Scheme for interpretation of approximately computed eigenvalues embedded in a continuous spectrum

It is assumed that a trapped mode (i.e., a function decaying at infinity that leaves small discrepancies of order $\varepsilon\ll1$ in the Helmholtz equation and the Neumann boundary condition) at some frequency $\kappa^0$ is found approximately in an acoustic waveguide $\Omega^0$. Under certain constraints, it is shows that there exists a regularly perturbed waveguide $\Omega^\varepsilon$ with the eigenfrequency $\kappa^\varepsilon=\kappa^0+O(\varepsilon)$. The corresponding eigenvalue $\lambda^\varepsilon$ of the operator belongs to the continuous spectrum and, being naturally unstable, requires “fine tuning” of the parameters of the small perturbation of the waveguide wall. The analysis is based on the concepts of the augmented scattering matrix and the enforced stability of eigenvalues in the continuous spectrum.