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Resonances in a Dirichlet quantum waveguide coupled to a cavity

Sylwia Kondej, Nikoloz Kurtskhalia

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.

Resonances in a Dirichlet quantum waveguide coupled to a cavity

Abstract

We consider a Dirichlet waveguide in () 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 , we show that the leading order term of the resonance behaves as . In the three-dimensional case, if the aperture is defined by a rectangular opening with volume proportional to , the resonant component behaves as . This shows that, in the analyzed class of models, the characteristic time scale associated with the resonances is generically of order , where denotes the volume of the aperture inducing the resonance.

Paper Structure

This paper contains 11 sections, 11 theorems, 162 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Description of model and preliminary results in two-dimensional system
  3. Geometry of planner waveguide and Hamiltonian of system
  4. Preliminary results for the special cases
  5. Embedded eigenvalues
  6. Resonances caused by an aperture in cavity
  7. Analytic continuation of $K^\varepsilon (z)$
  8. Asymptotic analysis of resonances
  9. Resonances in higher dimensional systems: A study of three-dimensional waveguides
  10. Resonances induced by a rectangular gap in a waveguide
  11. Final remarks and open questions

Key Result

Theorem 2.1

A number $z_0$ is an eigenvalue of $-\Delta^D_{I_\varepsilon}$ if and only if where the operator $K^\varepsilon(z)$ is considered as acting in $H^{1/2}(I_\varepsilon)$. Moreover,

Figures (2)

  • Figure 1: Geometry of the waveguide $\Sigma$ with a cavity $\mathcal{C}$ containing a gap $\bar{I}\varepsilon$.
  • Figure 2: Geometry of the waveguide $\Sigma$ with a box cavity $\mathcal{C}$ containing a rectangular gap $\bar{I}_\varepsilon$.

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • Remark 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Remark 3.5
  • Lemma 3.6
  • Lemma 4.1
  • ...and 8 more