\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 145, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/145\hfil Zeros of the Jost function] {Zeros of the Jost function for a class of exponentially decaying potentials} \author[D. Gilbert, A. Kerouanton\hfil EJDE-2005/145\hfilneg] {Daphne Gilbert, Alain Kerouanton} % in alphabetical order \address{Daphne Gilbert \hfill\break School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, \hfill\break Dublin 8, Ireland} \email{daphne.gilbert@dit.ie} \address{Alain Kerouanton \hfill\break School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, \hfill\break Dublin 8, Ireland} \email{alainkerouanton@hotmail.com \quad Fax:\,+35314024994} \date{} \thanks{Submitted October 4, 2005. Published December 8, 2005.} \subjclass[2000]{34L40, 35B34, 35P15, 33C10} \keywords{Jost solution; Sturm-Liouville operators; resonances; eigenvalues; \hfill\break\indent spectral singularities} \begin{abstract} We investigate the properties of a series representing the Jost solution for the differential equation $-y''+q(x)y=\lambda y$, $x \geq 0$, $q \in \mathrm{L}({\mathbb{R}}^{+})$. Sufficient conditions are determined on the real or complex-valued potential $q$ for the series to converge and bounds are obtained for the sets of eigenvalues, resonances and spectral singularities associated with a corresponding class of Sturm-Liouville operators. In this paper, we restrict our investigations to the class of potentials $q$ satisfying $|q(x)| \leq ce^{-ax}$, $x \geq 0$, for some $a>0$ and $c>0$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} We consider the differential equation $$\label{sturm} -y''+q(x)y=\lambda y \quad \mbox{for }x \geq 0,$$ where $q \in \mathrm{L}({\mathbb{R}}^{+})$ is real or complex-valued, with the boundary condition $$\label{boundary} y(0)\cos(\alpha)+y'(0)\sin(\alpha)=0 \quad \mbox{for some }\alpha \in [0,\pi).$$ In this paper, we consider the consequences of changes on the potential $q$ rather than on the boundary condition \eqref{boundary} and we therefore restrict ourself to the classical case $\alpha \in [0,\pi)$. For an analysis of Sturm-Liouville operators with real valued, exponentially decaying potentials and nonselfadjoint boundary conditions, see for example \cite{hruscev}. Let $z=\sqrt{\lambda}$, $\mathop{\rm Im} (z) > 0$. Since $q \in \mathrm{L}({\mathbb{R}}^{+})$, there exists a unique $\mathrm{L}^{2}({\mathbb{R}}^{+})$-solution $\chi(x,z)$ of \eqref{sturm} satisfying $\chi(x,z)=e^{izx}(1+o(1)) \quad \mbox{as }x \to +\infty,$ which is known as the Jost solution \cite{freiling}. Let $\phi(x,z^{2})$ be the solution of \eqref{sturm} satisfying $\phi(0,z^{2})=0$, $\phi'(0,z^{2})=1$. Then $\phi(x,z^{2})$ satisfies \eqref{boundary} with $\alpha=0$ and we have $\mathrm{W}_{0}\left(\chi(x,z), \phi(x,z^{2})\right)=\chi(0,z), \quad \mathop{\rm Im} (z)>0,$ where $\mathrm{W}_{0}$ denotes the Wronskian evaluated at $x=0$. Note that $\phi(x,z^{2})$ and $\chi(x,z)$ are linearly dependent if and only