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

Yuriy Golovaty

TL;DR

This work analyzes the spectral behavior of a Sturm-Liouville problem with a $\delta'$-like perturbation of mass density, formulating a family of self-adjoint realizations on varying spaces and a fixed-space non-self-adjoint matrix operator $\mathcal{A}_\varepsilon$. The authors establish norm resolvent convergence $\mathcal{A}_\varepsilon \to \mathcal{A}$ and prove Hausdorff convergence of the perturbed spectra to a real, discrete limit, whose structure reflects the union of left, middle, and right subproblems and may exhibit multiple eigenvalues with Jordan blocks. They further detail how eigenfunctions converge: simple eigenvalues converge to side-specific eigenfunctions, while mid-region perturbations yield jump-type limits at the interface with a calculable normalization. Overall, the paper provides a rigorous framework for spectral stability and eigenfunction limits under strong, $\delta'$-like mass perturbations, with implications for vibrating systems containing concentrated masses.

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 $δ'$-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.

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

TL;DR

This work analyzes the spectral behavior of a Sturm-Liouville problem with a -like perturbation of mass density, formulating a family of self-adjoint realizations on varying spaces and a fixed-space non-self-adjoint matrix operator . The authors establish norm resolvent convergence and prove Hausdorff convergence of the perturbed spectra to a real, discrete limit, whose structure reflects the union of left, middle, and right subproblems and may exhibit multiple eigenvalues with Jordan blocks. They further detail how eigenfunctions converge: simple eigenvalues converge to side-specific eigenfunctions, while mid-region perturbations yield jump-type limits at the interface with a calculable normalization. Overall, the paper provides a rigorous framework for spectral stability and eigenfunction limits under strong, -like mass perturbations, with implications for vibrating systems containing concentrated masses.

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 -like sequence . 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.

Paper Structure

This paper contains 6 sections, 7 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Statement of Problem
  3. Norm Resolvent Convergence of $\mathcal{A}_\varepsilon$
  4. Spectrum of $\mathcal{A}$
  5. Convergence of Spectra
  6. Some Remarks On Eigenfunction Convergence

Key Result

Proposition 1

$\sigma(\mathcal{A}_\varepsilon)=\sigma(T_\varepsilon)$.

Theorems & Definitions (15)

  • Proposition 1
  • proof
  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 5 more