Resonance widths for general Helmholtz resonators with straight neck. (English) Zbl 1366.81167

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)].


81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35P15 Estimates of eigenvalues in context of PDEs
35B34 Resonance in context of PDEs
81U05 \(2\)-body potential quantum scattering theory


Zbl 1337.35027
Full Text: DOI arXiv Euclid