The Sturm--Liouville problem with singular potential and the extrema of the first eigenvalue

Abstract

We get the infima and suprema of the first eigenvalue of the problem
\begin{gather*}
-y''+qy=\lambda y,\\
\left\{\begin{aligned}
y'(0)-k_0^2y(0)=0,\\ y'(1)+k_1^2y(1)=0,
\end{aligned}\right.
\end{gather*}
where \(q\) belongs to the set of constant-sign summable functions on \([0,1]\)
such that
\[
\int_0^1 q\,dx=1 \text{ or }\int_0^1 q\,dx=-1.
\]