Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds on eigenvalue ratios of quantum graphs

Evans M. Harrell, James B. Kennedy, Gabriel J. Ramos

Abstract

We study ratios of eigenvalues of the Laplacian on compact metric graphs. Our goals are threefold: First, we prove a sharp Ashbaugh--Benguria-type bound for the ratio of the first two eigenvalues on compact trees with Dirichlet conditions at all leaves, concretely showing that the ratio is maximized when the graph is an interval or an equilateral star. This improves a previous Payne--Pólya--Weinberger-type result due to Nicaise [Bull. Sci. Math., II. Sér. 111 (1987), 401--413]. Second, we extend this bound to a set of inequalities for the ratio of any pair of eigenvalues of such compact Dirichlet trees which respect the Weyl asymptotics up to an absolute constant. Third, we show that on non-trees, on which we also allow any mix of Neumann and Dirichlet conditions at the leaves, it is possible to recover bounds on the eigenvalue ratios depending only on the number of independent cycles and the number of Neumann leaves, in addition to the eigenvalue indices. This complements previously known counterexamples to analogues of the Ashbaugh--Benguria bound for general quantum graphs, by showing that the only way the bound can fail is through cycles and Neumann leaves, and by explicitly quantifying the extent to which it can fail.

Bounds on eigenvalue ratios of quantum graphs

Abstract

We study ratios of eigenvalues of the Laplacian on compact metric graphs. Our goals are threefold: First, we prove a sharp Ashbaugh--Benguria-type bound for the ratio of the first two eigenvalues on compact trees with Dirichlet conditions at all leaves, concretely showing that the ratio is maximized when the graph is an interval or an equilateral star. This improves a previous Payne--Pólya--Weinberger-type result due to Nicaise [Bull. Sci. Math., II. Sér. 111 (1987), 401--413]. Second, we extend this bound to a set of inequalities for the ratio of any pair of eigenvalues of such compact Dirichlet trees which respect the Weyl asymptotics up to an absolute constant. Third, we show that on non-trees, on which we also allow any mix of Neumann and Dirichlet conditions at the leaves, it is possible to recover bounds on the eigenvalue ratios depending only on the number of independent cycles and the number of Neumann leaves, in addition to the eigenvalue indices. This complements previously known counterexamples to analogues of the Ashbaugh--Benguria bound for general quantum graphs, by showing that the only way the bound can fail is through cycles and Neumann leaves, and by explicitly quantifying the extent to which it can fail.

Paper Structure

This paper contains 13 sections, 14 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A bound of Ashbaugh--Benguria-type
  3. General eigenvalue ratios for Dirichlet trees
  4. Bounds and existence of maximizers for general graphs
  5. Notation and preliminaries
  6. Metric graphs
  7. Function spaces and Laplacians
  8. Edge length perturbations
  9. Existence of minimizers and maximizers
  10. On Ashbaugh--Benguria-type bounds for Dirichlet trees
  11. General bounds for Dirichlet trees
  12. Bounds and optimizers for general graphs
  13. Proofs of the lemmas

Key Result

Theorem 1.1

Let $\Gamma$ be a compact Dirichlet tree, and let $k \geq 1$, then If $k=1$, the constant may be improved to $2+\sqrt{5} \approx 4.236$.

Figures (2)

  • Figure 1.1: A star $\Gamma$ with a single Dirichlet condition at the end of its long edge. (The white circle indicates a Dirichlet condition, black circles indicate Kirchhoff conditions.)
  • Figure 3.1: A pendant star ${\mathcal{S}}$ attached to a tree at the dummy vertex $w$.

Theorems & Definitions (34)

  • Theorem 1.1: Nicaise
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • ...and 24 more