Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide

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.
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