  The number state method is used to study soliton bands for three anharmonic
quantum lattices: i) The discrete nonlinear Schr\"{o}dinger equation, ii) The
Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these
systems is assumed to have $f$-fold translational symmetry in one spatial
dimension, where $f$ is the number of freedoms (lattice points). At the second
quantum level $(n=2)$ we calculate exact eigenfunctions and energies of pure
quantum states, from which we determine binding energy $(E_{\rm b})$, effective
mass $(m^{*})$ and maximum group velocity $(V_{\rm m})$ of the soliton bands as
functions of the anharmonicity in the limit $f \to \infty$. For arbitrary
values of $n$ we have asymptotic expressions for $E_{\rm b}$, $m^{*}$, and
$V_{\rm m}$ as functions of the anharmonicity in the limits of large and small
anharmonicity. Using these expressions we discuss and describe wave packets of
pure eigenstates that correspond to classical solitons.
