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A class of models of quantized, massless Bose fields, called the generalized spinboson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455 503) is considered. Theorems on the absence of ground states and the other eigenvectors of the model without infrared cutoff (but with ultraviolet cutoff ) are established with conditions in terms of correlation functions for some operators. Academic PressKey Words: massless quantum field; Fock space; infrared problem; generalized spin-boson model; ground state; eigenvector; correlation function; particle-field interaction; spin-boson model; Pauli Fierz model. Article ID jfan.1999.3472, available online at http:ÂÂwww.idealibrary.com on 470

We consider a massless scalar Bose field interacting with two particles, one of them infinitely heavy. Neither an infrared nor an ultraviolet cutoff is imposed. In case the charge of the particles is of the same sign and sufficiently small, we prove the existence of a ground state. energy of the atom is approximately still given by Balmer's formula (e 2 Z) 2 m eff /2(4π) 2 plus small radiative correction of order e 6 , where m eff is the effective mass as determined through the energy-momentum relation, see [18] and the discussion in the last section. Here we make a first step by proving the existence of a ground state for e sufficiently small.

We consider a model of a quantum mechanical system coupled to a (massless) Bose field, called the generalized spin-boson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455-503), without infrared regularity condition. We define a regularized Hamiltonian H(ν) with a parameter ν ≥ 0 such that H = H(0) is the Hamiltonian of the original model. We clarify a relation between ground states of H(ν) and those of H by formulating sufficient conditions under which weak limits, as ν → 0, of the ground states of H(ν)'s are those of H. We also establish existence theorems on ground states of H(ν) and H under weaker conditions than in the previous paper mentioned above.Mathematics Subject Classifications (1991): 81Q10, 47B25, 47N50

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