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

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 $\mathcal{O}\big((\mathrm{Re\,}k)^{-2}\big)$.