if $\chi(0,z)=0$ for some $z$ such that $\mathop{\rm Im} (z)>0$. The non-zero eigenvalues of the operator $\mathrm{L}_{0}$ associated with \eqref{sturm} and the Dirichlet boundary condition are therefore of the form $\lambda=z^{2}$, where $z$ is a zero of the Jost function $\chi(z)=\chi(0,z)$ satisfying $\mathop{\rm Im} (z)>0$. If $q$ is real-valued these zeros are situated on the segment line $z=it,~00$ then, whether $q$ is real or complex-valued, the Jost function $\chi(z)$ can be analytically extended to the half plane $\{z \in \mathbb{C}: \mathop{\rm Im} (z)>-a/2\}$ \cite[appendix II]{naim:invest,naim:linear2} and the part of the expansion in generalised eigenfunctions related to the continuous spectrum contains a spectral-type function of the form $$\label{spectral_type} \frac{1}{\pi}\Big( \frac{z}{\chi(z)\chi(-z)} \Big), \quad z>0.$$ The expansion in eigenfunctions and generalised eigenfunctions in the case of exponentially decaying, complex-valued potentials was established by Naimark \cite{naim:invest}. If $q$ is real-valued the spectral-type function \eqref{spectral_type} is actually the spectral density associated with $\mathrm{L}_{0}$ since, in this case, $\chi(-z)=\overline{\chi(z)}$ for $\mathop{\rm Im} (z)=0$. The latter was proved by Kodaira \cite{kodaira} for a real-valued potential $q$. If we set $\chi_{\pi/2}(x,z)=\frac{d}{d x}\chi(x,z) \quad \mbox{ and } \quad \chi_{\pi/2}(z)=\chi_{\pi/2}(0,z),$ then the non-zero eigenvalues of the operator $\mathrm{L}_{\alpha}$ associated with \eqref{sturm} and \eqref{boundary} are of the form $\lambda=z^{2}$, where $z$ is a zero of $\chi_{\alpha}(z)$ satisfying $\mathop{\rm Im} (z)>0$, with $$\label{chi_alpha} \chi_{\alpha}(x,z)=\chi(x,z)\cos(\alpha)+\chi_{\pi/2}(x,z)\sin(\alpha) \quad \mbox{and} \quad \chi_{\alpha}(z)=\chi_{\alpha}(0,z).$$ To see this note that $\chi(x,z)$ and $\phi_{\alpha}(x,z^{2})$ are linearly dependent if and only if $\chi_{\alpha}(z)=0$ , where $\phi_{\alpha}(x,z^{2})$ is a solution of \eqref{sturm} satisfying \eqref{boundary}, more precisely $\phi_{\alpha}(0,z^{2})=-\sin(\alpha)$, $\phi'(0,z^{2})=\cos(\alpha)$. If $q$ satisfies \eqref{exp_decay}, then $\chi_{\alpha}(z)$ can be analytically extended to the half-plane $\{\mathop{\rm Im} (z) >-a/2\}$ \cite[appendix II]{naim:invest,naim:linear2}. It is then likely that the zeros of $\chi_{\alpha}(z)$ situated just below the real axis will affect the behaviour of \eqref{spectral_type} \cite{east:anti,froese,hitrik}. Such a zero is called a resonance and, if $q$ is real valued and if the zero is situated on the semi-axis $-i t$, $00$ so that the spectral singularities cannot be associated with $\mathrm{L}^{2}({\mathbb{R}}^{+})$-solutions of \eqref{sturm}. Moreover, if $q$ also satisfies \eqref{exp_decay}, then the number of spectral singularities is finite \cite[appendix II]{naim:invest,naim:linear2}. The literature available on the study of eigenvalues, resonances and spectral singularities is already abundant but we propose here an