Asymptotic analysis of vibrating system containing stiff-heavy and flexible-light parts

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues and eigenfunctions in a strongly inhomogeneous vibrating medium with contrasting physical properties, revealing nonself-adjoint limit operators and loss of eigenfunction completeness.

Contribution

It provides complete asymptotic expansions for eigenvalues and eigenfunctions, and characterizes the limit operator's nonself-adjointness and Jordan cell structure in a singular perturbation setting.

Findings

Eigenvalues and eigenfunctions have explicit asymptotic expansions.

The limit operator is generally nonself-adjoint with Jordan cells.

Eigenfunction completeness may be lost in the limit.

Abstract

A model of strongly inhomogeneous medium with simultaneous perturbation of rigidity and mass density is studied. The medium has strongly contrasting physical characteristics in two parts with the ratio of rigidities being proportional to a small parameter $\epsilon$. Additionally, the ratio of mass densities is of order $\epsilon^{-1}$. We investigate the asymptotic behaviour of spectrum and eigensubspaces as $\epsilon\to 0$. Complete asymptotic expansions of eigenvalues and eigenfunctions are constructed and justified. We show that the limit operator is nonself-adjoint in general and possesses two-dimensional Jordan cells in spite of the singular perturbed problem is associated with a self-adjoint operator in appropriated Hilbert space $L_\epsilon$. This may happen if the metric in which the problem is self-adjoint depends on small parameter $\epsilon$ in a singular way. In particular, it leads to a loss of completeness for the eigenfunction collection. We describe how root spaces of the limit operator approximate eigenspaces of the perturbed operator.