Embedded eigenvalues for water-waves in a three-dimensional channel with a thin screen

We construct asymptotic expansions as $\varepsilon \to +0$ for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three dimensional channel with a thin screen of thickness $O(\varepsilon)$. The screen may be either submerged or surface-piercing. 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 $\Lambda_\dagger$ of the modified spectrum. The wetted part of the screen has a sharp edge. Depending on a certain integral characteristics $I$ of the screen profiles, we find two types of asymptotics, $\Lambda_\dagger - O(\varepsilon^2)$ and $\Lambda_\dagger - O(\varepsilon^4)$ 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 $[0, \Lambda_\dagger]$, while this interval contains exactly one eigenvalue, if $I \geq 0$. 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.