Resonances near spectral thresholds for multichannel discrete Schrödinger operators

Abstract: We study the distribution of resonances for discrete Hamiltonians of the form $H_0+V$ near the thresholds of the spectrum of $H_0$. Here, the unperturbed operator $H_0$ is a multichannel Laplace type operator on $\ell^2(\mathbb Z; \mathfrak{G}) \cong \ell^2(\mathbb Z)\otimes \mathfrak{G}$ where ${\mathfrak G}$ is an abstract separable Hilbert space, and $V$ is a suitable non-selfadjoint compact perturbation. We distinguish two cases. If ${\mathfrak G}$ is of finite dimension, we prove that resonances exist and do not accumulate at the thresholds in the spectrum of $H_0$. Furthermore, we compute exactly their number and give a precise description on their location in clusters in the complex plane. If ${\mathfrak G}$ is of infinite dimension, an accumulation phenomenon occurs at some thresholds. We describe it by means of an asymptotical analysis of the counting function of resonances. Consequences on the distribution of the complex and the embedded eigenvalues are also given.