Geodesic
Scattering anomalies in a~resonator above~the~thresholds of the continuous spectrum
Sbornik. Mathematics, Tome 206 (2015) no. 6, pp. 782-813

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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. Bibliography: 26 titles.
Keywords: Helmholtz resonator, scattering problem, thresholds of continuous spectrum, waves at near-threshold frequencies, almost complete reflection and passage. 
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     author = {S. A. Nazarov},
     title = {Scattering anomalies in a~resonator above~the~thresholds of the continuous spectrum},
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S. A. Nazarov. Scattering anomalies in a~resonator above~the~thresholds of the continuous spectrum. Sbornik. Mathematics, Tome 206 (2015) no. 6, pp. 782-813. http://geodesic.mathdoc.fr/item/SM_2015_206_6_a1/