alternative method that allows us to view them as a single mathematical object, namely as arising from the zeros of the Jost function. Our method is relatively simple and allows us, in particular, to investigate resonance-free regions for exponentially decaying potentials. More detailed results are obtained on the set of resonances for compactly supported and super-exponentially decaying potentials in \cite{froese, hitrik} and in \cite{east:anti} for a class of exponentially decaying potentials. The relationship between the Jost function and the classical Titchmarsh-Weyl function is briefly outlined in section 5. \section{The Series} It was shown by Eastham \cite{east:asympt,east:anti} that, for a real-valued integrable potential $q$, the Jost solution $\chi(x,z)$ can be represented in the form \eqref{east_series}. However, it is not difficult to show that the results below also hold when $q$ is complex-valued and integrable. We have $$\label{east_series} \chi(x,z)=e^{i xz} \Big( 1+ \sum_{n \geq 1}r_{n}(x,z) \Big),$$ with $$\label{east_series_terms} r_{0}(x,z)=1, \quad r_{n}(x,z)=\frac{i}{2z}\int_{x}^{+\infty}q(t)r_{n-1}(t,z)\left(1-e^{2i z(t-x)}\right)dt, \quad n \geq 1.$$ Also, $$\label{east_series_der} \frac{d}{d x}\chi(x,z)=e^{i xz} \Big( i z+ \sum_{n \geq 1}s_{n}(x,z) \Big),$$ with $$\label{east_series_der_terms} s_{n}(x,z)=-\frac{1}{2}\int_{x}^{+\infty} q(t)r_{n-1}(t,z)(1+e^{2i z(t-x)})dt \quad n \geq 1.$$ {}From \eqref{east_series_terms} we have \begin{gather*} r_{0}(x,z)=1, \\ r_{1}(x,z)=\frac{i}{2z}\int_{x}^{+\infty}q(t)\left(1-e^{2i z(t-x)}\right)dt \end{gather*} so that, for $\mathop{\rm Im} (z)>0$, $|r_{1}(x,z)| \leq \frac{1}{|z|}\int_{0}^{+\infty}|q(t)|dt.$ It is readily seen by induction on $n$ that $|r_{n}(x,z)| \leq \big( \frac{\|q\|_{1}}{|z|} \big)^{n}, \quad n \geq 0,\; x \geq 0,\; \mathop{\rm Im} (z)>0,$ where $\|\cdot\|_{1}$ is the $\mathrm{L}({\mathbb{R}}^{+})$-norm, from which it follows that $\big| 1+ \sum_{n \geq 1}r_{n}(x,z) \big| \leq \sum_{n \geq 0}\big( \frac{\|q\|_{1}}{|z|} \big)^{n}.$ The series in \eqref{east_series} therefore converges absolutely and uniformly for $x \geq 0$, $\mathop{\rm Im} (z)>0$ and $|z|>\|q\|_{1}$. Note that we supposed only that $q \in \mathrm{L}({\mathbb{R}}^{+})$. This result is similar to the one obtained by Rybkin \cite[theorem 3.1]{rybkin}. We now investigate the convergence of \eqref{east_series} for a class of exponentially decaying potentials. \section{Main Results} We suppose throughout this section that $$\label{exact_exp_decay} |q(x)|\leq ce^{-ax}, \quad x \geq 0,$$ holds for some $c>0$ and $a>0$. We first consider the case $\alpha=0$ and then examine the case $\alpha \in (0,\pi)$. In the latter case the details get rather cumbersome but, since we are aware of only few results concerning this case, we mention it anyway. Let $\delta > 0$ and let $\Lambda_{a,\delta}=\{ z \in \mathbb{C}: \mathop{\rm Im} (z)>-a/3,\, |z|>\delta \}.$ \begin{lemma}\label{lemma1} Suppose that \eqref{exact_exp_decay} holds and fix $\delta> 2c/a$. Then $|r_{n}(x,z)| \leq \frac{1}{n!