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

TL;DR

This paper develops a method to construct self-adjoint realizations of Hamiltonians with singular perturbations using Krein's resolvent formula, with applications to quantum field theory models like the Nelson model.

Contribution

It introduces a novel approach to define self-adjoint operators for Hamiltonians with singular creation and annihilation operators via resolvent formulas and limits.

Findings

Explicit characterization of the domain of the constructed Hamiltonian

Formula for the resolvent difference between the perturbed and unperturbed operators

Application to the Nelson model demonstrating the method's relevance

Abstract

Let $H:D(H)\subseteq{\mathscr F}\to{\mathscr F}$ be self-adjoint and let $A:D(H)\to{\mathscr F}$ (playing the role of the annihilator 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 Krein-type formula, we build self-adjoint realizations $\hat H$ of the formal Hamiltonian $H+A^{*}+A$ 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 $H+A^{*}_{n}+A_{n}+E_{n}$, 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.