Embedded Eigenvalues for Water-Waves in a Three-Dimensional Channel with a Thin Screen

Abstract

We construct asymptotic expansions as ε → +0 for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three dimensional channel with a thin screen of thickness O(ε). The screen may be either submerged or surface-piercing, and its wetted part has a sharp edge. The channel and the screen are mirror symmetric so that imposing the Dirichlet condition in the middle plane creates an artificial positive cut-off-value Λ† of the modified spectrum. Depending on a certain integral characteristic I of the screen profiles, we find two types of asymptotics of eigenvalues, λ = Λ† − O(ε) and λ = Λ† −O(ε) in the cases I > 0 and I = 0, respectively. We prove that in the case I < 0 there are no embedded eigenvalues in the interval [0,Λ†], while this interval contains exactly one eigenvalue, if I ≥ 0. For the justification of these result, the main tools are a reduction to an abstract spectral equation and the use of the max-min-principle.