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.
Categories: math-ph math.FA math.MP quant-ph