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.

Final version, to appear in Mathematical Physics, Analysis and Geometry

References in corpus (1)

• Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

Cited by in corpus (7)

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• Renormalization of spin$\unicode{x2013}$boson interactions mediated by singular form factors
• Extended State Space for describing renormalized Fock spaces in QFT
• Self-adjointness and domain of generalized spin-boson models with mild ultraviolet divergences
• On the resolvent of $H+A^{\ast }+A$