On the resolvent of $H+A^{\ast }+A$

Abstract

We present a much shorter and streamlined proof of an improved version of the results previously given in [A. Posilicano: On the Self-Adjointness of $H+A^{\ast }+A$, Math. Phys. Anal. Geom. (2020)] concerning the self-adjoint realizations of formal QFT-like Hamiltonians of the kind $H+A^{\ast }+A$, where $H$ and $A$ play the role of the free field Hamiltonian and of the annihilation operator respectively. We give explicit representations of the resolvent and of the self-adjointness domain; the consequent Krein-type resolvent formula leads to a characterization of these self-adjoint realizations as limit (with respect to convergence in norm resolvent sense) of cutoff Hamiltonians of the kind $H+A_n^{\ast }+A_n-E_n$, the bounded operator $E_n$ playing the role of a renormalizing counter term.

arXiv commentrevised version, denseness hypothesis of ran(A) removed

References in corpus (7)

• On Nelson-type Hamiltonians and abstract boundary conditions
• On the domain of the Nelson Hamiltonian
• The Renormalised Bogoliubov-Fröhlich Hamiltonian
• On a Direct Description of Pseudorelativistic Nelson Hamiltonians
• On the self-adjointness of H+A*+A
• Self-adjointness of a class of multi-spin-boson models with ultraviolet divergences
• Hamiltonians for polaron models with subcritical ultraviolet singularities