A note on a paper of Harris concerning the asymptotic approximation to the eigenvalues of $-y"+qy=\lambda y$, with boundary conditions of general form

TL;DR

This paper develops an asymptotic approximation for the eigenvalues of a Sturm-Liouville problem with a singular potential and general boundary conditions, extending classical results to more complex cases.

Contribution

It provides a novel asymptotic formula for eigenvalues of differential operators with singular potentials and general boundary conditions, broadening the scope of spectral analysis.

Findings

Derived an asymptotic eigenvalue approximation for singular potentials.

Extended classical eigenvalue results to include general boundary conditions.

Applicable to differential equations with integrable singularities in the potential.

Abstract

In this paper, we derive an asymptotic approximation to the eigenvalues of the linear differential equation $$ -y"(x)+q(x)y(x)=\lambda y(x), x\in (a,b) $$ with boundary conditions of general form, when $q$ is a measurable function which has a singularity in $(a,b)$ and which is integrable on subsets of $(a,b)$ which exclude the singularity.