\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 314, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/314\hfil Non-definite Sturm-Liouville problems]
{Complex oscillations of non-definite Sturm-Liouville problems}

\author[M. Kikonko, A. B. Mingarelli \hfil EJDE-2016/314\hfilneg]
{Mervis Kikonko, Angelo B. Mingarelli}

\address{Mervis Kikonko \newline
Department of Engineering Sciences and Mathematics,
Lule{\aa}  University of  Technology,
SE-971 87 Lule{\aa}, Sweden. \newline
Department of Mathematics and Statistics,
University of Zambia, P.O. Box 32379 Lusaka, Zambia}
\email{mervis.kikonko@ltu.se, mervis.kikonko@unza.zm}

\address{Angelo B. Mingarelli \newline
School of Mathematics and Statistics,
Carleton University, Ottawa, ON, Canada, K1S 5B6}
\email{angelo@math.carleton.ca}

\thanks{Submitted September 12, 2016. Published December 10, 2016.}
\subjclass[2010]{34C10, 34B25}
\keywords{Sturm-Liouville; non-definite; indefinite; spectrum; oscillation;
\hfill\break\indent  Dirichlet problem; turning point}

\begin{abstract}
 We expand upon the basic oscillation theory for general boundary problems
 of the form
 $$-y''+q(t)y=\lambda r(t)y, \quad t \in I = [a,b]
 $$
 where $q$ and $r$ are real-valued piecewise continuous functions
 and $y$ is required to satisfy a  pair of homogeneous separated boundary 
 conditions at the end-points.
 The \emph{non-definite case} is characterized by the indefiniteness of
 each of the quadratic forms
 $$
 B+\int_a^b (|y'|^2 +q|y|^2)\quad \text{and}\quad \int_a^b r|y|^2,
 $$
 over a suitable space where $B$ is a boundary term.
 In 1918 Richardson proved that, in the case of the Dirichlet problem,
 if $r(t)$ changes its sign exactly once and the boundary problem is
 non-definite then the zeros of the real and imaginary parts of any non-real
 eigenfunction interlace. We show that, unfortunately, this result is false
 in the case of two turning points, thus removing any hope for a general
 separation theorem for the zeros of the non-real eigenfunctions. Furthermore,
 we show that when a non-real eigenfunction vanishes inside $I$,  the absolute
 value of the difference between the total number of zeros of its real and
 imaginary parts is exactly 2.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

We are concerned here with Sturm-Liouville problems of the form
\begin{equation}
\label{Eqn1}
-y''+q(t)y=\lambda r(t)y
\end{equation}
where $-\infty <a \leq t \leq b < \infty$ and $y$ satisfies the boundary conditions
\begin{gather} \label{eqn1cond1}
y(a)\cos \alpha- y'(a)\sin \alpha=0, \\
\label{eqn1cond2}
y(b)\cos \beta+ y'(b)\sin \beta=0,
\end{gather}
$0\leq \alpha,\beta <\pi$, the potential function $q(t)$ and the weight 
function $r(t)$ are real-valued in general. The value of the parameter 
$\lambda \in \mathbb{C}$ for which there exists a solution $y(t,\lambda)$ 
which is non-identically zero on $[a,b]$ is called an \emph{eigenvalue} of 
problem \eqref{Eqn1}-\eqref{eqn1cond1}-\eqref{eqn1cond2}, and the corresponding 
function $y(t,\lambda)$ is called an \emph{eigenfunction} of the problem. 
The set consisting of all the eigenvalues of the problem is called the 
\emph{spectrum} of \eqref{Eqn1}-\eqref{eqn1cond1}-\eqref{eqn1cond2}. 
For the sake of simplicity we  assume occasionally that $q, r$ are both 
continuous or both piecewise continuous on $[a,b]$.

As alluded to in \cite{Everitt83}, the classical Sturm-Liouville oscillation 
theory of problems of the form \eqref{Eqn1}-\eqref{eqn1cond2} is concerned 
with the position and the number of zeros of solutions. The position of such 
zeros varies when the functions $q$, $r$, and/or the parameter $\lambda$ are changed. 
In particular, the weight function $r$ plays a critical role in the form and
nature of these results. For example, in \cite{Chen2016} the authors show that 
the oscillation of the weight function can drive away the real eigenvalues from 
the real line into the complex plane. It is also known \cite{Jabon84} that for 
a fixed weight function, an increase in the number of negative squares of
$$
B+\int_a^b (|y'|^2 +q|y|^2)
$$
by varying $q$, can lead to an increase in the number of non-real eigenvalues. 
For an historical overview of this subject until 1986, see \cite{abm86}.

