Examples of embedded eigenvalues for problems in acoustic waveguides

This short article discusses the spectrum of the Neumann Laplacian in the infinite domain Ω⊂ℝn, n⩾2 created by inserting a compact obstacle Pinto the uniform cylinder Ω0=(−∞, ∞)×Ω′. The main result is the existence of at least one embedded eigenvalue when Pis an (n−2)‐dimensional surface whose unit normal is parallel to Ω′ at each point of P. The special case when Pis symmetric about {0}×Ω′ is also treated. It is shown that there is at least one symmetric eigenvector and, when Pis sufficiently long, at least one antisymmetric eigenvector. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.