Spectrum of the semi-relativistic Pauli-Fierz model II

TL;DR

This paper proves the existence of a ground state for the semi-relativistic Pauli-Fierz Hamiltonian at zero mass by analyzing the limit of ground states as mass approaches zero, extending previous results for positive mass.

Contribution

It establishes the existence of the ground state for the semi-relativistic Pauli-Fierz model at zero mass, using novel estimates and convergence arguments.

Findings

Ground state exists for m=0 in the semi-relativistic Pauli-Fierz model.

Convergence of ground states as mass approaches zero is demonstrated.

A singular pull-through formula is effectively estimated.

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 $H_0$.