Membranes with thin and heavy inclusions: asymptotics of spectra

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues and eigenfunctions in 2D vibrating membranes with heavy inclusions, especially near resonance frequencies, revealing complex spectral structures due to non-self-adjoint limit operators.

Contribution

It introduces a novel asymptotic analysis of spectral properties for membranes with heavy inclusions, including the study of non-self-adjoint limit operators with Jordan chains.

Findings

Eigenvalues exhibit complex asymptotic behavior near resonance frequencies.

Limit operators have non-self-adjoint structure with Jordan chains of length 2.

Spectrum contains a countable set of eigenvalues with infinite multiplicity.

Abstract

We study the asymptotic behaviour of eigenvalues and eigenfunctions of 2D vibrating systems with mass density perturbed in a vicinity of closed curves. The threshold case in which resonance frequencies of the membrane and thin inclusion coincide or closely situated is investigated. The perturbed eigenvalue problem can be realized as a family of self-adjoint operators acting on varying Hilbert spaces. However the so-called limit operator which is ultimately responsible for the asymptotics of eigenvalues and eigenfunctions is non-self-adjoint and possesses the Jordan chains of length $2$. Apart from the lack of self-adjointness, the operator has non-compact resolvent. As a consequence, its spectrum has a complicated structure, for instance, the spectrum contains a countable set of eigenvalues with infinite multiplicity.