Essential Spectrum of a Self-Adjoint Operator on an Abstract Hilbert Space of Fock Type and Applications to Quantum Field Hamiltonians

AbstractWe establish general theorems on locating the essential spectrum of a self-adjoint operator of the form A⊗I+I⊗dΓ(S)+HI on the tensor product H⊗Fb(K) of a Hilbert space H and the abstract Boson Fock space Fb(K) over a Hilbert space K, where A is a self-adjoint operator on H bounded from below, dΓ(S) is the second quantization of a nonnegative self-adjoint operator S on K, and HI is a symmetric operator on H⊗Fb(K). We then apply the theorems to the generalized spin-boson model by A. Arai and M. Hirokawa (1997, J. Funct. Anal.151, 455–503) and a general class of models of quantum particles coupled to a Bose field including the Pauli–Fierz model in nonrelativistic quantum electrodynamics