Investigations of the Numerical Range of a Operator Matrix

We consider a $2\times2$ operator matrix $A$ (so-called generalized Friedrichs model) associated with a system of at most two quantum particles on ${\mathrm d}-$ dimensional lattice. This operator matrix acts in the direct sum of zero- and one-particle subspaces of a Fock space. We investigate the structure of the closure of the numerical range $W(A)$ of this operator in detail by terms of its matrix entries for all dimensions of the torus ${\mathbf T}^{\mathrm d}$. Moreover, we study the cases when the set $W(A)$ is closed and give necessary and sufficient conditions under which the spectrum of $A$ coincides with its numerical range.