Abstract. In this paper, we consider the spectrum of a model in quantum electrodynamics with a spatial cutoff. It is proven that (1) the Hamiltonian is self-adjoint; (2) under the infrared regularity condition, the Hamiltonian has a unique ground state for sufficiently small values of coupling constants. The spectral scattering theory is studied as well and it is shown that asymptotic fields exist and the spectral gap is closed.

In this paper we consider generalized spin-boson models with singular perturbations. It is proven that under the infrared regularity condition Hamiltonians have the unique ground state for sufficiently small values of coupling constants. In addition it is shown that the asymptotic creation and annihilation operators of massless boson field exist.

Ground states of the so called Yukawa model is considered. The Yukawa model describes a Dirac field interacting with a Klein-Gordon field. By introducing both ultraviolet cutoffs and spatial cutoffs, the total Hamiltonian is defined as a self-adjoint operator on a boson-fermion Fock space. It is shown that the total Hamiltonian has a positive spectral gap for all values of coupling constants. In particular the existence of ground states is proven.

An interaction system of a fermionic quantum field is considered. The state space is defined by a tensor product space of a fermion Fock space and a Hilbert space. It is assumed that the total Hamiltonian is a self-adjoint operator on the state space and bounded from below. Then it is proven that a subset of real numbers is the essential spectrum of the total Hamiltonian. It is applied to a Yukawa interaction system, which is a system of a Dirac field coupled to a Klein-Gordon, and the HVZ theorem is obtained.

Submitted by Stephen Fulling Keywords: Gibbs measure on path space Random-time Poisson process Ground state Boson number operator Nelson model a b s t r a c tWe give a formula in terms of a joint Gibbs measure on Brownian paths and the measure of a random-time Poisson process of the ground state expectations of fractional (in fact, any real) powers of the boson number operator in the Nelson model. We use this representation to obtain tight two-sided bounds. As applications, we discuss the polaron and translation invariant Nelson models.