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 $\mu\searrow 0.$
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