_0)$. \item Calculate $q(x)=\frac{C'_{2,0}(x)}{C_{1,0}(x)}$ where $C_{1,0}(x)$ and $C_{2,0}(x)$ are defined in \eqref{c10} and \eqref{c20}, respectively. \end{enumerate} \begin{remark}\label{rmk1} \rm It is obvious that the system of equations \eqref{dual0} are dual equations for indefinite Sturm-Liouville equation \eqref{orig}, corresponds to the system of equations \eqref{dual:string} in the classic Sturm-Liouville case (vibrating string). It means that the classical result is a particular case of our result; i.e., by inserting $q(x)\equiv 0$, $C_{1,0}(x)=-(L-x)$ and $C_{2,0}(x)=1$ in \eqref{dual0}, one can obtain \eqref{dual:string}. We can use the method stated in \cite{barcilon} to show that the systems of equations \eqref{dual0} and \eqref{dual1} with initial conditions \eqref{initial2} and \eqref{initial1}, respectively, satisfy the Lipschitz condition which guarantees the existence of a unique solution to the initial value problem. \end{remark} \begin{proposition} \label{prop1} Putting \eqref{q} in \eqref{phi} for \$0