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Nodal Deficiency of Neumann Eigenfunctions on a Symmetric Dumbbell Domain

Thomas Beck, Andrew Lyons

TL;DR

The paper analyzes Neumann Laplacian eigenfunctions on symmetric dumbbell domains Ωₑ with a shrinking neck, where the spectrum near a double eigenvalue μ splits into two branches due to symmetry and a neck-induced Sturm–Liouville problem. It derives first-order asymptotics for the eigenvalues and precise eigenfunction approximations in Ω₀ and the neck, establishing that the nodal deficiency of the perturbed dumbbell eigenfunctions is bounded below by twice the nodal deficiency of the end-domain eigenfunction, with equality when end nodal sets have no crossings. This leads to a practical criterion for identifying zero-nodal-deficiency eigenfunctions on the dumbbell and connects spectral flow to nodal geometry via the quantity Θ_μ. The approach combines variational Rayleigh-quotient methods, symmetry arguments, neck Sturm–Liouville analysis, and Faber–Krahn-based nodal-domain counting to characterize Courant sharpness, spectral partitions, and their stability under thin-neck perturbations.

Abstract

We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains. As the width of the connecting neck shrinks, these eigenfunctions converge to Neumann eigenfunctions defined over the ends of the dumbbell, together with a one-dimensional Sturm-Liouville solution in the neck. In this limit, the corresponding eigenvalues become degenerate, with multiplicity two. The nodal deficiency, defined as the difference between the eigenvalue index and the nodal domain count, is known by the Courant nodal domain theorem to be nonnegative. We show that, for small neck widths, the nodal deficiencies of the dumbbell eigenfunctions are no smaller than the nodal deficiencies of the limiting eigenfunctions in the ends, and we provide conditions under which equality is achieved. As a consequence, we establish a criterion for identifying eigenfunctions of zero nodal deficiency for the dumbbell domain.

Nodal Deficiency of Neumann Eigenfunctions on a Symmetric Dumbbell Domain

TL;DR

The paper analyzes Neumann Laplacian eigenfunctions on symmetric dumbbell domains Ωₑ with a shrinking neck, where the spectrum near a double eigenvalue μ splits into two branches due to symmetry and a neck-induced Sturm–Liouville problem. It derives first-order asymptotics for the eigenvalues and precise eigenfunction approximations in Ω₀ and the neck, establishing that the nodal deficiency of the perturbed dumbbell eigenfunctions is bounded below by twice the nodal deficiency of the end-domain eigenfunction, with equality when end nodal sets have no crossings. This leads to a practical criterion for identifying zero-nodal-deficiency eigenfunctions on the dumbbell and connects spectral flow to nodal geometry via the quantity Θ_μ. The approach combines variational Rayleigh-quotient methods, symmetry arguments, neck Sturm–Liouville analysis, and Faber–Krahn-based nodal-domain counting to characterize Courant sharpness, spectral partitions, and their stability under thin-neck perturbations.

Abstract

We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains. As the width of the connecting neck shrinks, these eigenfunctions converge to Neumann eigenfunctions defined over the ends of the dumbbell, together with a one-dimensional Sturm-Liouville solution in the neck. In this limit, the corresponding eigenvalues become degenerate, with multiplicity two. The nodal deficiency, defined as the difference between the eigenvalue index and the nodal domain count, is known by the Courant nodal domain theorem to be nonnegative. We show that, for small neck widths, the nodal deficiencies of the dumbbell eigenfunctions are no smaller than the nodal deficiencies of the limiting eigenfunctions in the ends, and we provide conditions under which equality is achieved. As a consequence, we establish a criterion for identifying eigenfunctions of zero nodal deficiency for the dumbbell domain.
Paper Structure (9 sections, 21 theorems, 115 equations, 5 figures)

This paper contains 9 sections, 21 theorems, 115 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Domain definition and statement of main results.
  3. Eigenfunction and eigenvalue estimates.
  4. Spectral partitions.
  5. Outline.
  6. An Eigenvalue Upper Bound
  7. An Eigenvalue Lower Bound
  8. Eigenvalue Branch Order
  9. Nodal Domains of the Perturbed Eigenfunctions

Key Result

Theorem 1.5

Under Assumptions ass:eig and ass:phi, there exists a constant $\epsilon_0>0$, depending only on the geometry of $\Omega_0$, the function $g$ defining $R_\epsilon$, and the eigenpair $(\mu,\phi)$, such that for $0<\epsilon<\epsilon_0$, with equality if the nodal set of $\phi$ in ${\Omega_L}$ contains no crossings.

Figures (5)

  • Figure 1: An example of the dumbbell domain $\Omega_\epsilon$.
  • Figure 2: The Courant sharp eigenfunctions on an almost square domain.
  • Figure 3: A pair of Courant sharp eigenfunctions for a rectangular dumbbell. Figure credit: Steven Zucca
  • Figure 4: An eigenfunction of a rectangular dumbbell with no nodal crossings. Figure credit: Steven Zucca
  • Figure 5: The partition of $\Omega_\epsilon$ given in Lemma \ref{['lem:Partition']}.

Theorems & Definitions (40)

  • Definition 1.1: The Separated Domain, $\Omega_{0}$
  • Definition 1.2: The Dumbbell Domain, $\Omega_\epsilon$
  • Theorem 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • Theorem 1.11
  • Lemma 2.1
  • ...and 30 more