Embedded eigenvalues of the Neumann problem in a strip with a box-shaped perturbation

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide Πl ε formed by the union of an infinite strip and a narrow box-shaped perturbation of size 2l×ε, where ε>0 is a small parameter. We prove the existence of the length parameter lk ε=πk+O(ε) with any k=1,2,3,. such that the waveguide Πlk ε ε supports a trapped mode with an eigenvalue λk ε=π2−4π4l2ε2+O(ε3) embedded into the continuous spectrum. This eigenvalue is unique in the segment [0,π2], and it is absent in the case l≠lk ε. The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main difficulty is caused by the rather specific shape of the perturbed wall ∂Πl ε, namely a narrow rectangular bulge with corner points, and we discuss available generalizations for other piecewise smooth boundaries.