}\big( \frac{2c}{|z| a} \big)^{n}e^{-nax}, \quad x \geq 0,~\mathop{\rm Im} (z) >-a/3,~n\geq 1$ and the series \eqref{east_series} converges absolutely and uniformly for $x \geq 0$, $z \in \Lambda_{a,\delta}$. \end{lemma} \begin{proof} We first prove by induction that $|r_{n}(x,z)| \leq \frac{1}{n!}\left(\frac{c}{|z| a}\right)^{n}\left( \frac{a+\mathop{\rm Im} (z)}{a+2\mathop{\rm Im} (z)} \right) \dots \left( \frac{na+\mathop{\rm Im} (z)}{na+2\mathop{\rm Im} (z)}\right)e^{-nax}, \quad n \geq 1.$ According to \eqref{east_series_terms} we have $r_{0}(x,z)=1$ and, from \eqref{east_series_terms} and \eqref{exact_exp_decay}, $r_{1}(x,z) \leq \frac{c}{2|z|}\int_{x}^{\infty} \left( e^{-at}+e^{-t(a+2\mathop{\rm Im}(z))+2x\mathop{\rm Im} (z)} \right)dt,$ which yields $|r_{1}(x,z)| \leq \frac{c}{a|z|} \left( \frac{a+\mathop{\rm Im} (z)}{a+2\mathop{\rm Im} (z)} \right)e^{-ax}.$ The result is therefore true for $n=1$. Suppose that it were true for $1 \leq k \leq n-1$, $n \geq 2$. According to \eqref{east_series_terms} we have $|r_{n}(x,z)| \leq \frac{1}{2|z|}\int_{x}^{\infty}|q(t)r_{n-1}(t,z)|\left( 1+e^{-2(t-x) \mathop{\rm Im} (z)}\right)dt,$ so that, from \eqref{exact_exp_decay} and the induction hypothesis, \begin{align*} |r_{n}(x,z)| \leq& \frac{c}{2|z|(n-1)!}\left( \frac{c}{|z|a}\right)^{n-1} \left( \frac{a+\mathop{\rm Im} (z)}{a+2\mathop{\rm Im} (z)} \right) \times\dots\\ &\times \left( \frac{(n-1)a+\mathop{\rm Im} (z)}{(n-1)a+2\mathop{\rm Im}(z)} \right)\int_{x}^{+\infty}e^{-nat}(1+e^{-2(t-x)\mathop{\rm Im} (z)})dt, \end{align*} which yields \begin{align*} &|r_{n}(x,z)|\\ & \leq \frac{1}{n!}\left(\frac{c}{|z| a}\right)^{n}\left( \frac{a+\mathop{\rm Im} (z)}{a+2\mathop{\rm Im} (z)} \right) \dots \left( \frac{(n-1)a+\mathop{\rm Im} (z)}{(n-1)a+2\mathop{\rm Im} (z)} \right) \left( \frac{na+\mathop{\rm Im} (z)}{na+2\mathop{\rm Im} (z)}\right)e^{-nax}, \end{align*} as required. The lemma is proved when we notice that $0< \frac{na+\mathop{\rm Im}(z)}{na +2\mathop{\rm Im} (z)} <2, \quad n \geq 1, \quad \mbox{and} \quad \frac{2c}{|z|a}<\frac{2c}{\delta a}<1$ if $\mathop{\rm Im} (z)>-a/3$ and $|z|>\delta > 2c/a$. \end{proof} We are now in position to identify a region in the $z$-plane where $\chi(z)$ cannot vanish. \begin{theorem}\label{thm1} Suppose \eqref{exact_exp_decay} holds and fix $\delta>2c/a$. Then, for $z \in \Lambda_{a,\delta}$, $|\chi(z)| \geq 2-\exp\left( \frac{2c}{\delta a} \right)$ In particular, if $\delta > \frac{2c}{a \ln(2)},$ then $\chi(z)$ cannot vanish inside the set $\Lambda_{a,\delta}$ and the operator $\mathrm{L}_{0}$ has \begin{itemize} \item[(i)] no eigenvalue $\lambda=z^{2}$ such that $z \in \Lambda_{a,\delta} \cap \{z:\mathop{\rm Im} (z)>0\}$, \item[(ii)] no spectral singularity $\lambda=z^{2}$ such that $z \in (-\infty,\delta)\cup(\delta,+\infty)$, \item[(iii)] no resonance inside $\Lambda_{a,\delta} \cap \{z:\mathop{\rm Im} (z)<0\}$. \end{itemize} \end{theorem} \begin{proof} According to lemma \ref{lemma1} we have, for $z \in \Lambda_{a,\delta}$, $|r_{n}(x,z)| \leq \frac{1}{n!