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Scattering anomalies in a resonator above the thresholds of the continuous spectrum. (English. Russian original) Zbl 1332.35248
Sb. Math. 206, No. 6, 782-813 (2015); translation from Mat. Sb. 206, No. 6, 15-48 (2015).
Author’s abstract: We consider the Dirichlet spectral problem for the Laplace operator in a multi-dimensional domain with a cylindrical outlet to infinity, a Helmholtz resonator. Using asymptotic analysis of the scattering matrix we demonstrate different types of reflection of high-amplitude near-threshold waves. One scattering type or another, unstable or stable with respect to variations of the resonator shapes, is determined by the presence or absence of stabilizing solutions at the threshold frequency, respectively. In a waveguide with two cylindrical outlets to infinity, we discover the effect of almost complete passage of the wave under “fine tuning” of the resonator.

MSC:
35P25 Scattering theory for PDEs
35B20 Perturbations in context of PDEs
35B34 Resonance in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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