Bounds on eigenvalue ratios of quantum graphs

TL;DR

This paper establishes sharp bounds on eigenvalue ratios of the Laplacian on compact quantum graphs, extending classical inequalities and identifying conditions for equality and failure.

Contribution

It proves a sharp eigenvalue ratio bound for compact trees, extends inequalities to all eigenvalue pairs, and characterizes how cycles and Neumann leaves affect bounds on non-trees.

Findings

Maximized eigenvalue ratio on trees occurs for intervals or equilateral stars.

Extended bounds apply to all eigenvalue pairs respecting Weyl asymptotics.

On non-trees, eigenvalue ratio bounds depend on cycles and Neumann leaves.

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--P\'olya--Weinberger-type result due to Nicaise [Bull. Sci. Math., II. S\'er. 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.