The paper is devoted to prove an exponential lower bound of the resonance (i.e. the absolute value of the imaginary part) of the first eigenvalue of a Helmholtz resonator woth straight neck.
More precisely, let \(\mathcal{C}, \mathcal{B} \subseteq \mathbb{R}^n\) be bounded, \(\overline{\mathcal{C}}\subseteq \mathcal{B}\). Let \(0\in D_1\subseteq\mathbb{R}^{n-1}\) be bounded with smooth boundary, and for \(\varepsilon>0\) let \(D_\varepsilon:=\varepsilon D_1\). Let \(\Omega(\varepsilon)\subseteq\mathbb{R}^n\) be the union of \(\mathcal{C}\), \(\mathbb R^n\setminus \overline{\mathcal{B}}\) and a tube through \(\mathcal{B}\) of length \(L>0\) and cross-section \(D_\varepsilon\) connecting \(\mathcal{C}\) and \(\mathbb{R}^n\setminus \overline{\mathcal{B}}\). Let \(P_\varepsilon\) be the Dirichlet Laplacian on \(\Omega(\varepsilon)\), and for \(\mu>0\) let \(P_\varepsilon(\mu):=U_\mu P_\varepsilon U_\mu^{-1}\), where \(U_\mu \varphi := \varphi(\cdot+i\mu f(\cdot))\) for some appropriate \(f\in C^\infty(\mathbb{R}^n;\mathbb{R}^n)\), and \(\rho(\varepsilon)\) an eigenvalue of \(P_\varepsilon(\mu)\). Then the main Theorem 2.2 shows that for all \(\delta>0\) there exists \(C_\delta>0\) such that for sufficiently small \(\varepsilon>0\) one has \[ |\mathrm{Im} \rho(\varepsilon)|\geq \frac{1}{C_\delta} e^{-2\alpha_0 (1+\delta)L/\varepsilon}. \] The proof relies on Carleman estimates combined with the results in [the last author and L. Nédélec, Ann. Henri Poincaré 17, No. 3, 645–672 (2016; Zbl 1337.35027)].