On the self-adjointness of H+A*+A

Let $H:D(H)\subseteq \mathcal{F}\to \mathcal{F}$ be self-adjoint and let $A:D(H)\to \mathcal{F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{\ast }$ is a singular perturbation of $H$), by a twofold application of a resolvent Krein-type formula, we build self-adjoint realizations $\widehat{H}$ of the formal Hamiltonian $H+A^{\ast }+A$ with $D(H)\cap D(\widehat{H})=\{0\}$. We give an explicit characterization of $D(\widehat{H})$ and provide a formula for the resolvent difference $(-\widehat{H}+z{)}^{-1}-(-H+z{)}^{-1}$. Moreover, we consider the problem of the description of $\widehat{H}$ as a (norm resolvent) limit of sequences of the kind $H+A_n^{\ast }+A_n+E_n$, where the $A_n\text{\hspace{0.17em}⁣}$'s are regularized operators approximating $A$ and the $E_n$'s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Krein’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.