A non-existence result due to small perturbations in an eigenvalue problem

TL;DR

This paper investigates how small perturbations in the function defining an eigenvalue problem can lead to non-existence of eigenfunctions, highlighting sensitivity in boundary-dependent eigenvalue problems.

Contribution

It demonstrates that approximating the function by step functions can result in the non-existence of eigenfunctions, revealing a subtle instability in such eigenvalue problems.

Findings

Eigenfunctions do not exist for step function approximations

Small perturbations can cause eigenvalue problem solutions to vanish

Highlights sensitivity of eigenvalue problems to boundary conditions

Abstract

We consider a well-posed eigenvalue problem on $(a,0)$, depending on a continuous function $m$. The boundary conditions in the points $a,0$ are depending on the eigenvalues. We divide $(a,0)$ into small intervals and approximate the function $m$ by a simple (step) function $m_S$, constant on each small interval. The eigenfunctions corresponding to $m_S$ do not exist.