  The system consisting of a fermion in the background of a wobbling kink is
studied in this paper. To investigate the impact of the wobbling on the
fermion-kink interaction, we employ the time-dependent perturbation theory
formalism in quantum mechanics. To do so, we compute the transition
probabilities between states given in terms of the Bogoliubov coefficients. We
derive Fermi's golden rule for the model, which allows the transition to the
continuum at a constant rate if the fermion-kink coupling constant is smaller
than the wobbling frequency. Moreover, we study the system replacing the shape
mode with a quasinormal mode. In this case, the transition rate to continuum
decays in time due to the leakage of the mode, and the final transition
probability decreases sharply for large coupling constants in a way that is
analogous to Fermi's golden rule. Throughout the paper, we compare the
perturbative results with numerical simulations and show that they are in good
agreement.