It is clear that, in the non-definite case, the weight function $r$ must 
take on both signs in the interval $(a,b)$, \cite{Rich18}. 
A point about which the weight function $r(t)$ actually changes its sign 
in the interval $(a,b)$ is called a {\bf turning point} of $r$.

We now focus on the the Dirichlet problem (i.e., $\alpha=\beta=0$ in 
\eqref{eqn1cond1}-\eqref{eqn1cond2}).

\begin{theorem}[{\cite[Theorem 3]{abm82}}] \label{Ming2}
Let $\lambda$ and $y(t,\lambda)$ be a non-real eigenvalue and associated 
non-real eigenfunction of the problem \eqref{Eqn1}. If $r(t)$ has precisely 
$n$ turning points in $(a,b)$ then $y(t,\lambda)$ may vanish at most $(n-1)$ 
times in $(a,b)$.
\end{theorem}

\begin{corollary}[{\cite[Corollary 1]{abm82}}] \label{Ming3}
Let $\lambda$ and $y(t,\lambda)$ be a non-real eigenvalue and associated non-real
 eigenfunction of the problem \eqref{Eqn1}-\eqref{eqn1cond2}. 
If $r(t)$ has exactly one turning point in $(a,b)$ then $y(t,\lambda)\neq 0$  
in $(a,b)$.
\end{corollary}

Below we show that the previous conclusion fails in the case of more than one
 turning point.

\begin{theorem}[See \cite{Rich18,abm82,abm86}] \label{Ming4}
Let $r$ be continuous and not vanish identically in any right neighborhood 
of $t=a$. If $r(t)$ changes its sign precisely once in $(a,b)$ then the roots
 of the real and imaginary parts $\psi$ and $\varphi$ of any non-real 
eigenfunction $y=\psi + i \varphi$ corresponding to a non-real eigenvalue, 
separate one another (or interlace).
\end{theorem}

Thus, Theorem~\ref{Ming2} implies that if the weight function has one turning 
point (i.e., $n=1$) then  no non-real eigenfunction can have a zero in $(a,b)$.
 Of course, Richardson's separation theorem, Theorem \ref{Ming4} above, 
also gives the same conclusion.
In the case where the weight function $r(t)$ has exactly two turning points, 
numerical results in \cite{Mervis2016} indicate that some non-real eigenfunctions
can vanish once in $(a,b)$, in agreement  with Theorem \ref{Ming2}.

In the sequel we present basic results in the non-definite case of 
Sturm-Liouville problems and give necessary examples in some cases. 
In Section \ref{sec1} we present a non-definite Sturm-Liouville problem in
 which the weight function has more than one turning point in $(a,b)$ which 
then violates Richardson's separation theorem, Theorem~\ref{Ming4}. 
This shows that we cannot easily generalize the separation theorem to the 
case of more than one turning point.
In Section \ref{sec2}, we consider the 
case in which the weight function $r(t)$ has two turning points in $(a,b)$ 
with the assumption that $r(t)$ does not vanish identically on a subinterval
 of $(a,b)$. We prove that the absolute value of the difference between the 
total number of zeros of the real and imaginary parts of a given non-real 
eigenfunction (corresponding to a non-real eigenvalue) of a problem of 
the form \eqref{Eqn1}-\eqref{eqn1cond2} is equal to 2. 
The main stimulus for the work covered in Section \ref{sec2} arose out of the 
numerical results presented in the paper \cite{Mervis2016}.

\section{Failure of the interlacing property} \label{sec1}

In this section we show that Richardson's separation theorem, 
Theorem~\ref{Ming4}, fails for a weight function having more than one turning
 point. We do this by exhibiting a non-definite Sturm-Liouville problem whose
 weight function has more than one turning point in $(a,b)$ having a non-real 
eigenfunction that vanishes there.

Let $\lambda = \sigma +i \tau$, $y(t)$ be some non-real eigenvalue-eigenfunction 
pair of the \emph{complex} coefficient Sturm-Liouville equation
\begin{equation} \label{counter}
-y''+\exp (it)y=\lambda y,
\end{equation}
satisfying the boundary conditions
\begin{equation} \label{countercond1}
y(a)=y(b)=0.
\end{equation}
The existence of such eigenvalues is due to Hilb, see \cite{abm86}.

