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

Abstract

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide $Π_{l}^{\varepsilon}$ obtained from a straight unit strip by a low box-shaped perturbation of size $2l\times\varepsilon,$ where $\varepsilon>0$ is a small parameter. We prove the existence of the length parameter $l_{k}^{\varepsilon}=πk+O\left( \varepsilon\right) $ with any $k=1,2,3,...$ such that the waveguide $Π_{l_{k}^{\varepsilon}}^{\varepsilon }$ supports a trapped mode with an eigenvalue $λ_{k}^{\varepsilon}% =π^{2}-4π^{4}l^{2}\varepsilon^{2}+O\left( \varepsilon^{3}\right) $ embedded into the continuous spectrum. This eigenvalue is unique in the segment $\left[ 0,π^{2}\right] $ and is absent in the case $l\neq l_{k}^{\varepsilon}.$ The detection of this embedded eigenvalue is based on a criterion for trapped modes involving an artificial object, the augmented scattering matrix. The main technical difficulty is caused by corner points of the perturbed wall $\partialΠ_{l}^{\varepsilon}$ and we discuss available generalizations for other piecewise smooth boundaries.