Hamiltonians for polaron models with subcritical ultraviolet singularities

TL;DR

This paper develops a rigorous mathematical framework for polaron models with subcritical ultraviolet singularities, establishing the existence of a well-defined Hamiltonian as a limit of cutoff models.

Contribution

It constructs explicit self-adjoint Hamiltonians for polaron models with ultraviolet singularities, extending the understanding of their mathematical structure.

Findings

Hamiltonians are shown to be self-adjoint without cutoff.

Cutoff Hamiltonians converge strongly to the non-cutoff Hamiltonian.

The approach applies to all subcritical (superrenormalisable) interactions.

Abstract

We treat the ultraviolet problem for polaron-type models in nonrelativistic quantum field theory. Assuming that the dispersion relations of particles and the field have the same growth at infinity, we cover all subcritical (superrenormalisable) interactions. The Hamiltonian without cutoff is exhibited as an explicit self-adjoint operator obtained by a finite iteration procedure. The cutoff Hamiltonians converge to this operator in the strong resolvent sense after subtraction of a perturbative approximation for the ground-state energy.