}\left( \frac{2c}{\delta a} \right)^{n}e^{-nax},\quad x \geq 0,$ so that $\big| \sum_{n \geq 1} r_{n}(x,z) \big| \leq \sum_{n \geq 1}\frac{1}{n!}\left( \frac{2c}{\delta a} \right)^{n}e^{-nax}=\exp\left( \frac{2c}{\delta a}e^{-ax}\right)-1.$ Since $|\chi(x,z)|=e^{-x \mathop{\rm Im} (z)} \Big|1+\sum_{n \geq 1}r_{n}(x,z)\Big| \geq e^{-x \mathop{\rm Im} (z)}\Big\{1-\Big| \sum_{n \geq 1}r_{n}(x,z) \Big| \Big\},$ we obtain $|\chi(z)| \geq 2-\exp\left( \frac{2c}{\delta a}\right).$ In particular, $\chi(z)$ does not vanish if $2-\exp\left( \frac{2c}{\delta a}\right)>0,$ i.e. if $\delta > \frac{2c}{a\ln(2)},$ from which $(i)$, $(ii)$ and $(iii)$ follow. \end{proof} Note that, under the hypotheses of theorem \ref{thm1}, if $\lambda=z^{2}$ is an eigenvalue of $\mathrm{L}_{0}$ then $z$ can only be located on the semi disk $\{z \in \mathbb{C}: |z| \leq \delta, \mathop{\rm Im} (z) >0\}$ and, if $q$ is real-valued, on the segment line $z=it,~02c/a$. Then $|s_{n}(x,z)| \leq \frac{|z|}{n!}\big( \frac{2c}{|z| a} \big)^{n}e^{-nax},\quad x \geq 0, \quad \mathop{\rm Im}(z)>-a/3,\quad n \geq 1$ and the series \eqref{east_series_der} converges absolutely and uniformly for $x \geq 0$, $z \in \Lambda_{a,\delta}$. \end{lemma} \begin{proof} From \eqref{east_series_terms}, \eqref{east_series_der} and \eqref{east_series_der_terms}, we have $\frac{d}{d x}\chi(x,z)=e^{i zx}\Big(iz+\sum_{n \geq 1} s_{n}(x,z) \Big)$ and $|s_{n}(x,z)| \leq \frac{|z|}{2|z|}\int_{x}^{+\infty}|q(t)r_{n-1}(t,z)|\left(1+e^{-2\mathop{\rm Im} (z) (t-x)} \right)dt, \quad n \geq 1.$ Arguing as in lemma \ref{lemma1}, we obtain the stated result. \end{proof} The bounds we obtain for $\alpha \in (0,\pi/2) \cup (\pi/2,\pi)$ are not as tight as the ones obtained in theorem \ref{thm1}, which is rather natural as, for $\alpha \in (0,\pi/2) \cup (\pi/2,\pi)$, it is possible to find resonances far below the real axis or large eigenvalues, depending on the value of $\alpha$. We refer to the first example in the next section for an illustration of this phenomenon. \begin{theorem}\label{thm2} Suppose that \eqref{exact_exp_decay} holds and let $\delta$ be such that $\delta > \frac{2c}{a\ln(2)}.$ Then $(i)$, $(ii)$ and $(iii)$ of theorem \ref{thm1} hold as they stand for the operator $\mathrm{L}_{\pi/2}$ and $(i)$, $(ii)$ and $(iii)$ of theorem \ref{thm1} continue to hold for the operator $\mathrm{L}_{\alpha}$, $\alpha \in (0,\pi/2)\cup(\pi/2,\pi)$, provided we replace $\delta$ by $\max\{\delta, \delta_{\alpha}\}$, where $\delta_{\alpha} = |\cot(\alpha)| \frac{\exp\left(\frac{2c}{\delta a}\right)}{2-\exp\big(\frac{2c}{\delta a}\big)}.$ \end{theorem} \begin{proof} We first suppose that $\alpha=\pi/2$. According to \eqref{chi_alpha}, \eqref{east_series_der} and lemma \ref{lemma2} we have, for $z \in \Lambda_{a,\delta}$, $$\label{bound_Lpi/2} |\chi_{\pi/2}(z)| \geq |z|-|z|\Big\{\exp\big( \frac{2c}{\delta a}\big)-1 \Big\} =|z|\Big\{2-\exp\big(\frac{2c}{\delta a}\big)\Big\}.