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

TL;DR

This paper investigates the spectral properties of diagonal combs, a class of infinite metric graphs, revealing how their eigenvalue growth rates change with volume and establishing bounds using spectral geometry techniques.

Contribution

It introduces a theoretical framework for eigenvalue asymptotics on infinite metric graphs, specifically analyzing the transition from discrete spectrum to eigenvalue growth rates.

Findings

Eigenvalues grow slower than quadratic in infinite volume regime

Polynomial bounds on the $k$-th eigenvalue are established

Eigenvalue growth becomes quadratic when the graph has finite volume

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.