On the self-adjointness of H+A*+A

Abstract

Let $H:D(H)\subseteq \mathcal{F}\to \mathcal{F}$ be self-adjoint and let $A:D(H)\to \mathcal{F}$ (playing the role of the annihilator operator) be $H$-bounded. Assuming some additional hypotheses on $A$ (so that the creation operator $A^{\ast }$ 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 $H+A^{\ast }+A$ 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 $H+A_n^{\ast }+A_n+E_n$, 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.

arXiv commentFinal version, to appear in Mathematical Physics, Analysis and Geometry

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Cited by in corpus (9)

• Ground states and associated path measures in the renormalized Nelson model
• Absence of Ground States in the Renormalized Massless Translation-Invariant Nelson Model
• Self-adjointness of a class of multi-spin-boson models with ultraviolet divergences
• The resolvent of the Nelson Hamiltonian improves positivity
• Renormalization of spin$\unicode{x2013}$boson interactions mediated by singular form factors
• Boundary Conditions that Remove Certain Ultraviolet Divergences
• Extended State Space for describing renormalized Fock spaces in QFT
• On the resolvent of $H+A^{\ast }+A$
• Self-adjointness and domain of generalized spin-boson models with mild ultraviolet divergences