Threshold analysis for a family of $2 \times 2$ operator matrices

TL;DR

This paper analyzes the spectral properties of a family of 2x2 operator matrices related to a two-particle lattice system, identifying conditions for threshold eigenvalues and virtual levels based on coupling constants and specific momentum points.

Contribution

It introduces a detailed threshold analysis for a class of operator matrices modeling two-particle lattice interactions, providing necessary and sufficient conditions for spectral threshold phenomena.

Findings

Identifies a specific set of momentum points where threshold phenomena occur.

Establishes a critical coupling constant value for spectral threshold behavior.

Provides criteria for the existence of eigenvalues or virtual levels at thresholds.

Abstract

We consider a family of $2 \times 2$ operator matrices ${\mathcal A}_\mu(k),$ $k \in {\Bbb T}^3:=(-\pi, \pi]^3,$ $\mu>0$, acting in the direct sum of zero- and one-particle subspaces of a Fock space. It is associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\Bbb Z}^3,$ interacting via annihilation and creation operators. We find a set $\Lambda:=\{k^{(1)},...,k^{(8)}\} \subset {\Bbb T}^3$ and a critical value of the coupling constant $\mu$ to establish necessary and sufficient conditions for either $z=0=\min\limits_{k\in {\Bbb T}^3} \sigma_{\rm ess}({\mathcal A}_\mu(k))$ ( or $z=27/2=\max\limits_{k\in {\Bbb T}^3} \sigma_{\rm ess}({\mathcal A}_\mu(k))$ is a threshold eigenvalue or a virtual level of ${\mathcal A}_\mu(k^{(i)})$ for some $k^{(i)} \in \Lambda.$