Optimal lower bound of the resonance widths for a Helmholtz tube-shaped resonator

The study of the resonances of theHelmholtz resonator has been broadly described in previous works (see [11] and references therein). Here, for a simple 2-dimensional resonator in the shape of a tube, we analyze the transition zone where oscillations start to appear. Following a careful analysis, we obtain an optimal lower bound of the width of the resonances.