We investigate resonances in non-compact quantum graphs with standard and
general coupling conditions. For the rational ratio of the lengths of the
edges of the graph, there may be eigenvalues embedded in the continuous
spectrum of the corresponding operator. These eigenvalues may become
resonances if the edge lengths are perturbed. We are interested in the
behaviour of the imaginary part of the second derivative of the square root
of energy $k$ with respect to the parameter giving the lengths of the edges
of the graph in the vicinity of the former eigenvalue of the graph. We
introduce two methods how to obtain the second derivative of $k$ and explain
their usage on examples.
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The talk is based on the papers
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M. Lee, M. Zworski, A Fermi golden rule for quantum graphs, J. Math. Phys.
57 (2016), 092101,
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P. Exner, J. Lipovsky, Pseudo-orbit approach to trajectories of resonances
in quantum graphs with general vertex coupling: Fermi rule and high-energy
asymptotics, J. Math. Phys. 58 (2017), 042101.