Next, let $\kappa=\mu+ i \eta$, $z$, be a non-real eigenvalue-eigenfunction 
pair of the problem
\begin{equation} \label{counter3}
-z''+\exp(it)z=\kappa z,
\end{equation}
satisfying the new set of boundary conditions, namely,
\begin{equation}\label{counter4}
z(b)=z(2b)=0,\quad  z'(b)=y'(b),
\end{equation}
where  $y$ already satisfies \eqref{counter}-\eqref{countercond1} 
(and, of course, $y'(b)\neq 0$).
Separating real and imaginary parts in \eqref{counter} and in \eqref{counter3} 
we get
\begin{gather}
- y'' + (\cos t - \sigma) y = i\, (\tau - \sin t) y,\quad 
 y(a)=y(b)=0,\label{counter5} \\
-z''+(\cos t-\mu)z  =i\,(\eta-\sin t)z, \quad z(b)=z(2b)=0, \label{counter15}
\end{gather}
with $z$ being normalized by setting $z'(b)=y'(b)$.

Now, on the interval $[a, 2b]$, consider the equation
\begin{equation}
\label{countermain}
-W''+\Big(\cos t-\rho(t)\Big)W=i \Big(r(t)-\sin t \Big)W,
\end{equation}
where,
\begin{gather*}
\rho(t) = \begin{cases}
  \sigma, &\text{if } t\in (a,b)\\
  \mu, &\text{if }t\in (b,2b),
\end{cases} \\
r(t) = \begin{cases}
  \tau, &\text{if }t\in (a,b)\\
  \eta, &\text{if }t\in (b,2b).
\end{cases} 
\end{gather*}
Then $\rho, r$ are real piecewise continuous functions on $[a, 2b]$. 
In addition, we know that $\tau -\sin t$ must change its sign at least
 once in $(a,b)$ since $i$ is a non-real eigenvalue in \eqref{counter5}. 
So, the function $r(t)$ changes its sign on $(a,b)$ on account 
of \eqref{counter5} and then again on $(b,2b)$ on account of \eqref{counter15}.
 Now the function
\[
W(t) = \begin{cases}
  y(t), &\text{if } t\in [a,b]\\
  z(t), &\text{if } t\in [b,2b]
\end{cases} 
\]
 satisfies the boundary conditions
\begin{equation} \label{countercondtns}
W(a)=W(2b)=0.
\end{equation}

\noindent\textbf{Claim:} 
$W$ is an eigenfunction of the Sturm-Liouville problem 
\eqref{countermain}-\eqref{countercondtns} having the complex eigenvalue, $i$. 
In addition, $W(b)=0$.

\begin{proof}
That $W(b)=0$ is clear from the definition. Clearly, $W'(t)$ exists in the two 
intervals $(a,b)$ and $(b,2b)$. We show that $W'(b)$ exists. 
Let $W_{\pm}'(b)$ be the right/left derivatives of $W$ at  $t=b$. 
By the definition of $W(t)$, we know that
$W_+'(b)=z'(b)=y'(b)=W_- '(b)$, so we conclude that $W'(b)$ exists, 
and that the function $W(t)$ is thus continuously differentiable on the 
interval $(a,2b)$. Finally, $W'$ is an absolutely continuous function on 
each of $(a,b)$ and $(b, 2b)$ since $y', z'$ have this property. 
It follows that, in fact, $W'$ is itself absolutely continuous on $[a,2b]$,
 and therefore $W$ is an eigenfunction of 
\eqref{countermain}-\eqref{countercondtns},
corresponding to the non-real eigenvalue $i$ that vanishes at an interior point 
(i.e, $t=b$).
\end{proof}

We have therefore proved the following theorem.

\begin{theorem}
There exists a regular non-definite Sturm-Liouville problem on a  finite 
interval $I$ having a non-real eigenfunction $y(t, \lambda)$, corresponding 
to a non-real eigenvalue $\lambda$, such that $y(t,\lambda)=0$ for some $t$ 
in the interior of $I$.
\end{theorem}

Of course, this shows that the expected interlacing property of the zeros of 
the real and imaginary parts of a non-real eigenfunction cannot hold, in general.


\section{Zeros of real and imaginary parts of non-real eigenfunctions in the 
two-turning point case}
\label{sec2}

When the weight function $r(t)$ has two turning points, a non-real 
eigenfunction may vanish at most once in the interval $(a,b)$, by 
Theorem \ref{Ming2}. If $r(t)$ has two turning points, then the two turning points divide the interval $(a,b)$ into three subintervals and if a non-real eigenfunction vanishes once in the interval $(a,b)$, it will vanish in the middle interval. Moreover, we establish the difference between the number of zeros of the real and imaginary parts of a non-real eigenfunction corresponding to a non-real eigenvalue of a problem of the form \eqref{Eqn1}-\eqref{eqn1cond2}.

