  Quantized nonlinear lattice models are considered for two different classes,
boson and fermionic ones. The quantum discrete nonlinear Schroedinger model
(DNLS) is our main objective, but its so called modified discrete nonlinear
(MDNLS) version is also included, together with the fermionic polaron (FP)
model. Based on the respective dynamical symmetries of the models, a method is
put forward which by use of the associated boson and spin coherent state
vectors (CSV) and a factorization ansatz for the solution of the Schroedinger
equation, leads to quasiclassical Hamiltonian equations of motion for the CSV
parameters. Analysing the geometrical content of the factorization ansatz made
for the state vectors invokes the study of the Riemannian and symplectic
geometry of the CSV manifolds as generalized phase spaces. Next, we investigate
analytically and numerically the behavior of mean values and uncertainties of
some physically interesting observables as well as the modifications in the
quantum regime of processes such as the discrete self trapping (DST), in terms
of the Q-function and the distribution of excitation quanta of the lattice
sites. Quantum DST in the symmetric ordering of lattice operators is found to
be relatively enhanced with respect to the classical DST. Finally, the meaning
of the factorization ansatz for the lattice wave function is explained in terms
of disregarded quantum correlations, and as a quantitative figure of merit for
that ansatz a correlation index is introduced.
