Existence of the ground state for the Pauli-Fierz model with a variable mass

The Pauli-Fierz model with a variable mass $v$ is considered. An ultraviolet cutoff and an infrared regularity condition are imposed on a quantized radiation field. It is shown that the ground state exists for arbitrary values of coupling constants, when $v(x)\le c⟨x{⟩}^{-\beta }$, $\beta >3$.