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

Abstract

In the present paper we consider a family of $2\times 2$ operator matrices ${\mathcal{A}}_{\mu }(k),$ $k\in {\mathbb{T}}^3:=(-\pi ,\pi {]}^3,$ $\mu >0,$ associated with the Hamiltonian of a system consisting of at most two particles on a three-dimensional lattice ${\mathbb{Z}}^3,$ 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 $z=0=\min {\sigma }_{\mathrm{ess}}({\mathcal{A}}_{\mu }(\overline{0}))$ and the operator ${\mathcal{A}}_{\mu }(\overline{\pi })$ has a virtual level at the point $z=18=\max {\sigma }_{\mathrm{ess}}({\mathcal{A}}_{\mu }(\overline{\pi }))$, where $\overline{0}:=(0,0,0),\overline{\pi }:=(\pi ,\pi ,\pi )\in {\mathbb{T}}^3.$ 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.

8 pages

References in corpus (1)

• On the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case