is emphasized that the uniqueness in this problem is only up to the eigenfunctions with complex frequencies, i.e., up to the quasi-normal modes.

Thus imposing the radiation conditions with real $k$ we remove the quasi-normal modes from our consideration only. However this cannot prevent the excitation of these modes in the real physical problem. Hence, when dealing with systems unbounded in space (open systems) one has always to investigate the consequences of qnm excitation. An example of such a problem will be considered in Sec. III.

As a very simple and physically motivated example of quasi-normal modes we consider here the oscillations of electromagnetic field outside a perfectly conducting sphere of radius $a$. In this case the electric and magnetic fields are expressed in terms of two scalar functions $f_{kl}^{\rm{TE}}(r)$ and $f_{kl}^{\rm{TM}}(r)$ (Debye potentials [@Stratton]) which are the radial parts of the solutions to the scalar wave equation (\[2-1\]). Outside the perfectly conducting sphere placed in vacuum the solution to the Helmholtz equation (\[2-3\]) obeying the radiation conditions (\[2-5\]) has the form $(d=3)$ $$\label{2-6} f_{kl}(r)=C\, h^{(1)}_l\left ( \frac{\omega}{c}\,r
\right ){,}\quad r>a\,{,}$$ where $ h^{(1)}_l(z)$ is the spherical Hankel function of the first kind [@AS]. At the surface of perfectly conducting 