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

We consider the polaron of the spinless semi-relativistic Pauli-Fierz model. The Hamiltonian of the model is defined by $H(\mathbf{P}) = \sqrt{(\mathbf{P}-d\Gamma(\mathbf{k}) + e\bA)^2 + M^2} + d\Gamma(\omega_m)$, 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 $m\geq 0$. Let $E(\mathbf{P})$ be the lowest energy of $H(\mathbf{P})$. In this paper, we prove the inequality $E(\mathbf{P} - \mathbf{k}) - E(\mathbf{P}) + \omega_m(\mathbf{k}) \geq m$, for all $\mathbf{P}, \mathbf{k}\in\mathbb{R}^3$.