Resonances from perturbations of quantum graphs with rationally related edges

TL;DR

This paper investigates how perturbations in the edge lengths of quantum graphs with rationally related edges can transform embedded eigenvalues into resonances, providing both general analysis and simple examples.

Contribution

It introduces a detailed analysis of the transition from embedded eigenvalues to resonances in quantum graphs under perturbations of rationally related edges.

Findings

Embedded eigenvalues can turn into resonances under perturbations.

The effect is analyzed both generally and through simple examples.

Perturbations can significantly alter the spectral properties of quantum graphs.

Abstract

We discuss quantum graphs consisting of a compact part and semiinfinite leads. Such a system may have embedded eigenvalues if some edge lengths in the compact part are rationally related. If such a relation is perturbed these eigenvalues may turn into resonances; we analyze this effect both generally and in simple examples.