On the Resolvent of H+A $$^{*}$$ +A

We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of \(H+A^{*}+A\). Math. Phys. Anal. Geom. 23 (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind \(H+A^{*}+A\), where H and A play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Kreĭn-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind \(H+A^{*}_{n}+A_{n}-E_{n}\), the bounded operator \(E_{n}\) playing the role of a renormalizing counter term. These abstract results apply to various concrete models in Quantum Field Theory.