  The eigenvalue problem for the dressed bound-state of unstable multilevel
systems is examined both outside and inside the continuum, based on the N-level
Friedrichs model which describes the couplings between the discrete levels and
the continuous spectrum. It is shown that a bound-state eigenenergy always
exists below each of the discrete levels that lie outside the continuum.
Furthermore, by strengthening the couplings gradually, the eigenenergy
corresponding to each of the discrete levels inside the continuum finally
emerges. On the other hand, the absence of the eigenenergy inside the continuum
is proved in weak but finite coupling regimes, provided that each of the form
factors that determine the transition between some definite level and the
continuum does not vanish at that energy level. An application to the
spontaneous emission process for the hydrogen atom interacting with the
electromagnetic field is demonstrated.
