On spectrum of strings with $\delta'$-like perturbations of mass density

TL;DR

This paper investigates how eigenvalues and eigenfunctions of a Sturm-Liouville problem are affected by a $ abla$-like perturbation in the weight function, establishing resolvent and spectral convergence results.

Contribution

It introduces a framework for analyzing the spectral behavior of Sturm-Liouville operators with $ abla$-like perturbations, including norm resolvent and Hausdorff spectral convergence.

Findings

Eigenvalues and eigenfunctions converge under $ abla$-like perturbations.

Norm resolvent convergence of the operator family is established.

Spectral convergence in the Hausdorff sense is proven.

Abstract

We study the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem for the Sturm-Liouville operator with general boundary conditions and the weight function perturbed by the so-called $\delta'$-like sequence $\varepsilon^{-2}h(x/\varepsilon)$. The eigenvalue problem is realized as a family of non-self-adjoint matrix operators acting on the same Hilbert space and the norm resolvent convergence of this family is established. We also prove the Hausdorff convergence of the perturbed spectra.