On the concentration of the point spectrum on the

S. A. Nazarov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 38, Tome 348 (2007), pp. 98-126

Résumé

For the linearized theory of water-waves, we find out families of submersed or surface-piercing bodies in an infinite three-dimensional channel which depend on the small parameter $\varepsilon>0$ and have the following property: For any positive $d$ and integer $J$, there exists $\varepsilon(d,J)>0$ such that, for $\varepsilon\in(0,\varepsilon(d,J)]$, 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 a finite energy.