  The present paper is devoted to the study of resonances for a $1$D
Schr\"{o}dinger operator with truncated periodic potential. Precisely, we
consider the half-line operator $H^{\mathbb N}=-\Delta +V$ and $H^{\mathbb
N}_{L}= -\Delta + V1_{[0, L]}$ acting on $\ell^{2}(\mathbb N)$ with Dirichlet
boundary condition at $0$ with $L \in \mathbb N$. We describe the resonances of
$H^{\mathbb N}_{L}$ near the boundary of the essential spectrum of $H^{\mathbb
N}$ as $L \rightarrow +\infty$ under a special assumption.\\ The present paper
is in a series of our research papers on resonances
