Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups

TL;DR

This paper investigates the spectral properties of Laplace operators on metric graphs, providing bounds on negative eigenvalues, conditions for positivity preservation of heat semigroups, and insights into resolvents.

Contribution

It offers new bounds on eigenvalues and conditions for positivity preservation, advancing understanding of Laplacians on metric graphs.

Findings

Upper bound on the number of negative eigenvalues

Lower bound on the spectrum of Laplace operators

Sufficient condition for positivity preserving heat semigroup

Abstract

The main objective of the present work is to study the negative spectrum of (differential) Laplace operators on metric graphs as well as their resolvents and associated heat semigroups. We prove an upper bound on the number of negative eigenvalues and a lower bound on the spectrum of Laplace operators. Also we provide a sufficient condition for the associated heat semigroup to be positivity preserving.