Weak coupling limit with a removal of an ultraviolet cutoff for a Hamiltonian of particles interacting with a massive scalar field

A Hamiltonian of an interaction system between N-particles and a massive scalar field is considered. The Hamiltonian with an ultraviolet cutoff is defined as a self-adjoint operator acting in a Hilbert space. Renormalized Hamiltonians are defined by subtracting renormalization terms from the Hamiltonian. It is shown that N-body Schrödinger Hamiltonians can be derived from taking a weak coupling limit and removing the ultraviolet cutoff simultaneously for the renormalized Hamiltonians. In particular, in the case where the space dimension equals three, the Yukawa potential appears in the N-body Schrödinger Hamiltonian. It is also shown that, in the case where the space dimensions are one or two, infimum of the spectra of the renormalized Hamiltonians converge to those of the N-body Schrödinger Hamiltonians.