On the Self-Adjointness of H+A∗+A

Abstract

Let H: dom (H) ⊆ F→ F be self-adjoint and let A: dom (H) → 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 Ĥ of the formal Hamiltonian H + A∗ + A with dom (H) ∩ dom (Ĥ) = { 0 }. We give an explicit characterization of dom (Ĥ) and provide a formula for the resolvent difference (− Ĥ + z) − 1− (− H+ z) − 1. Moreover, we consider the problem of the description of Ĥ as a (norm resolvent) limit of sequences of the kind H+An∗+An+En, 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.