A Short Remark on the Polaron in the Semi-relativistic Pauli-Fierz Model

Abstract

We consider the polaron of the spinless semi-relativistic Pauli-Fierz model. The Hamiltonian of the model is defined by $H(\mathbf{P}) = \sqrt{(\mathbf{P}-dΓ(\mathbf{k}) + e\bA)^2 + M^2} + dΓ(ω_m)$, where $\mathbf{P}\in {\mathbb{R}}^3$ is the momentum of the polaron, $d\Gamma (\cdot )$ denotes the second quantization operator and ${\omega }_m=|\mathbf{k}|+m$ denotes the dispersion relation of the photon with virtual mass $m\ge 0$. Let $E(\mathbf{P})$ be the lowest energy of $H(\mathbf{P})$. In this paper, we prove the inequality $E(\mathbf{P}-\mathbf{k})-E(\mathbf{P})+{\omega }_m(\mathbf{k})\ge m$, for all $\mathbf{P},\mathbf{k}\in {\mathbb{R}}^3$.