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 ${\Pi }_l^{\epsilon }$ obtained from a straight unit strip by a low box-shaped perturbation of size $2l\times \epsilon ,$ where $\epsilon >0$ is a small parameter. We prove the existence of the length parameter $l_k^{\epsilon }=\pi k+O\left(\epsilon \right)$ with any $k=1,2,3,...$ such that the waveguide ${\Pi }_{l_k^{\epsilon }}^{\epsilon }$ supports a trapped mode with an eigenvalue ${\lambda }_k^{\epsilon }$ embedded into the continuous spectrum. This eigenvalue is unique in the segment $\left[0,{\pi }^2\right]$ and is absent in the case $l\ne l_k^{\epsilon }.$ 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 $\partial {\Pi }_l^{\epsilon }$ and we discuss available generalizations for other piecewise smooth boundaries.

36 pages, 6 figures

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Cited by in corpus (1)

• Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues