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

Abstract

We consider a family of $2\times 2$ operator matrices ${\mathcal{A}}_{\mu }(k),$ $k\in {\mathbb{T}}^3:=(-\pi ,\pi {]}^3,$ $\mu >0$, 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 ${\mathbb{Z}}^3,$ interacting via annihilation and creation operators. We find a set $\Lambda :=\{k^{(1)},...,k^{(8)}\}\subset {\mathbb{T}}^3$ and a critical value of the coupling constant $\mu$ to establish necessary and sufficient conditions for either $z=0=\underset{k\in {\mathbb{T}}^3}{\min }{\sigma }_{\mathrm{ess}}({\mathcal{A}}_{\mu }(k))$ ( or $z=27/2=\underset{k\in {\mathbb{T}}^3}{\max }{\sigma }_{\mathrm{ess}}({\mathcal{A}}_{\mu }(k))$ is a threshold eigenvalue or a virtual level of ${\mathcal{A}}_{\mu }(k^{(i)})$ for some $k^{(i)}\in \Lambda .$

9 pages

References in corpus (1)

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