\begin{theorem}\label{Mervis1}
Let $q, r \in C[a,b]$ and assume that the weight function $r$ has precisely 
two turning points in the interval $(a,b)$, and that it does not vanish 
identically in any subinterval of $(a,b)$.
Let $\lambda$ be a non-real eigenvalue of problem
\begin{gather}\label{sec1eqn1}
-y''+q(t)y=\lambda r(t)y, \\
\label{sec1cond1}
y(a)=y(b)=0
\end{gather}
and let $y(t,\lambda)$ be a corresponding, necessarily non-real,  
eigenfunction having exactly one zero in $(a,b)$.  Then the absolute value 
of the difference between the total number of zeros of the real and imaginary 
part of $y$ in $(a,b)$ is two.
\end{theorem}

\begin{proof}
Since $\lambda \notin \mathbb{R}$,  classical arguments imply that
\begin{equation}\label{sec1eqn2}
\int_a^b r|y|^2dt =0.
\end{equation}
We define a function $f$ by $f(t)=\int_a^t r\,|y|^2\,dx$, (see \cite{abm82}).
 It then follows that
\begin{equation}\label{sec1eqn3}
f(a)=f(b)=0.
\end{equation}
By hypothesis there exists $k \in (a,b)$ such that $y(k)=0$. 
Hence, $y$ is a non-real eigenfunction of \eqref{sec1eqn1} on the 
interval $[a,k]$ satisfying $y(a)=0=y(k)$ and, as a result, $f(a)=0=f(k)$ 
must hold. Similarly, $y(k)=0=y(b)$ forces $f(k)=f(b)=0$. 
Let $c_1$ and $c_2$ be the two turning points of $r$ with $a < c_1<c_2<b$.

We now claim that $k \in (c_1, c_2)$. Without loss of generality, 
let us assume that $k \in (a,c_1)$. Then there is a number $t_1 \in (a,c_1)$ 
such that $0=f'(t_1)=r(t_1)|y(t_1)|^2$. This means that $y(t_1)=0$, since
 $r\neq 0$ in $(a,c_1)$. Hence $t_1$ is another zero of $y$, contradicting 
the assumption that $k$ is the only zero of $y$ in $(a,b)$. 
The same argument holds if $k \in (c_2,b)$. So, this case is impossible.

In the second case, without loss of generality, we assume that $k=c_1$. 
Then the open interval $(a,c_1)$ is turning point free, by hypothesis. 
This means that $r(t)\neq 0$ for any $t \in (a,c_1)$. But then the Dirichlet 
problem for \eqref{sec1eqn1} must be definite on $[a,c_1]$. 
Classical Sturm-Liouville Theory now implies that all its Dirichlet eigenvalues 
on $[a,c_1]$ must be real. But this is impossible as we started with a non-real 
eigenvalue!  A similar argument applies in the case where $k=c_2$. 
Thus, this case cannot occur. Since the first two cases are impossible, 
it must be the case that $k \in (c_1,c_2)$, as stated.

Now, $k$ divides the interval $(a,b)$ into two intervals $(a,k)$ and $(k,b)$ 
on each of  which $r(t)$ has one turning point. So, in particular, 
our non-real eigenfunction $y$ satisfies a Dirichlet problem for \eqref{sec1eqn1} 
on the interval $[a,k]$, where $r$ has one turning point (namely, $c_1$) in $(a,k)$. 
By Richardson's Theorem \ref{Ming4}, the zeros of the real and imaginary 
parts of $y$ must  interlace in the interval $(a,k)$. Similarly, the same
 argument  applied to $[k,b]$ yields that the zeros of the real and imaginary 
parts of $y$ must interlace in the interval $(k,b)$. 
This means that the zeros of the real and imaginary parts of $y$ interlace
 on \emph{almost} the whole interval $(a,b)$ except near, and at, the only 
zero of $y(t)$, i.e., where $ t=k$.

We write $y(t)=\varphi (t) + i\, \psi (t)$. If $\varphi (t)$ has $n$ zeros 
in $(a,k)$, then $\psi (t)$ has $n-1$ zeros in $(a,k)$, since the zeros 
interlace in $(a,k)$. Similarly, if $\varphi (t)$ has $m$ zeros in $(k,b)$, 
then $\psi (t)$ has  $m-1$ zeros in $(k,b)$. Recall that both 
$\varphi(k)=\psi(k)=0$ by hypothesis. Adding the total number of zeros we 
find that $\varphi$ has $n+m+1$ zeros in $(a,b)$ while $\psi (t)$ must have 
$n+m-1$ zeros in $(a,b)$. The difference in the number of zeros being equal 
to two, the proof is complete.
\end{proof}

\section{Conclusion}\label{conclusion}

We have proved two main results that further develop the work that 
Richardson  started some 100 years ago. First, we show that Richardson's 
separation theorem (1918) for the zeros of the real and imaginary part 
of a non-real eigenfunction (corresponding to a non-real eigenvalue) 
of a non-definite Sturm-Liouville Dirichlet problem in the case of one 
turning point is false, in general, by exhibiting a counterexample in the 
case of two turning points. The counterexample shows that a complex 
eigenfunction can actually vanish in the interior of the interval of definition!

