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

Abstract

The present paper addresses questions on resonances for a $1$D Schrödinger operator with truncated periodic potential. Precisely, we consider the half-line operator $H^{\mathbb{N}}=-\Delta +V$ and $H_L^{\mathbb{N}}=-\Delta +V{1}_{[0,L]}$ acting on ${\ell }^2(\mathbb{N})$ with Dirichlet boundary condition at $0$ with $L\in \mathbb{N}$. We describe the resonances of $H_L^{\mathbb{N}}$ near the boundary of the essential spectrum of $H^{\mathbb{N}}$ as $L\to +\infty$ in the generic case.

28 pages, 3 figures

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