initial solutions in time accounting the fact that outgoing waves take away the energy. In other words, we are dealing here with the radiation of the energy with the amplitude decaying in time.

Indeed, if we remove the requirement of reality of the wave vector $k$, then the homogeneous wave equation (2.1) and the Helmholtz equation (2.3) will have nonzero solutions with [*complex frequencies*]{}, these solutions obeying the radiation conditions (\[2-5\]) and a common boundary condition on a compact surface $S$ (for instance, Dirichlet or Neumann conditions). In quantum mechanics the radiation condition with a complex wave number $k$ is known as the Gamov condition which singles out the resonance states in the spectrum of the Hamiltonian [@Gamov; @G1; @G2].

When introducing the radiation conditions and proving the respective uniqueness theorem in the text books [@RR] only the real wave vector $k$ is considered. The possibility of existence of quasi-normal modes with complex frequencies satisfying the radiation conditions at spatial infinity with a complex $k$ is not mentioned usually. We are aware only of alone textbook where the eigenfunctions with complex frequencies are noted in this context. It is the article written by Sommerfeld in the book [@FM], where it 