Resonances for 1D half-line periodic operators: I. Generic case

The present paper addresses questions on 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 $H^{\mathbb N}_L = -\Delta + V 1_{[0,L]}$ 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 $L \rightarrow +\infty$ in the generic case.