Perturbation analysis of embedded eigenvalues for water-waves

The starting point of our study is the knowledge that certain surface piercing bodies support a trapped mode, i.e. an embedded eigenvalue in the continuous spectrum. In the framework of the two-dimensional theory of linear water waves, we investigate the question whether a trapped mode still exists after the small perturbation of the body contours. The perturbation of the obstacle is performed by a linear combination of appropriate profile functions. The coefficients of the profile functions and a perturbation parameter of the eigenvalue form a parameter space which controls the embedded eigenvalue as well as the geometry of the water domain. Based on the concept of enforced stability of embedded eigenvalues in the continuous spectrum, we will show that the trapped mode is preserved in the small perturbation, if the profile functions fulfil problem dependent orthogonalisation and normalisation conditions. The argumentation relies on a sufficient condition for the existence of a trapped mode and the notion of the augmented scattering matrix. With the help of asymptotic analysis, we will derive a fixed point equation in the parameter space to determine the appropriate profiles of perturbation. We study the solvability of this equation by the Banach contraction principle.