Clusters of resonances for a non-selfadjoint multichannel discrete Schr\"odinger operator

TL;DR

This paper analyzes the distribution and clustering of resonances for a multichannel discrete Schrödinger operator with non-selfadjoint perturbations, providing exact counts and detailed location descriptions near spectral thresholds.

Contribution

It introduces a precise method to count and locate resonances in clusters for non-selfadjoint multichannel discrete Schrödinger operators near spectral thresholds.

Findings

Exact number of resonances computed

Resonance locations described in clusters

Resonance distribution near spectral thresholds

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 $\ell^2(\mathbb Z; \mathbb C^N) \cong \ell^2(\mathbb Z)\otimes \mathbb C^N$ and $V$ is a non-selfadjoint compact perturbation. We compute the exact number of resonances and give a precise description on their location in clusters around some special points in the complex plane.