Spectrum of the semi-relativistic Pauli–Fierz model II

Abstract

We consider the ground state of the semi-relativistic Pauli–Fierz Hamiltonian

Here $\text{A(x)}$ denotes the quantized radiation field with an ultraviolet cutoff function and $H_f$ the free field Hamiltonian with dispersion relation $|\text{k}|$. 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 $\sqrt{{\text{k}}^2+m^2}$ $(m>0)$. 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.