Spectrum of the semi-relativistic Pauli–Fierz model II

Abstract

We consider the ground state of the semi-relativistic Pauli–Fierz Hamiltonian $$ H = |\textbf{p} - \textbf{A(x)}| + H\_f + V\textbf{(x)}. $$ Here $\textbf{A(x)}$ denotes the quantized radiation field with an ultraviolet cutoff function and $H\_f$ the free field Hamiltonian with dispersion relation $|\textbf{k}|$. The Hamiltonian $H$ describes the dynamics of a massless and semi-relativistic charged particle interacting with the quantized radiation field with an ultraviolet cutoff function. In 2016, the first two authors proved the existence of the ground state $\Phi\_m$ of the massive Hamiltonian $H\_m$ is proven. Here, the massive Hamiltonian $H\_m$ is defined by $H$ with dispersion relation $\sqrt{\textbf{k}^2+m^2}$ $(m>0)$. In this paper, the existence of the ground state of $H$ is proven. To this aim, we estimate a singular and non-local pull-through formula and show the equicontinuity of ${a(k)\Phi\_m}{0\<m\<m\_0}$ with some $m\_0$, where $a(k)$ denotes the formal kernel of the annihilation operator. Showing the compactness of the set ${\Phi\_m}{0\<m\<m\_0}$, the existence of the ground state of $H$ is shown.