Trapping of a wave in a curved cylindrical acoustic waveguide with constant cross-section

Cylindrical acoustic waveguides with constant cross-section  $ \omega $ are considered, specifically, a straight waveguide $ \Omega ={\mathbb{R}}\times \omega \subset {\mathbb{R}}^d$ and a locally curved waveguide $ \Omega ^\varepsilon $ that depends on a parameter $ \varepsilon \in (0,1]$. For $ d>2$, in two different settings ( $ \varepsilon =1$ and $ \varepsilon \ll 1$), the task is to find an eigenvalue $ \lambda ^\varepsilon $ that is embedded in the continuous spectrum $ [0,+\infty )$ of the waveguide $ \Omega ^\varepsilon $ and, hence, is inherently unstable. In other words, a solution of the Neumann problem for the Helmholtz operator $ \Delta +\lambda ^\varepsilon $ arises that vanishes at infinity and implies an eigenfunction in the Sobolev space $ H^1(\Omega ^\varepsilon )$. In the first case, it is assumed that the cross-section $ \omega $ has a double symmetry and an eigenvalue arises for any nontrivial curvature of the axis of the waveguide $ \Omega ^\varepsilon $. In the second case, under an assumption on the shape of an asymmetric cross-section $ \omega $, the eigenvalue $ \lambda ^\varepsilon $ is formed by scrupulous fitting of the curvature $ O(\varepsilon )$ for small $ \varepsilon >0$.