and spatial behaviour is a direct consequence of radiation conditions (\[2-5\]) and it is typical for eigenfunctions describing oscillations in external unbounded regions, the physical content and details of oscillation process being irrelevant. The eigenfunctions corresponding to complex eigenvalues are called [*quasi-normal modes*]{} keeping in mind their unusual properties [@Nollert]. The physical origin of such features is obvious, in fact we are dealing here with [*open systems*]{} in which the energy can be radiated to infinity. Therefore in open systems field cannot acquire a stationary state.

Quasi-normal modes do not obey the standard completeness condition and the notion of norm cannot be defined for them [@Nollert]. Therefore these eigenfunctions cannot be used for expansion of the classical field with the aim to quantize it and to introduce the relevant Fock operators. The treatment of these problems can be found, for example, in [@Ching; @Leung; @Chang].

It is worthy to investigate the spatial form of the of quasi-normal modes with allowance for their time dependence. Indeed, these solutions have the character of propagating waves that are eventually going to spatial infinity. Let us take the point which is sufficiently far from the region with nontrivial dynamics in the system under study. 