$$ It follows that $\chi_{\pi/2}(z)$ cannot vanish inside $\Lambda_{a,\delta}$ if $\delta > 2c/a\ln(2)$, and the first part of the theorem is proved.\\ Suppose now that $\alpha \in (0,\pi/2)\cup(\pi/2,\pi)$. From \eqref{east_series} and lemma \ref{lemma1} we get $|\chi(z)| \leq 1+ \sum_{n \geq 1} |r_{n}(0,z)| \leq \exp\left( \frac{2c}{\delta a} \right).$ On the other hand, according to \eqref{chi_alpha}, $|\chi_{\alpha}(z)| \geq |\sin(\alpha)\chi_{\pi/2}(z)|-|\cos(\alpha)\chi(z)|$ so that, with \eqref{bound_Lpi/2}, we obtain $|\chi_{\alpha}(z)| \geq |z\sin(\alpha)|\big\{2-\exp\big(\frac{2c}{\delta a}\big)\big\}-|\cos(\alpha)|\exp\big(\frac{2c}{\delta a}\big).$ {}From the equality above, it is not hard to see that $\chi_{\alpha}(z)>0$ for $|z|> |\cot(\alpha)| \frac{\exp\left(\frac{2c}{\delta a}\right)}{2-\exp\big(\frac{2c}{\delta a}\big)},$ from which the last part of the theorem follows. \end{proof} Let $\delta'=\max\left\{\delta,\delta_{\alpha} \right\}$. Under the hypotheses of theorem \ref{thm2}, the eigenvalues $\lambda=z^{2}$ must be such that $z \in \{z \in \mathbb{C} : |z| \leq \delta' \}$, the resonances situated on $\{z \in \mathbb{C} : -a/3<\mathop{\rm Im} (z)<0\}$ must be inside the set $\{z \in \mathbb{C} : -a/3<\mathop{\rm Im} (z)<0, |z| \leq \delta'\}$ and the spectral singularities $\lambda=z^{2}$ must satisfy $-\delta'< z < \delta'$. \section{Examples} \subsection*{The case $q \equiv 0$} Let $q \equiv 0$ in \eqref{sturm}. Then the Jost solution is $\chi(x,z)=e^{i zx}$ so that $\chi_{\alpha}(x,z)=\cos(\alpha)e^{izx}+iz\sin(\alpha)e^{i zx}, \quad \alpha \in (0,\pi).$ Hence the only zero of $\chi_{\alpha}(z)$ is \begin{itemize} \item $z=0$ if $\alpha=\pi/2$ \item $z_{\alpha}=i \cot(\alpha)$ if $\alpha \in (0,\pi/2)\cup(\pi/2,\pi)$. \end{itemize} If $\alpha \in (0,\pi/2)$ then $\mathop{\rm Im} (z_{\alpha})>0$, so that $\lambda_{\alpha}=-\cot^{2}(\alpha)$ is an eigenvalue and, if $\alpha \in (\pi/2,\pi)$, then $\mathop{\rm Im} (z_{\alpha}) <0$ so that $z_{\alpha}=i\cot(\alpha)$ is a resonance. If we suppose that $\alpha$ is strictly complex then $z_{\alpha}=-\frac{\sinh(2\mathop{\rm Im}(\alpha))}{\cosh(2\mathop{\rm Im} (\alpha))-\cos(2\mathop{\rm Re}(\alpha))}+i\frac{\sin(2\mathop{\rm Re} (\alpha))}{\cosh(2\mathop{\rm Im}(\alpha))-\cos(2\mathop{\rm Re}(\alpha))},$ so that $\lambda_{\alpha}=z_{\alpha}^{2}$ is an eigenvalue if $\sin(2\mathop{\rm Re}(\alpha))>0$, and $z_{\alpha}$ is a resonance if $\sin(2\mathop{\rm Re}(\alpha))<0$ and $\lambda_{\alpha}=z_{\alpha}^{2}$ is a spectral singularity if $\sin(2\mathop{\rm Re}(\alpha))=0$. \subsection*{The Jost-Bessel function} If we take $q(x)=be^{-dx}$ in \eqref{sturm}, with $b,~d \in \mathbb{C}$ and $\mathop{\rm Re} (d)>0$, then it can be proved by induction \cite{kerouanton} that, in the notation of \eqref{east_series}, \begin{align*} \chi(x,z)&=e^{izx}\Big\{1+\sum_{n \geq 1}r_{n}(x,z)\Big\}\\ &= e^{izx} \Big\{ 1+\sum_{n \geq 1}\frac{(bd^{-2}e^{-dx})^{n}}{n!