Dyson expansion for form-bounded perturbations, and applications to the polaron problem

TL;DR

This paper develops an abstract Dyson expansion for form-bounded perturbations and applies it to polaron models, demonstrating properties of the vacuum expectation value and ground state energy related to momentum.

Contribution

It introduces a Dyson expansion applicable to form-bounded perturbations and extends results on polaron models without probabilistic methods.

Findings

Vacuum expectation value of heat semi-group is completely monotone in momentum squared

Ground state energy is a concave function of momentum squared

Results apply to a broad class of polaron-type models

Abstract

We present an abstract Dyson expansion for perturbations that are merely relatively form-bounded, and apply it to the polaron problem. For a large class of polaron-type models, including the Fr\"ohlich and Nelson models, we prove that the vacuum expectation value of the heat semi-group is a completely monotone function of the square of the total momentum. Consequently, the ground state energy is a concave function of the square of the momentum, a result recently proved for the Fr\"ohlich model in \cite{polzer} using a probabilistic approach via Wiener integrals.