Eigenvalue approximations for linear periodic differential equations with a singularity

Abstract

We consider the second order, linear differential equation \begin{equation*}y''(x) + (\lambda.q(x)) y(x) = 0 \tag{1}\end{equation*} where $q$ is a real-valued, periodic function with period a. Our object in this paper is to derive asymptotic estimates for the eigenvalues of (1) on $[0;a]$ with periodic and semi­periodic boundary conditions. Our approach to regularizing (1) follows that used by Atkinson [1], Everitt and Race [4], and Harris and Race [6]. We illustrate our methods by calculating asymptotic estimates for the periodic and semi­periodic eigenvalues of (1) in the case where $q(x) = 1/|1-x|$.