In the large polaron model of H. 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.

The Nelson Hamiltonian is unitarily equivalent to a Hamiltonian defined through a closed, semibounded quadratic form, the unitary transformation being explicitly known and due to Gross. In this paper we study mapping properties of the Grosstransform in order to characterize regularity properties of vectors in the form domain of the Nelson Hamiltonian. Since the operator domain is a subset of the form domain, our results apply to vectors in the domain of the Hamiltonian was well. -This work is a continuation of our previous work on the Fröhlich Hamiltonian.

We study the motion of neutral test particles in the gravitational field of two charged black holes described by the extreme Reissner-Nordstrøm dihole metric where the masses and charges of the black holes are chosen such that the gravitational attraction is compensated by the electrostatic repulsion. We investigate circular orbits in the equatorial plane between the two black holes with equal masses as well as the case of circular orbits outside this symmetry plane. We show that the first case reduces to an effective two-body problem with a behavior similar to a system described by the Reissner-Nordstrøm spacetime. The main focus is directed to the second case with circular orbits outside the equatorial plane.

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