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

TL;DR

This paper proves the existence of embedded eigenvalues (trapped modes) in a perturbed acoustic waveguide with a box-shaped obstacle, using augmented scattering matrix techniques and analyzing corner point effects.

Contribution

It establishes conditions for the existence of trapped modes with embedded eigenvalues in a waveguide with a box-shaped perturbation, including explicit eigenvalue asymptotics.

Findings

Existence of eigenvalues embedded in the continuous spectrum for specific waveguide lengths.

Eigenvalues are unique within the segment [0, π²] and absent otherwise.

Method involves augmented scattering matrix and analysis of corner points.

Abstract

We consider the spectral Neumann problem for the Laplace operator in an acoustic waveguide $\Pi_{l}^{\varepsilon}$ 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}=\pi k+O\left( \varepsilon\right) $ with any $k=1,2,3,...$ such that the waveguide $\Pi_{l_{k}^{\varepsilon}}^{\varepsilon }$ supports a trapped mode with an eigenvalue $\lambda_{k}^{\varepsilon}% =\pi^{2}-4\pi^{4}l^{2}\varepsilon^{2}+O\left( \varepsilon^{3}\right) $ embedded into the continuous spectrum. This eigenvalue is unique in the segment $\left[ 0,\pi^{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 $\partial\Pi_{l}^{\varepsilon}$ and we discuss available generalizations for other piecewise smooth boundaries.