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 $H^{\mathbb{N}}=-\mathrm{\Delta }+V$ and $H^{\mathbb N}L = -\Delta + V 1{[0,L]}$actingon$\ell^{2}(\mathbb N)$ with Dirichlet boundary condition at $0$ with $L\in \mathbb{N}$. We describe the resonances of $H_L^{\mathbb{N}}$ near the boundary of the essential spectrum of $H^{\mathbb N}$as$L \rightarrow +\infty$ in the generic case.