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Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs

James B. Kennedy, Delio Mugnolo, Matthias Täufer

Abstract

We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete spectrum for the Neumann Laplacian. In this regime, we establish polynomial upper and lower bounds on the $k$-th eigenvalue, revealing that the eigenvalues grow at a rate strictly slower than quadratic. However, once the diagonal combs transition to finite volume, their growth accelerates to a quadratic rate. Our methodology involves employing spectral geometric principles tailored for metric graphs, complemented by deriving estimates for the $k$-th eigenvalue on compact metric graphs.

Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs

Abstract

We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete spectrum for the Neumann Laplacian. In this regime, we establish polynomial upper and lower bounds on the -th eigenvalue, revealing that the eigenvalues grow at a rate strictly slower than quadratic. However, once the diagonal combs transition to finite volume, their growth accelerates to a quadratic rate. Our methodology involves employing spectral geometric principles tailored for metric graphs, complemented by deriving estimates for the -th eigenvalue on compact metric graphs.
Paper Structure (6 sections, 11 theorems, 39 equations, 1 figure)

This paper contains 6 sections, 11 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and main result
  3. Proofs
  4. Upper bounds
  5. Lower bounds
  6. Weyl asymptotics for finite volume comb graphs

Key Result

Proposition 1

The volume $|\mathcal{G}_\alpha|$ is finite if and only if $\alpha \in (1, \infty)$. The spectrum of the Kirchhoff Laplacian $\Delta^{\mathrm{N}}_{\mathcal{G}_\alpha}$ on $\mathcal{G}_\alpha$ is purely discrete if and only if $\alpha \in (\frac{1}{2}, \infty)$, whereas for $\alpha \in (0,\frac{1}{2}

Figures (1)

  • Figure 2.1: The diagonal comb $\mathcal{G}_\alpha$.

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1: See DufKenMug22, Theorem 3.4 (1)--(2)
  • Remark 1
  • Theorem 2.1
  • Remark 2
  • Theorem 2.2
  • Theorem 2.3
  • ...and 15 more