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 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 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+Aₙ-Eₙ H + A n ∗ + A n - E n , the bounded operator Eₙ E n playing the role of a renormalizing counter term. These abstract results apply to various concrete models in Quantum Field Theory.