Geodesic
Asymptotic behaviour of an eigenvalue in the continuous spectrum of a~narrowed waveguide
Sbornik. Mathematics, Tome 203 (2012) no. 2, pp. 153-182

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The existence of an eigenvalue embedded in the continuous spectrum is proved for the Neumann problem for Helmholtz's equation in a two-dimensional waveguide with two outlets to infinity which are half-strips of width $1$ and $1-\varepsilon$, where $\varepsilon>0$ is a small parameter. The width function of the part of the waveguide connecting these outlets is of order $\sqrt{\varepsilon}$; it is defined as a linear combination of three fairly arbitrary functions, whose coefficients are obtained from a certain nonlinear equation. The result is derived from an asymptotic analysis of an auxiliary object, the augmented scattering matrix. Bibliography: 29 titles.
Keywords: acoustic waveguide, water waves in a channel, eigenvalues in the continuous spectrum, asymptotic behaviour, augmented scattering matrix. 
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     title = {Asymptotic behaviour of an eigenvalue in the continuous spectrum of a~narrowed waveguide},
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G. Cardone; S. A. Nazarov; K. Ruotsalainen. Asymptotic behaviour of an eigenvalue in the continuous spectrum of a~narrowed waveguide. Sbornik. Mathematics, Tome 203 (2012) no. 2, pp. 153-182. http://geodesic.mathdoc.fr/item/SM_2012_203_2_a0/