Ground states of a general class of quantum field Hamiltonians (Q2703038)

The paper is an exposition in full detail of the authors' results in [J. Funct. Anal. 151, 455-503 (1997; Zbl 0898.47048)], along with a few extensions, on the existence of the ground state in Fock space \({\mathcal F}= {\mathcal H}\otimes {\mathcal F}_b\) for spin-boson models, where the quantum system (``spin'') of the Hamiltonian \(A\) in \({\mathcal H}\) is coupled to a free Bose field \( \phi \) of the fispersion law \(\omega(k)\) in \({\mathcal F}_b\) by an interaction linear in the Bose field NEWLINE\[NEWLINE H=A\otimes I+I\otimes d\Gamma (\hat \omega)+\sum S_i\otimes \phi (g_i). NEWLINE\]NEWLINE The main assumptions are (i) \(\omega ^{-1/2}g_i\in L^2\) and (ii) a certain relative boundedness property of the interaction; neither discreteness of the energy spectrum, nor boundedness, are required for the ''spin'' operators \(A\), \(S_i\). Also, infrared singularity \(\omega ^{-1}g_i\notin L^2\) is not excluded. The infrared problem is approached by regularisation \(H(\nu)\) of the Hamiltonian \(H\) via a mass gap \(\nu \) in the bare-boson spectrum \( \omega ^{\left( \nu \right) }=\nu +\omega \) (instead of low-momentum cut-off), which ensures that \(H(\nu)\rightarrow H\) in strong- (intead of norm-) resolvent sense, and the ground state is constructed as weak limit point of ground states of \(H (\nu)\). In case of regularity the existence is proven, while examples are given to show that a ground state may exist in infrared singular models (Wigner-Weisskopf).