Spectrum of the semi-relativistic Pauli-Fierz model I

HVZ type theorem for semi-relativistic Pauli-Fierz Hamiltonian, $$\HHH=\sqrt{(p\otimes \one -A)^2+M^2}+V\otimes \one +\one\otimes \hf,\quad M\geq 0,$$ in quantum electrodynamics is studied. Here $H$ is a self-adjoint operator in Hilbert space $\LR\otimes \fff\cong \int^\oplus_{\RR^d}\fff {\rm d}x$, and $A=\int^\oplus_{\RR^d} A(x) {\rm d}x$ a quantized radiation field and $\hf$ the free field Hamiltonian defined by the second quantization of a dispersion relation $\omega:\RR^d\to \RR$. It is emphasized that massless case, $M=0$, is included. Let $E=\inf \sigma (\HHH)$ be the bottom of the spectrum of $\HHH$. Suppose that the infimum of $\omega$ is $m>0$. Then it is shown that $\sigma_{\rm ess}(\HHH)=[E+m, \infty)$. In particular the existence of the ground state of $\HHH$ can be proven.