On the Self-Adjointness of H+A\u2217+A

Abstract

Let H:text {dom}(H)subseteq mathfrak {F}to mathfrak {F} be self-adjoint and let A:text {dom}(H)to mathfrak {F} (playing the role of the annihilation 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 Kreı̆n-type formula, we build self-adjoint realizations widehat H of the formal Hamiltonian H + A∗ + A with text {dom}(H)cap text {dom}(widehat H)={0}. We give an explicit characterization of text {dom}(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}+A_{n}+E_{n}, where the An’s are regularized operators approximating A and the En’s are suitable renormalizing bounded operators. These results show the connection between the construction of singular perturbations of self-adjoint operators by Kreı̆n’s resolvent formula and nonperturbative theory of renormalizable models in Quantum Field Theory; in particular, as an explicit example, we consider the Nelson model.