Topological Resonances on Quantum Graphs

TL;DR

This paper provides a mathematical foundation for topological resonances in quantum graphs, analyzing the set of metrics with compactly supported eigenfunctions and estimating its dimension based on graph properties.

Contribution

It formalizes the concept of topological resonances on quantum graphs and estimates the dimension of the metric set associated with compactly supported eigenfunctions.

Findings

Dimension estimates of the semi-algebraic set related to metrics

Analysis of topological resonances near the real axis

Discussion of the case of trees

Abstract

In this paper, we try to put the results of Smilansky and al. on "Topological resonances" on a mathematical basis.A key role in the asymptotic of resonances near the real axis for Quantum Graphs is played by the set of metrics for which there exists compactly supported eigenfunctions. We give several estimateof the dimension of this semi-algebraic set, in particular in terms of the girth of the graph. The case oftrees is also discussed.