Obviously, it has sense to say about the value of the quasi-normal mode at a given point only after arrival at this point of the wave described by this mode. The maximal value of the quasi-normal mode is observed just at the moment of its arrival at this point. At the later moments the quasi-normal mode is dumping due to its characteristic time dependence. Indeed, taking into account all this we obtain for the maximal observed value of the quasi-normal mode (\[2-12\]) the following physically acceptable expression $$\label{2-13}\left ( e^{-i \omega t}f_{k1}^{\text{TE}}(r)\right
)_{\text{max-obs}}=-i \,C\,\frac{a}{r}\left ( 1-\frac{a}{r} \right
), \quad r\geq a\,{.}$$ Thus, in our consideration the problem of unbounded (exponential) rising of quasi-normal modes, when $r\to \infty$, does not arise.

In order to associate with resonance phenomenon a single square integrable eigenfunction, rather sophisticated methods are used, for example, complex scaling [@scaling] (known also as the complex-coordinate method or as the complex-rotational method).

The necessary condition for appearing the quasi-normal modes is imposing the radiation conditions at spatial infinity on the field functions. When other conditions are used at spatial infinity, the quasi-normal modes do not arise. For example, if we demand that the solution to the wave equation (\[2-1\]) 