}\left(\frac{1}{(1-2i z/d)} \dots \frac{1}{(n-2i z/d)} \right) \Big\}. \end{align*} This formula for the Jost solution is independently confirmed in \cite{east:anti}, where it is noted that when $q$ is real valued, \eqref{sturm} is satisfied by the Bessel function $\mathrm{J}_{-2i z/d}\left\{ (2i d^{-1}\sqrt{b})e^{-dx/2} \right\},$ which is in $\mathrm{L}^{2}({\mathbb{R}}^{+})$ for $\mathop{\rm Im} (z)>0$ (see also \cite[\S 4.14]{titch1} and \cite[\S 2.13]{watson}).\\ If $d>0$ and $b>0$, then as in \cite{east:anti} $\mathrm{L}_{0}$ had no eigenvalues and also no antibound states in the segment line $z=i t$, $-d/2-1/3, |z|>\delta\}$, so that the estimate obtained in the last example is consistent with the bound obtained in theorem \ref{thm1}. Note that the bounds obtained in theorem \ref{thm1} with $a=1$ and $c=1$ also apply, for example, to the complex valued potential $q(x)=\frac{x-i}{x+i}e^{(-1+2i)x}.$ \section{Jost function and Titchmarsh-Weyl function} We suppose in the first instance that $q\in \mathrm{L}({\mathbb{R}}^{+})$ is real valued and give a brief account of the relationship between the Jost function and the Titchmarsh-Weyl function, since the eigenvalues and more generally the spectrum of the operator $\mathrm{L}_{\alpha}$ have traditionally been studied using the properties of the Titchmarsh-Weyl function $m_{\alpha}(\lambda)$. Let $\phi_{\alpha}(x,\lambda)$ be defined as above and let $\theta_{\alpha}(x,\lambda)$ be the solution of \eqref{sturm} satisfying $\theta_{\alpha}(0,\lambda)=\cos(\alpha), \quad \theta_{\alpha}'(0,\lambda)=\sin(\alpha).$ Since Weyl's limit-point case applies at $+\infty$, it is known that there exists a unique linearly independent $\mathrm{L}^{2}({\mathbb{R}}^{+})$-solution $\psi_{\alpha}$ of \eqref{sturm} such that $\psi_{\alpha}(x,\lambda)=\theta_{\alpha}(x,\lambda)+m_{\alpha}\phi_{\alpha}(x,\lambda), \quad x \geq 0, ~\mathop{\rm Im} \lambda>0,$ which is known as the Weyl solution \cite{titch1}. The function $m_{\alpha}(\lambda)$ is analytic in the upper half plane $\{\lambda \in \mathbb{C}:~ \mathop{\rm Im} (\lambda)>0\}$ and satisfies $\mathop{\rm Im} \left(m_{\alpha}(\lambda) \right)>0 \quad \mbox{ for \mathop{\rm Im} (\lambda)>0},$ so that $\lim_{\mathop{\rm Im} \lambda \to 0+}m_{\alpha}(\lambda)$ exists and is finite Lebesgue almost everywhere. The eigenvalues of $\mathrm{L}_{\alpha}$ are the poles of $m_{\alpha}$. On the other hand, it is readily seen that $\chi(x,z)=\mathrm{W}_{0}( \chi,\phi_{\alpha})\theta_{\alpha}(x,z^{2}) +\mathrm{W}_{0}(\theta_{\alpha},\chi)\phi_{\alpha}(x,z^{2}), \quad \mathop{\rm Im} (z)>0,$ so that we have formally $\psi_{\alpha}(x,z^{2})=\frac{1}{\mathrm{W}_{0}( \chi,\phi_{\alpha})}\chi\left(x, z \right).