##
Singular Eigenvalue Problems for Second Order Linear
Ordinary Differential Equations}

##
*
Arpad Elbert, Kusano Takasi,
Manabu Naito
*

** Address.**
A. Elbert, Mathematical Institute of the Hungarian Academy of Sciences,
Budapest, P.O.B. 127, H--1364, Hungary

K. Takasi, Department of Applied Mathematics,
Faculty of Science, Fukuoka University,
8-19-1 Nanakuma, Jonan-ku, Fukuoka, 814-80 Japan

M. Naito, Department of Mathematical Sciences,
Faculty of Science, Ehime University, Matsuyama, Japan

** E-mail:**
elbert@math-inst.hu

tkusano@ssat.fukuoka-u.ac.jp

mnaito@solaris.math.sci.ehime-u.ac.jp

**Abstract.**
We consider linear differential equations of the form
\begin{equation*}
(p(t)x^{\prime})^{\prime}+\lambda q(t)x=0~~~(p(t)>0,~q(t)>0) \tag{A}
\end{equation*}
on an infinite interval $[a,\infty)$ and study the problem of finding those values of $\lambda$ for which
\eqref{kuseA} has principal solutions $x_{0}(t;\lambda)$ vanishing at $t = a$. This problem may well be called a singular
eigenvalue problem, since requiring $x_{0}(t;\lambda)$ to be a principal solution can be considered as a
boundary condition at $t=\infty$. Similarly to the regular eigenvalue problems for \eqref{kuseA} on compact intervals,
we can prove a theorem asserting that there exists a sequence $\{\lambda _{n}\}$ of eigenvalues such that
$\displaystyle 0<\lambda _{0}<\lambda _{1}<\cdots<\lambda _{n}<\cdots$,
$\displaystyle\lim_{n\to\infty}\lambda _{n}=\infty$,
and the eigenfunction $x_{0}(t;\lambda _{n})$ corresponding to $\lambda = \lambda _{n}$ has exactly $n$~zeros
in $(a,\infty),~n=0,1,2,\dots$. We also show that a similar situation holds for nonprincipal solutions of \eqref{kuseA}
under stronger assumptions on $p(t)$ and $q(t)$.

**AMS classification.**
34B05, 34B24, 34C10

**Key words.**
Singular eigenvalue problem, Sturm-Liouville equation,
zeros of nonoscillatory solutions