Spectrum of the semi-relativistic Pauli-Fierz model I

A HVZ type theorem for the semi-relativistic Pauli-Fierz Hamiltonian,H=(p⊗1-A)²+M²⊗1+V⊗1+1⊗H_f,M≥0, in quantum electrodynamics is studied. Here H is a self-adjoint operator in Hilbert space L²(R^d)⊗F≅∫R^d⊕Fdx, A=∫R^d⊕A(x)dx is a quantized radiation field and H_f is the free field Hamiltonian defined by the second quantization of a dispersion relation ω:R^d→R. It is emphasized that massless case, M=0, is included. Let E=inf σ(H) be the bottom of the spectrum of H. Suppose that the infimum of ω is m>0. Then it is shown that σ_ess(H)=[E+m, ∞). In particular the existence of the ground state of H can be proven.