Self-adjointness of a class of multi-spin-boson models with ultraviolet divergences

TL;DR

This paper establishes the mathematical self-adjointness and resolvent formulas for multi-spin-boson quantum models with ultraviolet divergences, extending single-spin results and ensuring stability under form factor approximations.

Contribution

It provides explicit self-adjointness domains and resolvent expressions for complex multi-spin-boson models with ultraviolet divergences, generalizing previous single-spin analyses.

Findings

Explicit self-adjointness domain derived

Resolvent operator expressed via concatenated propagators

Stability under form factor approximations proven

Abstract

We study a class of quantum Hamiltonian models describing a family of $N$ two-level systems (spins) coupled with a structured boson field of positive mass, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences (hence, non-normalizable). Spin-spin interactions which do not modify the total number of excitations are also included. Generalizing previous results in the single-spin case, we provide explicit expressions for the self-adjointness domain and the resolvent operator of such models, both of them carrying an intricate dependence on both the spin-field and spin-spin coupling via a family of concatenated propagators. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors via normalizable ones, for example an ultraviolet cutoff.