Self-adjointness and domain of the Fröhlich Hamiltonian

Abstract

In the large polaron model of Herbert Fröhlich, 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 Fröhlich 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 analyse the smoothness of vectors in the domain of the Hamiltonian with respect to the position of the electron.