$ It follows that $$\label{m_chi} m_{\alpha}(z^{2})=\frac{\mathrm{W}_{0}(\theta_{\alpha},\chi)}{\mathrm{W}_{0}( \chi,\phi_{\alpha})}=\frac{\mathrm{W}_{0}(\theta_{\alpha},\chi)}{\chi_{\alpha}(z)}, \quad \mathop{\rm Im} (z)>0, \quad \mathop{\rm Re} (z)>0$$ and the poles of $m_{\alpha}(z^{2})$ are the zeros of $\chi_{\alpha}(z)$. Since $\mathrm{W}_{0}(\theta_{\alpha},\chi)$ and $\chi_{\alpha}(z)$ are analytic in the upper half plane $\{z \in \mathbb{C}:\mathop{\rm Im} (z)>0\}$, we can analytically extend $m_{\alpha}(\lambda)$ using \eqref{m_chi}. The extended Titchmarsh-Weyl function is meromorphic on $\mathbb{C} \setminus [0,+\infty)$. If $q \in \mathrm{L}({\mathbb{R}}^{+})$ is allowed to be complex valued and if $\mathop{\rm Im} (q) \leq 0$, a similar situation prevails \cite{sims} and we can construct a Titchmarsh-Weyl function which is analytic on $\{\lambda \in \mathbb{C}: \mathop{\rm Im} (\lambda)>0 \}$ and can be analytically extended to a function meromorphic on $\mathbb{C} \setminus [0,+\infty)$. For additional information and references on the relationship between the Jost solution and the Titchmarsh-Weyl function, we refer to \cite{hruscev}. \begin{thebibliography}{00} \bibitem{east:asympt} M. S. P. Eastham, The Asymptotic Solution of Linear Differential Systems, Applications of the Levinson Theorem, London Mathematical Society Monographs New Series, Clarendon Press, Oxford, 1989. \bibitem{east:anti} M. S. P. Eastham, Antibound states and exponentially decaying Sturm-Liouville potentials, J. London Math. Soc. (2) 65 (2002) 624-638. \bibitem{freiling} G. Freiling, V. Yurko, Inverse Sturm-Liouville Problems and their Applications, Nova Science Publishers, New York, 2001. \bibitem{froese} R. Froese, Asymptotic distribution of resonances in one dimension, Journal of Differential Equations 137 (1997) 251-272. \bibitem{hitrik} M. Hitrik, Bounds on scattering poles in one dimension, Comm. Math. Phys. 208 (1999) 381-411. \bibitem{hruscev} S. V. Hru\v{s}\v{c}ev, Spectral singularities of dissipative Schr\"{o}dinger operators with rapidly decaying potentials, Indiana University Math. Journal Vol. 33 No. 4 (1984) 613-638. \bibitem{kerouanton} A. Kerouanton, Self- and nonself-adjoint Sturm-Liouville operators with exponentially decaying potentials, PhD thesis, Dublin Institute of Technology (2005). \bibitem{kodaira} K. Kodaira, The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices, Amer. J. Math. 71 (1949) 921-945. \bibitem{naim:invest} M. A. Naimark, Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, American Mathematical Translation Series 2 Volume 16 (1966). \bibitem{naim:linear2} M. A. Naimark, Linear Differential Operators, Part II, Harrap, London, 1968. \bibitem{rybkin} A. Rybkin, Some new and old asymptotic representations of the Jost solution and the Weyl m-function for Schr\"{o}dinger operators on the line, Bull. London Math. Soc. 34 (2002) 61-72. \bibitem{sims} A. R. Sims, Secondary conditions for linear differential operators of the second order, Journal of Mathematics and Mechanics, Vol. 6, No. 2 (1957). \bibitem{titch1} E. C. Titchmarsh, Eigenfunction Expansions, Part I, Second Edition, Clarendon, Oxford, 1962. \bibitem{watson} G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1962. \end{thebibliography} \end{document}