Resonances in a Dirichlet quantum waveguide coupled to a cavity

We consider a Dirichlet waveguide in $\mathbb{R}^n$ ($n = 2,3$) with an attached cavity. We show that if the cavity admits a small gap, then the original embedded eigenvalues turn into resonances. The main question we address is how the size of the gap affects the resonant properties, in particular the imaginary part of the resonant pole. For example, in the case of a two dimensional waveguide with a gap of size $\varepsilon$, we show that the leading order term of the resonance behaves as $\mathcal O (\varepsilon^2)$. In the three-dimensional case, if the aperture is defined by a rectangular opening with volume proportional to $\varepsilon^2$, the resonant component behaves as $\mathcal{O}(\varepsilon^4)$. This shows that, in the analyzed class of models, the characteristic time scale associated with the resonances is generically of order $\mathcal{O}((\mathrm{vol}_\varepsilon)^{-2})$, where $\mathrm{vol}_\varepsilon$ denotes the volume of the aperture inducing the resonance.