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

Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{*}$ is a singular perturbation of $H$), by a twofold application of a resolvent Krein-type formula, we build self-adjoint realizations $\hat H$ of the formal Hamiltonian $H+A^{*}+A$ with $D(H)\cap D(\hat H)=\{0\}$. We give an explicit characterization of $D(\hat H)$ and provide a formula for the resolvent difference $(-\hat H+z)^{-1}-(-H+z)^{-1}$. Moreover, we consider the problem of the description of $\hat H$ as a (norm resolvent) limit of sequences of the kind $H+A^{*}_{n}+A_{n}+E_{n}$, where the $A_{n}\!$'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.