  We study the spectral properties of discrete Schr\"odinger operator $$
\widehat H_\mu=\widehat H_0 + \mu \widehat{V},\qquad \mu\ge0, $$ associated to
a one-particle system in $d$-dimensional lattice $\mathbb{Z}^d, $ $d=1,2,$
where the non-perturbed operator $\hat H_0$ is a self-adjoint
Laurent-Toeplitz-type operator generated by $\hat e:\mathbb{Z}^d\to\mathbb{C}$
and the potential $\hat V$ is the multiplication operator by $\hat
v:\mathbb{Z}^d\to\mathbb{R}.$ Under certain regularity assumption on $\hat e$
and a decay assumption on $\hat v$, we establish the existence or non-existence
and also the finiteness of eigenvalues of $\hat H_\mu.$ Moreover, in the case
of existence we study the asymptotics of eigenvalues of $\hat H_\mu$ as
$\mu\searrow 0.$
