  The energy spectrum of the quantum Klein-Gordon lattice is computed
numerically for different nonlinear contributions to the Hamiltonian. In
agreement with the studies on the effective Hubbard Hamiltonian for boson
quasi-particles (see for instance Refs.\onlinecite{AGRANOVICH,Eilbeck}) a
pairing of the phonon states is found when the nonlinearity of the lattice is
significant. On the opposite, when the nonlinear contribution is weak or
moderate, which is common in materials the effective Hamiltonian is not
appropriate because it neglects all the energy terms that do not conserve the
boson number. Then for a realistic modelling of the hybridization between the
free phonon and the phonon bound pairs, the Klein-Gordon Hamiltonian is
required since it is derived from the potential energy of the atoms and thus it
does not involve any arbitrary quanta conservation. Actually, when the
nonlinearity is weak we prove that the binding energy of the phonon bound pairs
vanishes at the center of the lattice Brillouin zone whereas at the edge, it
may be comparable to the phonon band width.Consequently, the signature of a
weak nonlinearity is found to be a pseudogap that opens in the spectrum region
of the two phonon energy, at the edge of the lattice Brillouin zone. Our
results are shown to be valid for all the lattice dimensions and for some model
parameters that are relevant for the optical phonon spectra.
