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


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