Double-sided estimates for eigenfrequencies in the John problem for freely floating body

The two-dimensional problem on oblique incident waves and a freely floating cylinder is reduced to the study of the spectrum of a suitable self-adjoint operator in Hilbert space. Using tools from spectral measure theory we estimate the difference between eigenfrequencies of the original problem and a problem on an inert body, which does not react to the buoyancy forces. We give the localization of eigenfrequencies of the freely floating body, and in addition derive a sufficient condition for the existence of the point spectrum in the corresponding boundary value problem.