Spectrum of the semi-relativistic Pauli-Fierz model II

We consider the semi-relativistic Pauli-Fierz Hamiltonian $$H_m=\mathrm{\mid }\mathbf{p}-\mathbf{A}(\mathbf{x})\mathrm{\mid }+H_{f,m}+V(\mathbf{x}),m\ge 0,$$ and prove the existence of the ground state of $H_m$ for $m=0$. Here $\mathbf{A}(\mathbf{x})$ denotes a quantized radiation field and $H_{f,m}$ the free field Hamiltonian with the dispersion relation $\sqrt{\mathrm{\mid }\mathbf{k}{\mathrm{\mid }}^2+m^2}$ with $m\ge 0$. This paper is the sequel of [HH16], where the existence of the ground state ${\mathrm{\Phi }}_m$ of $H_m$ for $m>0$ is proven. In order to show the existence of the ground state for $m=0$ we estimate a singular and non-local pull-through formula and show the equicontinuity of set $\{a(k){\mathrm{\Phi }}_m{\}}_{0<m<m_0}$ with some $m_0$, where $a(k)$ denotes the formal kernel of the annihilation operator. Taking a subsequence $m_j$, we can conclude that ${\lim }_{m_j\to 0}{\mathrm{\Phi }}_{m_j}={\mathrm{\Phi }}_0≠0$ and ${\mathrm{\Phi }}_0$ is the ground state of $H_0$.