  A model operator $H$ corresponding to the energy operator of a system with
non-conserved number $n\leq 3$ of particles is considered. The precise location
and structure of the essential spectrum of $H$ is described. The existence of
infinitely many eigenvalues below the bottom of the essential spectrum of $H$
is proved if the generalized Friedrichs model has a virtual level at the bottom
of the essential spectrum and for the number $N(z)$ of eigenvalues below $z<0$
an asymptotics established. The finiteness of eigenvalues of $H$ below the
bottom of the essential spectrum is proved if the generalized Friedrichs model
has a zero eigenvalue at the bottom of its essential spectrum.
