  A model operator $H$ associated with the energy operator of a system
describing three particles in interaction, without conservation of the number
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 for the
case where an associated generalized Friedrichs model has a resonance at the
bottom of its essential spectrum. An asymptotics for the number $N(z)$ of
eigenvalues below the bottom of the essential spectrum is also established. The
finiteness of eigenvalues of $H$ below the bottom of the essential spectrum is
proved if the associated generalized Friedrichs model has an eigenvalue with
energy at the bottom of its essential spectrum.
