Eigenvalues and virtual levels of a family of $2 \times 2$ operator matrices

TL;DR

This paper investigates the spectral properties of a family of 2x2 operator matrices modeling a two-particle system on a 3D lattice, identifying a critical parameter value where virtual levels occur at spectrum edges.

Contribution

It establishes the existence of a unique parameter value where virtual levels appear at spectrum edges for the operator family, and provides threshold energy expansions.

Findings

Existence of a critical parameter  where virtual levels occur.

Identification of virtual levels at spectrum edges for specific parameter values.

Derivation of threshold energy expansions for the Fredholm determinant.

Abstract

In the present paper we consider a family of $2 \times 2$ operator matrices ${\mathcal A}_\mu(k),$ $k \in {\Bbb T}^3:=(-\pi, \pi]^3,$ $\mu>0,$ associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\Bbb Z}^3,$ interacting via creation and annihilation operators. We prove that there is a value $\mu_0$ of the parameter $\mu$ such that only for $\mu=\mu_0$ the operator ${\mathcal A}_\mu(\overline{0})$ has a virtual level at the point $z=0=\min\sigma_{\rm ess}({\mathcal A}_\mu(\overline{0}))$ and the operator ${\mathcal A}_\mu(\overline{\pi})$ has a virtual level at the point $z=18=\max\sigma_{\rm ess}({\mathcal A}_\mu(\overline{\pi}))$, where $\overline{0}:=(0,0,0), \overline{\pi}:=(\pi,\pi,\pi) \in {\Bbb T}^3.$ The absence of the eigenvalues of ${\mathcal A}_\mu(k)$ for all values of $k$ under the assumption that $\mu=\mu_0$ is shown. The threshold energy expansions for the Fredholm determinant associated to ${\mathcal A}_\mu(k)$ are obtained.