On the domain of the Nelson Hamiltonian
Marcel Griesemer, Andreas W\"unsch
TL;DR
This paper investigates the regularity properties of vectors in the Nelson Hamiltonian's form domain by analyzing the Gross-transform, extending previous work on related Hamiltonians.
Contribution
It characterizes the mapping properties of the Gross-transform to understand the regularity of vectors in the Nelson Hamiltonian's form domain.
Findings
Mapping properties of the Gross-transform are established.
Regularity properties of vectors in the form domain are characterized.
Results extend to vectors in the operator domain.
Abstract
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 Gross-transform 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 Fr\"ohlich Hamiltonian.
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