CONCENTRATION OF THE POINT SPECTRUM ON THE CONTINUOUS ONE IN PROBLEMS OF LINEAR WATER-WAVE THEORY

摘要

For the linear theory of water waves, we. find out families of submerged or surface-piercing bodies in an infinite three-dimensional canal, which depend on a small pa.ram.eter ε > 0 and have the following property: for any positive d and natural J, there exists ε(d, J) > 0 such that, for ε ∈ (0. ε(d,J)), the segment [0, d] of the continuous spectrum, of the problem, contains at least J eigenvalues. These eigenvalues are associated, with trapped modes, i.e., solutions of the homogeneous problem, which decay exponentially at infinity and possess finite energy.