Asymptotic stability of the spectra of generalized indefinite strings

TL;DR

This paper investigates the asymptotic stability of spectra in generalized indefinite strings (GISs), establishing conditions for spectral stability through perturbation theory and linear relations.

Contribution

It introduces a unitarily equivalent linear relation for GISs and proves spectral stability under perturbations, advancing understanding of spectral behavior in indefinite string models.

Findings

Spectra of GISs are asymptotically stable under certain conditions.

Solutions depend continuously on distributions and measures.

Convergence of linear relations for GISs is established.

Abstract

This paper focuses on the asymptotic stability of the spectra of generalized indefinite strings (GISs). A unitarily equivalent linear relation is introduced for GISs. It is shown that the solutions of the corresponding differential equations are continuously dependent on distributions and measures under certain conditions. Using these results, the convergence of unitarily equivalent linear relations for GISs is discussed. By the perturbation theory to closed linear relations, an asymptotic stability result concerning the spectra of linear relations for GISs is obtained.