Spectrum of the semi-relativistic Pauli-Fierz model II

Abstract

We consider the semi-relativistic Pauli-Fierz Hamiltonian $$ H_m = |{\bf p}-{\bf A}({\bf x})| + H_{f,m} + V({\bf x}),\quad m\geq0, $$ and prove the existence of the ground state of $H_m$ for $m=0$. Here ${\bf A}({\bf x})$ denotes a quantized radiation field and $H_{f,m}$ the free field Hamiltonian with the dispersion relation $\sqrt{|{\bf k}|^2+m^2}$ with $m\geq0$. This paper is the sequel of [HH16], where the existence of the ground state $\Phi_m$ 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 $\{a(k)\Phi_m\}_{0<m<m_0}$ with some $m_0$, where $a(k)$ denotes the formal kernel of the annihilation operator. Taking a subsequence $m_j$, we can conclude that $\lim_{m_j\to0}\Phi_{m_j}=\Phi_0\not=0$ and $\Phi_0$ is the ground state of $H_0$.