Self-Adjointness and Domain of the Froehlich Hamiltonian

TL;DR

This paper investigates the self-adjointness domain of the Froehlich Hamiltonian in the large polaron model, providing an explicit characterization using a dressing transform and analyzing the smoothness of vectors in the domain.

Contribution

It offers a new explicit description of the Hamiltonian's domain via a dressing transform, advancing understanding of the operator's mathematical structure.

Findings

Explicit domain characterization using a dressing transform

Analysis of vector smoothness in the Hamiltonian's domain

Application of a new operator bound to the Froehlich model

Abstract

In the large polaron model of H. Froehlich, the electron-phonon interaction is a small perturbation in form sense, but a large perturbation in operator sense. This means that the form-domain of the Hamiltonian is not affected by the interaction but the domain of self-adjointness is. In the particular case of the Froehlich model, we are nevertheless able, thanks to a recently published new operator bound, to give an explicit characterization of the domain in terms of a suitable dressing transform. Using the mapping properties of this dressing transform, we analyze the smoothness of vectors in the domain of the Hamiltonian with respect to the position of the electron.