On semiclassical and universal inequalities for eigenvalues of quantum graphs

TL;DR

This paper investigates eigenvalue inequalities for quantum graphs, revealing how their sharp constants depend on graph topology and identifying conditions where classical inequalities hold or fail.

Contribution

It introduces topology-dependent inequalities for quantum graph spectra and clarifies when classical bounds are applicable, including counterexamples.

Findings

Sharp constants depend on graph topology

Classical inequalities hold for trees

Counterexamples show classical form can fail

Abstract

We study the spectra of quantum graphs with the method of trace identities (sum rules), which are used to derive inequalities of Lieb-Thirring, Payne-P\'olya-Weinberger, and Yang types, among others. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false.