Then we show that if a non-real eigenfunction (corresponding to a non-real 
eigenvalue) of a non-definite Sturm-Liouville Dirichlet problem in the case 
of two  turning points vanishes in the interior of the interval under 
consideration then the absolute value of the difference between the total 
number of zeros of the real and imaginary parts of this eigenfunction must 
be equal to 2.

Many questions in this area remain unanswered. For instance, one observation 
on the spectrum of a non-definite Sturm-Liouville problem is that if the problem
\begin{gather}\label{rem3eqn1}
-y''+q(t)y=\lambda r(t)y, \\
\label{rem3eqn2}
y(a)=y(b)=0
\end{gather}
has a non-real eigenvalue, $c + i d$, $d \neq 0$ and a real eigenvalue, say 
$\gamma$, then $c \neq \gamma$. In other words we claim that, in the 
non-definite case, there cannot exist a non-real eigenvalue whose real 
part is also an eigenvalue. Whether this is an accident or a result of a more 
general yet unproven theorem, is unknown, but we conjecture that it is so and 
leave this for future research.

Furthermore, there is a need to prove general results on the behaviour 
of the real and imaginary parts of non-real eigenfunctions in the case 
where the weight function has a finite number of turning points. 
For further open questions on the non-real spectrum of non-definite 
problems see the monograph, \cite{Atkinson2010}.


\subsection*{Acknowledgments}
This research is supported, in part, by the International Science Programme
 in Mathematical Sciences, Uppsala University, Sweden.

We wish to thank Professor L-E Persson (Lule\aa\ University of Technology) 
for the suggestions that improved the final version of this manuscript. 
We also wish to thank the Division of Mathematical Sciences and Statistics, 
Lule\aa\ University of Technology, Sweden, for the financial support during 
the research visit of the first author. Furthermore, we wish to thank 
the International Science Program  based at Upssala University, 
Sweden for funding the PhD studies and financial assistance towards 
the research visits by the first author.

Furthermore, we wish to thank the careful referee for the corrections
that improved the final version of this paper.


\begin{thebibliography}{0}

\bibitem{Atkinson2010} F. V. Atkinson, A. B. Mingarelli;
 \emph{Multiparameter eigenvalue problems},
  Sturm-Liouville theory, CRC Press, 2011.

\bibitem{Jabon84} F. V. Atkinson, D.~Jabon;
 \emph{Indefinite sturm-liouville problems},
  Proc.1984 workshop on Spectral Theory of Sturm-Liouville Differential
  Operators, Argon National Laboratory, Argon, Illinois 60439, 1984,
  pp.~31--45.

\bibitem{Everitt83} W. N. Everitt, M. K. Kwong, A. Zettl;
 \emph{Oscillation of eigenfunctions of
  weighted regular sturm-liouville problems}, J. London Math. Soc. \textbf{2}
  (1983), no.~27, 106--120.

\bibitem{Mervis2016} M. Kikonko;
 \emph{On a non-definite sturm-liouville problem in the two-turning
  point case - analysis and numerical results}, J.Appl.Math.Phys \textbf{4}
  (2016), 1787--1810.

\bibitem{abm82} A. B. Mingarelli;
 \emph{Indefinite sturm-liouville problems in ``ordinary and
  partial differential equations''}, Everitt, W.N. ed., Lecture notes in
  Mathematics 964, Springer-Verlag, Berlin-New York, 1982, pp.~519--528.

\bibitem{abm86} A. B. Mingarelli;
\emph{A survey of the regular weighted sturm-liouville problem-the
  non-definite case}, Applied Differential Equations (1986), 109--137.

\bibitem{Chen2016} J. Qi, B. Xie,  S. Chen;
 \emph{The upper and lower bounds on non-real
  eigenvalues of indefinite sturm-liouville problems}, Proc. American
  Mathematical Society \textbf{144} (2016), 547--559.

\bibitem{Rich18} R. G. D. Richardson;
\emph{Contributions to the study of oscillation properties of
  the solutions of linear differential equations of the second order}, American
  Journal of Mathematics \textbf{40} (1918), no.~3, 283--316.

\end{thebibliography}

\end{document}