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Membranes with thin and heavy inclusions: asymptotics of spectra

Yuriy Golovaty

TL;DR

The paper analyzes spectral asymptotics for a 2D membrane with mass concentrated near a thin curve, leading to a limit problem governed by a non-self-adjoint operator with Jordan blocks. By formulating the limit as a matrix operator $\mathcal{P}$ on a fixed space and employing quasimodes, Dirichlet-to-Neumann maps, and careful coupling across $\gamma$, it identifies three spectral regimes determined by the relation of the limit spectrum to $\sigma(A)$ and $\sigma(B)$. It establishes detailed asymptotics: for $\lambda_0\in\sigma(B)\setminus\sigma(A)$, eigenvalues split as $\lambda_0+\lambda_1^{(k)}\varepsilon+O(\varepsilon^2)$; for $\lambda_0\in\sigma(A)\cap\sigma(B)$, half-integer powers $\lambda_0\pm\omega\varepsilon^{1/2}+O(\varepsilon)$ appear due to Jordan chains, along with additional integer-power corrections; and for $\lambda_0\in\sigma(A)\setminus\sigma(B)$, integer-power corrections arise from a finite-dimensional spectral matrix. The results illuminate how concentrated mass along a curve induces non-self-adjoint limit behavior and complex spectral structures in singular perturbations of membranes, with broad implications for coupled PDE systems and effective boundary-coupling models.

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.

Membranes with thin and heavy inclusions: asymptotics of spectra

TL;DR

The paper analyzes spectral asymptotics for a 2D membrane with mass concentrated near a thin curve, leading to a limit problem governed by a non-self-adjoint operator with Jordan blocks. By formulating the limit as a matrix operator on a fixed space and employing quasimodes, Dirichlet-to-Neumann maps, and careful coupling across , it identifies three spectral regimes determined by the relation of the limit spectrum to and . It establishes detailed asymptotics: for , eigenvalues split as ; for , half-integer powers appear due to Jordan chains, along with additional integer-power corrections; and for , integer-power corrections arise from a finite-dimensional spectral matrix. The results illuminate how concentrated mass along a curve induces non-self-adjoint limit behavior and complex spectral structures in singular perturbations of membranes, with broad implications for coupled PDE systems and effective boundary-coupling models.

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 . 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.

Paper Structure

This paper contains 11 sections, 16 theorems, 118 equations, 4 figures.

Table of Contents

  1. Introduction and Statement of Problem
  2. Formal construction of the limit operator
  3. Properties of the limit operator
  4. Spectrum of the limit operator
  5. Structure of generalized eigenspaces
  6. Case $\lambda\in \sigma(A)\setminus\sigma(B)$.
  7. Case $\lambda\in \sigma(B)\setminus\sigma(A)$.
  8. Case $\lambda\in \sigma(A)\cap\sigma(B)$.
  9. Asymptotics of eigenvalues in the case $\lambda_0\in \sigma(B)\setminus \sigma(A)$
  10. Asymptotics of eigenvalues in the case $\lambda_0\in \sigma(A)\cap\sigma(B)$
  11. Asymptotics of eigenvalues in the case $\lambda_0\in\sigma(A)\setminus\sigma(B)$

Key Result

Lemma 3.1

The spectrum of $B$ consists of a countable set of real eigenvalues of infinite multiplicity. Moreover, $\lambda$ belongs to $\sigma(B)$ if and only if $\lambda$ is an eigenvalue of the Sturm-Liouville problem

Figures (4)

  • Figure 1: Membrane with heavy and thin inclusion
  • Figure 2: Plot of the function $\zeta$.
  • Figure 3: Bifurcation of the eigenvalue $\lambda_0\in \sigma(B)\setminus \sigma(A)$.
  • Figure 4: Bifurcation of the eigenvalue $\lambda_0\in \sigma(A)\cap\sigma(B)$.

Theorems & Definitions (28)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 18 more