Resonances in a Dirichlet quantum waveguide coupled to a cavity

TL;DR

This paper studies how small gaps in a Dirichlet waveguide with a cavity cause embedded eigenvalues to turn into resonances, with the resonance behavior depending on the gap size and geometry.

Contribution

It provides asymptotic analysis of resonance behavior in waveguides with small apertures, revealing how the resonance width scales with the aperture volume in different dimensions.

Findings

Resonance width in 2D scales as the square of the gap size.

In 3D, the resonance width scales as the fourth power of the aperture volume.

The characteristic time scale of resonances is inversely related to the square of the aperture volume.

Abstract

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.