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Splitting of the poles of a Helmholtz resonator. (English. Russian original) Zbl 0827.35095

Russ. Acad. Sci., Izv., Math. 43, No. 2, 233-260 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 5, 44-74 (1993).
Summary: It is shown that in a neighborhood of a two-fold eigenvalue of the Neumann problem there are two poles of the Green’s function of the Helmholtz resonator. Asymptotic expressions for them with respect to a small parameter \(\varepsilon\) (the linear size of the hole) are constructed, and the principal terms of the asymptotics are written out for the corresponding scattering and radiation problems.

MSC:

35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A45 Diffraction, scattering
70K30 Nonlinear resonances for nonlinear problems in mechanics
47A40 Scattering theory of linear operators
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