Pseudo-orbit approach to trajectories of resonances in quantum graphs with general vertex coupling: Fermi rule and high-energy asymptotics

TL;DR

This paper develops a pseudo-orbit approach to analyze resonance trajectories in quantum graphs with general vertex couplings, deriving formulas for resonance shifts and asymptotic behaviors at high energies.

Contribution

It introduces a new pseudo-orbit framework for quantum graph resonances, extending the Fermi rule to general couplings and analyzing high-energy asymptotics.

Findings

Resonance pole positions are expanded up to second order in Taylor series.

Resonances with $ ext{δ}_s'$ coupling approach the real axis at a rate of $ig( ext{Re} ext{ }kig)^{-2}$.

The derived formulas generalize the Fermi rule for various vertex couplings.

Abstract

The aim of the paper is to investigate resonances in quantum graphs with a general self-adjoint coupling in the vertices and their trajectories with respect to varying edge lengths. We derive formulae determining the Taylor expansion of the resonance pole position up to the second order which represent, in particular, a counterpart to the Fermi rule derived recently by Lee and Zworski for graphs with the standard coupling. Furthermore, we discuss the asymptotic behavior of the resonances in the high-energy regime in the situation where the leads are attached through $\delta$ or $\delta_\mathrm{s}'$ conditions, and we prove that in the case of $\delta_\mathrm{s}'$ coupling the resonances approach to the real axis with the increasing real parts as $\mathcal{O}\big((\mathrm{Re\,}k)^{-2}\big)$.