\documentclass[reqno]{amsart}
\usepackage{hyperref}
\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